Data evaluation for surface-sensitive label-free methods to obtain real-time kinetic and structural information of thin films: A practical review with related software packages.

Interfacial layers are important in a wide range of applications in biomedicine, biosensing, analytical chemistry and the maritime industries. Given the growing number of applications, analysis of such layers and understanding their behavior is becoming crucial. Label-free surface sensitive methods are excellent for monitoring the formation kinetics, structure and its evolution of thin layers, even at the nanoscale. In this paper, we review existing and commercially available label-free techniques and demonstrate how the experimentally obtained data can be utilized to extract kinetic and structural information during and after formation, and any subsequent adsorption/desorption processes. We outline techniques, some traditional and some novel, based on the principles of optical and mechanical transduction. Our special focus is the current possibilities of combining label-free methods, which is a powerful approach to extend the range of detected and deduced parameters. We summarize the most important theoretical considerations for obtaining reliable information from measurements taking place in liquid environments and, hence, with layers in a hydrated state. A thorough treamtmaent of the various kinetic and structural quantities obtained from evaluation of the raw label-free data are provided. Such quantities include layer thickness, refractive index, optical anisotropy (and molecular orientation derived therefrom), degree of hydration, viscoelasticity, as well as association and dissociation rate constants and occupied area of subsequently adsorbed species. To demonstrate the effect of variations in model conditions on the observed data, simulations of kinetic curves at various model settings are also included. Based on our own extensive experience with optical waveguide lightmode spectroscopy (OWLS) and the quartz crystal microbalance (QCM), we have developed dedicated software packages for data analysis, which are made available to the scientific community alongside this paper.


Introduction
The term -biosensing‖ does not have a unique definition in the literature. The various definitions that have been proposed do have in common that a biological entitymolecule (e.g., a protein), virus or cellmust be involved, either as a receptor to capture an analyte, or as the analyte (e.g., the components of blood). In a typical surface sensitive label-free biosensor measurement, one component, the biorecognition element or also called as bioreceptor (an antibody, enzyme, DNA, aptamer, mammalian cell or microorganism) is at a fix position bound to the solid sensor (transducer) surface. Another component, the analyte, is driven over the receptor-sensitized surface in liquid phase inside a fluidic system [26,27]. Based on the transducer type, the adsorbed layer generates an optical, mechanical, or electrical signal, in which a wealth of valuable information may be contained. Intense research has been carried out to derive important physicochemical properties of the adsorbed layers from these signals.
What makes the quantitative analysis of biosensor response especially difficult is that these processes are running simultaneously and influenced by multiple experimental parameters. In such biosensing systems, the main processes are convection, diffusion and reaction (adsorption or association). It is crucial to qualify that in which régime such systems work (e.g., reaction-limited versus diffusion-limited, rapid collection versus full retention) and provide robust approximate relations conveying how various relevant quantities scale as the system parameters change [26]. These systems are similar to heterogeneous catalytic reactions, dealt with by the chemical engineering community before the advent of immunoassay and biosensor technologies. Today, a wide range of label-free biosensor technologies capable of high-resolution kinetic measurements have been developed and many instruments are available commercially and are being continuously improved pushing higher throughput and lower detection limits as well as enabling nanostructural analysis and measurement in complex biofluids. The analytical performance of label-free methods is especially exploited in biointeraction analysis (BIA) and became essential tools in drug discovery. In this field most of the examined solid-liquid interfacial processes are on the boundary of physisorption and chemisorption. Although non-covalent interactions are driving the adsorption of biomolecules on solid surfaces (forming thin films or establishing affinitybased ligand-analyte pairs), hydrogen bonds are involved, as they inevitably will be in biolayers.
In this review we summarize the most important techniques, their theoretical foundations, and the characteristic nanostructural and kinetic features that can be obtained (e.g., thickness, refractive index, viscosity and water content of adsorbed layers, kinetic and equilibrium adsorption constants). Our main aim is to provide the basic equations and experimentally relevant examples for those who are dealing with kinetic data generated by surface-sensitive methods. We focus on two highly representative examples of optical and mechanical sensors, OWLS and QCM. In the endeavor of detecting various characteristics simultaneously on an examined system, combination of the highly sensitive label-free methods serves as a particularly powerful solution. We discuss the currently available possibilities of combinations and review the gain that this approach can provide, especially in the characterization of heavily hydrated interfacial layers.
Based on our own extensive experience we developed data analysis packages for OWLS and QCM techniques. We show through representative examples the main features of these

Detection principles and major representatives
Optical methods detect refractive index (RI) changes, mechanical methods detect mass changes and electrochemical methods measure changes in charge distribution 13 . A summary of representatives of these methods is shown in Fig. 1. Although the variety of biosensors is quite large, only a few achieved significant success, as manifested by commercial instruments widely used in laboratories around the world -SPR, OWLS, dual polarization interferometry (DPI) and QCM, which all have an extensive literature both regarding their theory and applications [28,[32][33][34][35][36].
Optical methods measure RI changes originating from the close vicinity of the sensor surface. Sensing is established through the evanescent electromagnetic (EM) field formed by the EM wave coupled into an optical transducer film. The evanescent field extends over the film surface and dies away with an exponentially decaying field intensity. Most commonly, the evanescent field is generated by surface plasmons of a metal layer excited using an irradiating light beam (SPR-based methods, Fig. 1 A) or by an EM wave propagating in a waveguide film by a series of total internal reflections (waveguide-based methods, Fig.  1 B, C). The great advantage of these techniques is that they offer high sensitivity by probing a large surface area only in the close vicinity of the surface. Local optical density changes in the evanescent field change the velocity and phase of the EM wave. In general, the detected response signal R is proportional to the total RI shift Δn at up to a distance z from the surface [37]: Here, σ denotes the field decay length (penetration depth). Measuring the polarization state of an incident EM wave reflected from a substrate surface is an alternative optical method for probing tiny RI changes on the surface. This principle is exploited in ellipsometry, shown in Fig. 1 D. Although ellipsometry is capable of measuring the kinetics of biolayer formation, it has remained a complementary technique rather than a standalone biosensor, due to its moderate sensitivity and the complicated optical models needed for evaluating the raw data. Biosensors with mechanical transduction principle are often termed acoustic wave devices as their detection method is based on the generation of standing acoustic waves by mechanical oscillation of the transducer. The most successful representation of mechanical transduction is the QCM technique ( Fig. 1 E), which offers sensitive measurement with robust interrogation of the piezoelectric quartz crystal sensor chip. Other mechanical methods, like microcantilevers ( Fig. 1 F) have failed to provide stable and reproducible kinetic signals and have not become popular for kinetic measurements, whereas QCM has amply demonstrated its potential, especially for investigating the highly hydrated layers formed by synthetic polymers or biomaterials [38][39][40].
Translating the presence of biomolecules to an electrical signal is an alternative label-free detection method 13 . Field-effect transistor (FET) technology is quite promising as a transducer 22 . The biosensor-related realization is usually termed a BioFET. Charged biomolecules adsorbing on a semiconductor channel change the local surface charge density and thus the conductivity of the semiconductor, resulting in a shift in the measured electrical current (Fig. 1 G). The above list of label-free methods capable of monitoring adsorption kinetics is not complete. Such additional techniques (e.g., streaming potential measurement [41]) is outside the scope of this review. Fig. 1 Detection principle of label-free biosensor methods. All subfigures involve a visualization of the most relevant part of the biosensor positioned in the top as well as a graph about the raw measurement data that includes the biointeraction signal. A. SPR biosensor. The surface plasmons are excited by the incident light incoupled by a prism. The intensity of reflected light is measured by a photodetector. At a specific angle Θ where surface plasmons are excited, intensity minimum is recorded in the angle-spectrum. B. OWLS. Discrete waveguide modes (TE and TM polarizations) are excited by an incident laser light coupled into the waveguide film via an optical grating. Resonance peaks are detected at two incident angles α termed incoupling angles, corresponding to the two waveguide mode polarizations. C. DPI. The end face of the sensor chip is broadly illuminated by laser light coupled into the two waveguide films (top: sensing, down: reference waveguide) in which TE and TM modes are excited. The beams coupled out from the waveguides interfere with each other resulting in a 2D interference pattern detected in the far field on a CCD array. D. Spectroscopic ellipsometry. The ellipsometric angles Ψ and Δ characterizing the polarization state of light reflected from the sample surface are detected as a function of the incident light wavelength. E. QCM. The shear mode oscillation of the quartz crystal generates a standing acoustic plane wave. Resonance peaks are detected as a function of frequency and the resonance frequency is detected together with the band width of the resonance peak. F. Microcantilevers [42]. An incident laser beam is reflected from the cantilever towards a position sensitive detector (PSD) measuring the cantilever deflection due to an added mass as a function of time. G. BioFET [43]. The source (S) and drain (D) electrodes are connected with a silicon nanowire (SiNW) semiconductor channel which is the sensing element of the device. In the case of an n-type SiNW, the binding of a negatively charged biomolecule results in a drop in the measured electrical current. The diagrams and graphs are adapted from the below listed references.

Surface plasmon resonance (SPR) biosensor and biointeraction analysis (BIA)
SPR is probably the most common label-free biosensor techniqueit was the earliest to be commercialized and has been aggressively marketed. SPR biosensors exploits the chargedensity oscillation excited by EM waves at the interface of two media which have dielectric constants of opposite signs, like a metal (gold) and a dielectric (glass) 13,23 . For a basic SPR setup, see Fig. 1 A. Due to the enormous literature about SPR-based biosensing [44][45][46], we do not feel that it is necessary to go into further details here. A few additional comments are made in Section 2.1.2. One important point that has received less attention than would be appropriate is that, because of the intrinsically low sensitivity of SPR (about an order of magnitude less than that is achievable by waveguide-based methods [47]) it was found necessary to -fill the evanescent field‖ by coating the gold layer in which the surface plasmons resonate with a thick layer of a dextran hydrogel to which the receptors could readily be immobilized. This greatly increased the number of binding events that could be sensed and compensated for the low intrinsic sensitivity. The development was favored by the circumstance that a major product of the company, Pharmacia AB, which was the first to successfully develop a commercial SPR instrument (Biacore) was the Sephadex range of dextrans. Hence there was great in-house expertise in the manipulation of such materials. A parallel advantage was that binding took place in a quasi-three-dimensional medium (the volume fraction of the dextran being very small), allowing the homogeneous binding kinetics familiar to biochemists to be used for data analysis. It was not initially realized that the hydrogel exerts Hofmeister-like effects [48] on the medium, which can significantly chemically alter ligand-receptor interactions compared with those in pure water. At the same time diffusion within the hydrogel is not unhindered as in bulk solution, which affects the overall binding kinetics through its influence on transport [49]. On the other hand, the hydrogel somewhat mimics cytoplasm, in which many of the reactions studied with biosensors take place in reality, hence the hydrogel might be considered a more realistic medium than aqueous buffer. Of note, immobilization can heavily alter the local environment of the immobilized molecules changing their binding properties compared to interaction values measured in bulk solution. A recent systematic study using GCI illuminated that the k a rate and K d values of the binding events are heavily influenced, but the k d values not [50]. This should be kept in mind when comparing the results of surface sensitive technologies to other methods measuring binding events in bulk solutions (For example, isothermal titration calorimetry [50][51][52]). The above observation about the tuned binding constants has been also observed in living cells when the local environment of the receptors were varied by enzymatic digestion of the glycocalyx [53].

J o u r n a l P r e -p r o o f
In the nearly 700 million USD per year market of SPR biosensors [54], the industry leader position is occupied by Biacore, currently owned by the company Cytiva (Little Chalfont, UK; formerly part of GE Healthcare Life Sciences). The flagship instruments are Biacore S200 [55] in terms of leading sensitivity (reported baseline noise of < 0.015 RU ≈ 0.15 pg mm -2 ) as well as Biacore 8K+ [56] regarding throughput (samples can be measured on 12 x 384well microplates in one run on an 8-channel sensor chip).

Optical waveguide lightmode spectroscopy (OWLS)
If an EM wave is confined into a thin layer which has a thickness less than the wavelength of incident light and RI higher than its surrounding media, and the wave is coupled into the layer allowing constructive interference between the waves reflected from the two interfaces facing each other, discrete waveguide modes can propagate 13 . Versatile incoupling and readout methods have made waveguide-based transduction technology a leader in the field of biosensors, which also demonstrates great promise for the development of new generation biosensors (see later Section 2.2.5). Unlike SPR, which early on became dominated by large commercial instrument makers who strove after ease of use rather than biophysicochemical insight, many OWLS users have built their own equipment and the commercial instrument companies making OWLS apparatus are small and have worked closely with cutting-edge researchers to develop their instruments. This happy circumstance has undoubtedly facilitated the development of the new generation of biosensing technology (Section 2.2.5).
In OWLS, the evanescent EM field is generated by coupling an external laser beam into the optical waveguide film of a planar sensor chip, as illustrated in Fig. 1 B. For the so-called monomode OWLS, the waveguide (F) is typically a 160-200 nm thick high-RI (e.g., SiO 2 -TiO 2 ) dielectric film (S) deposited on silica or alkali-free glass (e.g., AF45 glass) using the sol-gel technique or physical vapor deposition. The light is coupled into the waveguide through an optical grating embossed or written into the waveguide film. The read-out is based on scanning the angle of the incident beam by a precision goniometer and recording the discrete incoupling angles (α) at which waveguide modes are excited, i.e. at which propagating standing electromagnetic waves are formed in the waveguiding layer. In monomode OWLS, the zeroth order mode of each of the two possible polarization states can be excited, namely the transverse electric (TE) and transverse magnetic (TM) modes. The excitation of modes is detected by continuously measuring the intensity of light leaving the ends of the optical chip. It is preferable to place a photodiode at both ends in order to compensate for any variation of grating alignment. Furthermore, measuring two α values independently, with the two oppositely placed photodiodes, makes it possible to determine the exact values of the incident angles by determining and taking into account the autocollimation angle (that corrects the perpendicular incident angle to 0°), use of which cannot be avoided due to the high angular precision required by OWLS. This enables OWLS to work without the need for any calibration, making it unique among the evanescent wave-based labelfree techniques. If excited, intensity peaks appear in the angle-spectrum (Fig. 1 B). The incoupling angle depends on the effective RIs of the waveguide, N TE and N TM . The definition of N TE and N TM involves the optogeometrical parameters of all the layers through which light passes, including the RI of air (n 0 ), substrate (n S ) and covering sample medium (n C ), as well as the RI and thickness of the waveguide film (n F , d F ) and adsorbed layer (n A , d A ). Any RI variation in the evanescent field that may originate, e.g., from a new liquid medium or biomolecule adsorption, results in change of N and detunes α. The incoupling angles are recorded sequentially, hence surface events can be monitored in real time. By recording α TE and α TM , one can determine N TE and N TM , and thus the n A and d A parameters of the target layer. The relation between N TE and N TM as well as n A and d A is described by the 4-layer mode equation. The exact values of n F and d F are usually determined from the baseline part of the recording, during which analyte-free solution flows over the surface. If n C and n S are known, n F and d F can be calculated using the 3-layer mode equation.
The RI of many pure liquids, including water, have been extensively researched and tabulated values and interpolation formulae are available for all relevant wavelengths and temperatures. The RI of unknown liquids and solutions can be independently measured by a refractometer, or can be calculated from Eq. (2) if the RI of the medium in which the adsorbing molecules are dissolved (n medium ), the concentration of the bulk solution (c B ) and the RI increment dn/dc of the solution are known: (2) Here, dn/dc is defined as the slope of a n B vs. c B graph and its value can be simply determined by refractometric measurements of analyte solutions prepared at different concentrations. Early work with proteins suggested that a -universal‖ value of 0.1840 mL g -1 obtained [57], but this was later shown to be erroneous on both theoretical and experimental grounds [58]: dn/dc depends on both the actual protein (its atomic and bond composition) and the solventusually an aqueous buffer. A considerable compilation of dn/dc values of different biomolecules has been made [59]. Using the determined n A and d A values, the surface mass density M can be calculated by the de Feijter's formula as follows [57]: The two guided lightmode polarization states enable the measurement of two independent waveguide characteristics N TE and N TM and thus the determination of two optogeometrical parameters of the adlayer, n A and d A . The fact that the TE and TM modes are sensitive to the overall orientation of components in the adlayer enables nanoscale conformational changes to be examined (details in Section 5). These capabilities make the OWLS a powerful technique for the simultaneous analysis of both layer formation kinetics and its nanostructure [60][61][62]. Such a structural analysis is not possible by a conventional SPR biosensor, as only the TM mode can propagate and hence only a single polarized mode (N TM ) can be monitored. Thus, n A and d A cannot be separated and M is determined according to an approximate -calibration‖ of the SPR signal (1 RU ≈ 1 ng cm -2 ) [63]. The OWLS 210 biosensor system of MicroVacuum Ltd. [64] (Budapest, Hungary) offers a detection limit < 1 ng cm -2 and the independent measurement of n A and d A . OWLS has been used for the kinetic and structural analysis of protein adsorption [65][66][67][68][69], DNA [70][71][72] and synthetic polymer [73,74] layers as well as J o u r n a l P r e -p r o o f measuring the kinetics of living mammalian [75,76] and bacterial [77,78] cell adhesion. OWLS is popular in membrane protein and lipid studies. For instance, with this technique it is possible to investigate the formation of lipid bilayers from vesicles [79], lipid bilayer structures [80], the affinity and kinetics of protein-lipid interactions [81,82], and structural changes during such interactions [83].

Dual polarization interferometry (DPI)
Similar to OWLS, dual polarization interferometry (DPI) is a real-time and quantitative analytical technique that allows the RI and thickness of thin films and molecular layers adsorbed on a waveguide surface to be measured [84][85][86]. As with OWLS, DPI is a very sensitive method with a dimensional resolution of an order of ångströms perpendicular to the plane of the waveguide [84]. The principle of the DPI technique is that a laser light is divided into two beams which enter into two vertically stacked single-mode planar waveguides, a sensing and a reference one; recombination of the two beams in the far field creates a twodimensional interference pattern. To excite alternately the two orthogonal polarization modes of the waveguides, the linearly polarized laser is rotated. The thickness of the adsorbed layer and the RI can be calculated by measuring the interferogram for both polarizations. If the polarization is switched rapidly, conformational changes associated with molecular interactions can be observed in real-time. DPI has been used in membrane protein and lipid studies. For instance, with this technique it is possible to investigate the formation of lipid structures [87].
A typical DPI setup uses three flow-channels, one of which is a reference channel. In the reviewed studies about detection, protein structure changes, binding events, etc., the most used DPI instruments are the AnaLight Bio200 and AnaLight 4D from Farfield Group Ltd. (UK) [85,[88][89][90], although this company has been declared insolvent and dissolved. It had declared that the AnaLight Bio200 gives quantitative data on real-time changes in thickness with a resolution of 0.1 Å and surface density with a resolution of 0.1 pg mm -2 of the immobilized layer [91], implying that the instrument can detect small molecules with a molecular weight below 100 Da with a mass resolution of 5 Da. Indeed, bisphenol A specific aptamer was immobilized on the sensor chip surface through biotin-avidin interaction, and detection limits of 1.7 mM on the thickness signals and 2.5 mM on the mass signals were demonstrated [92]. The software provided with the instrument facilitates comprehensive analysis of both structural change and interaction kinetics in quantitative units. Although DPI can only handle layers formed during the measurement, the multiple path length dual polarization interferometer (MPL-DPI) has the ability to characterize layers coated ex situ, which increases the number of condensed matter systems that can be studied; sample preparation may be easier too [93].

Quartz crystal microbalance (QCM)
QCM employs a vibrating element (resonator) whose mechanical oscillation generates bulk acoustic waves (BAW) propagating inside the resonator and extending beyond its surface. The resonator is an AT-cut quartz crystal disk mounted between gold electrodes on its two sides. Applying AC through the resonator induces shear mode oscillation within the crystal via the piezoelectric effect, generating standing acoustic plane waves with a propagation direction normal to the surface. This shear wave propagates through the crystal and extends beyond its boundary with an exponentially decaying intensity. Besides vibrating at the fundamental resonance frequency, overtone oscillations are also generated. The crystal structure permits only odd overtone resonance frequencies (f n ) to be excited, defined by: (4) where n denotes the overtone number (n = 1, 3, 5, 7, etc.), d Q is the thickness of the quartz disk, v is the velocity and λ is the oscillation wavelength. The penetration depth of the acoustic wave is the largest at the fundamental frequency (around 250 nm) and decays with the overtone number [94]. If an adlayer is formed on the surface, it detunes the resonance frequency. A viscoelastic adlayer generate energy dissipation during the oscillation at the corresponding odd overtone characterized by another measured parameter, the dissipation factor (D n ). For the measurement of f n and D n , most QCM instruments use one of two different readout methods [95,96]: (ⅰ) an impulse excitation (ring-down) method measuring the decay of the crystal oscillation after the excitation is turned off; or (ⅱ) an impedance analysis method measuring the impedance frequency spectrum with a network analyzer. Both techniques realize simultaneous measurement of the shift in f n and D n at each measurement epoch, where the shift at time t is defined as: Δf n = f(t)-f(t=0), ΔD n = D(t)-D(t=0), respectively.
The best-known QCM system, the QCM-D (QCM with dissipation monitoring, Biolin Scientific AB [97] (Gothenburg, Sweden)) measures using the ring-down procedure [98,99]. The applied driving voltage is intermittently terminated inducing a decay in the crystal oscillation. Fitting the recorded amplitude results in fit parameters of decay time (τ) and f n , allowing the calculation of D n : (5) where U dissipated the energy dissipated in one oscillation cycle and U stored the energy stored in the oscillating system. An example for the impedance analysis method is the QCM-I system manufactured by MicroVacuum Ltd. [100]. Here, impedance spectra (conductance as a function of frequency) for the different overtones are measured, and the resonance peaks are fitted (see Fig. 1 E), where the fit parameters are the resonance frequency and the half-width at half-maximum (Γ). Here, the dissipation factor can be expressed as [96]: (6) J o u r n a l P r e -p r o o f The oscillation decay and the impedance spectrum measured in the time and frequency domain, respectively, are the Fourier transform of each other [95]. As such, D provided by the QCM-D and QCM-I readout methods are equivalent and can be evaluated using the same methods [95,96].
According to the simplest evaluation model of a thin and rigid adlayer, the detected Δf n normalized with its corresponding overtone number (Δf n /n) is proportional to the added surface mass per unit area: (7) called the Sauerbrey equation [101], where C is a mass sensitivity constant dependent on the physical properties of the quartz crystal and M S is the surface mass density calculated with the Sauerbrey equation. The criteria for applying the Sauerbrey equation have been thoroughly discussed elsewhere [101,102]. Although they assume experimentally unrealistic conditions (e.g., the medium is vacuum), the Sauerbrey equation has been found to be usable for the evaluation of measurements made in a liquid medium, where the adlayer is rigid and its thickness is only a few nm. However, when the adlayer is thick or is viscoelastic, the Sauerbrey equation significantly underestimates the mass deposited on the sensor surface. Many biomaterials are viscoelastic due to their tendency to bind a large amount of water. In order to accurately evaluate data measured on such biolayers, a more complex model explicitly incorporating viscoelasticity is needed, which involves ΔD n for data evaluation. The viscoelastic case and the Sauerbrey limit can be well estimated by qualitative data analysis methods. As such, if there is a small dependence of Δf n /n on n or if the magnitude of ΔD n is below 2×10 -6 or, according to Reviakine et al., ΔD n /(-Δf n /n) << 4×10 -7 Hz, the adlayer can be treated as rigid [99,[102][103][104].
To calculate the realistic mass of a viscoelastic adlayer beyond the Sauerbrey limit, a model considering the measured ΔD n values is needed. Such a continuum mechanical model was developed by Voinova and co-workers. The model assumes that as a Kelvin-Voigt material, the viscoelastic adlayer can be described by an elastic (spring) and viscous (dashpot) element, holding that the crystal is fully elastic. The Voinova model [105] is mostly referred to as the standard Kelvin-Voigt model and it serves as the core model of the data analysis software of QCM-D, currently called Dfind (Biolin Scientific AB), previously called QTools). This model is a specialized case of a more comprehensive model [99,106]. In practice, the Voinova model takes both the measured Δf n and ΔD n for the calculation of three adlayer parameters: thickness (d A ), shear viscosity (η A ) and shear elastic modulus (μ A ). The model can be extended to consider multiple adlayers. It requires knowledge of the adlayer mass density (ρ A ). The equations of this Kelvin-Voigt model are: where ω is the angular frequency (ω n = 2πf n ) and i is the imaginary unit. The method of determining adlayer parameters is based on the calculation of Δf n and ΔD n from known initial parameter values (d A 0 , η A 0 , μ A 0 ) first. The calculated Δf n c and ΔD n c values are then compared to the measured Δf n m and ΔD n m at each overtone involved in the analysis. Using standard deviation values std Δfn m and std ΔDn m that characterize the measured Δf n m and ΔD n m a  2 error function can be computed as follows: Until the computed is not accepted (minimum is not reached) an iterated set of parameter values are chosen, and the above calculation method is repeated. If a minimum is found, the decision of accepting the parameter values can be made by visually checking the fit or by assessing the value of . This fit method can be performed independently, point-by-point, at each measurement time resulting in a set of Δf n fit and ΔD n fit as well as d A fit , η A fit , μ A fit in time [105,107,108]. Once the parameters are calculated, the surface mass can be determined based on the calculated d A and given ρ A values: Unfortunately environmental noise (e.g., mechanical irregularities in peristaltic pumping) significantly interfere with the fundamental frequency and dissipation, thus they are usually not considered in the evaluation [103,109]. The standard Kelvin-Voigt model considers a complex shear modulus written as G* = G' + iG'' = μ A + iωη A , where the G' storage modulus characterizes the elasticity and the G'' loss modulus characterizes the viscous behavior of the material. The model assumes a frequency-independent G' and applies a simple linear frequency-dependence for G''. In reality, both G' and G'' depend on the frequency and the dependence is not likely to be linear. Nevertheless, some studies applied the standard Kelvin-Voigt model for characterizing various types of viscoelastic adlayers apparently successfully [103,104,107,110,111]. Some works invoked a linear frequency-dependence of both μ A and η A for improving the fit of the model to the data [108,112,113]. Other approaches for dealing with the different shapes of frequency-dependence use a power law for the expression of μ A and η A , e.g. by Johannsmann, based on his thoroughly detailed model using an equivalent circuit approach for the viscoelastic system description [95,96] (a software called QTM applying this model is available to download [114]). Another model involving power-law analysis is connected to the group of Shull [115,116] and has been applied in some recent studies [116][117][118][119]. The current version of the Dfind software also allows use of an extended Kelvin-Voigt model involving power-law frequency dependence [120]. An excellent guide to choosing the proper analysis method was published by Reviakine and co-workers [99]. Other data analysis methods considering various model cases (e.g., fully elastic, fully viscous and viscoelastic adlayer; thin and thick films) can be found [120][121][122].
QCM has been extensively used for revealing the properties of soft biomaterials. The application of QCM in combination with optical techniques to characterize hydrated thin layers is further detailed in Section 5. Also, in a number of studies QCM was applied for extracting quantitative kinetic information in order to investigate, e.g., the dynamics of IgG oligomer formation [123], DNA hybridization [124], the interaction of proteins with cell surface carbohydrates [125], polydopamine-coated surfaces [126] as well as biopharmaceuticals with Protein A-modified surfaces [127]. The QSense Pro QCM-D instrument (Biolin Scientific AB) provides simultaneous measurement of eight independent sensors with a maximum mass and dissipation sensitivity of ~ 0.5 ng cm -2 and ~ 0.04×10 -6 , respectively [128].

Novel label-free optical biosensor techniques in kinetic measurements
In addition to the OWLS technique, recently introduced waveguide-based instruments such as the resonant waveguide grating (RWG) and grating-coupled interferometry (GCI) biosensors have shown the potentials of the measurement principle in terms of high throughput and achieving even higher sensitivities [20,129,130]. The measurement principles of four novel optical techniques are visualized in Fig. 2. An important limitation of these techniques is, however, that water, trapped in the analyte layer, cannot be detected, because they monitor RI contrast relative to the aqueous background. This fact means that only the -dry‖ mass of analyte molecules can be effectively determined [67,131].  [132], copyright 2020 Royal Society of Chemistry; (C) copyright Sartorius AG; (D) from Ref. [133], copyright 2017 Springer Nature; (D) from Ref. [134], copyright 2019 American Physical Society]

Grating coupled interferometry (GCI)
The highest sensitivity among label-free biosensors is provided by techniques exploiting interference-based detection methods [20,135]. GCI is one of the most successful realization of these methods, which has just recently started to demonstrate its capabilities in kinetic studies.
The main difference between an OWLS and GCI setup is that GCI couples a second laser beam into the waveguide layer in which the two beams interfere with each other and the characteristics of this interference is monitored. Fig. 2 A presents the arrangement of a GCI setup. A GCI chip can incorporate multiple fluidically separated channels (i.e., flow cells); two such channels can be seen in the figure. This is a very useful feature to facilitate the simultaneous acquisition of kinetic curves corresponding to slightly different conditions [136]. The sensor chip has three separate diffraction gratings embedded in the waveguide film, made from Ta 2 O 5 . The waveguide sits on a glass substrate and the top of the waveguide is coated by a SiO 2 film, which covers the entire waveguide except the area used for sensing. A laser beam called the measuring beamis coupled into the waveguide film through the 1 st grating. A guided mode is launched and in the region of the sensing window its evanescent field is in contact with the sample solution. The role of the 2 nd grating is to couple the reference beam into the waveguide film. The measuring and reference beams interfere with each other inside the waveguide and the combined beam carries the interference characteristics. The SiO 2 film is used to prevent the evanescent field of the combined beam from penetrating into the medium and undergoing a phase shift caused by RI changes in the medium. The 3 rd grating serves as an outcoupler of the combined beam, which is directed toward a photodetector. In contrast to DPI, where the interference pattern is spatially analyzed, GCI monitors the phase shift as a function of time. This detection method is realized by a time-varying phase modulation provided by a liquid crystal modulator (LCM) positioned in the path of the reference light before incoupling. The phase modulation of the reference beam translates the phase information of interference into a time-varying intensity change (see Fig. 2 A, bottom graph). Any RI changes in the evanescent field lead to a phase shift in the guided wave resulting in a change of the time-dependent signal characteristics. The Δυ phase shift is determined by fitting the measured curves at each measurement time and calculated as a difference from the baseline: Δυ = υ(t)-υ(t 0 ). Δυ is proportional to RI change [20,137,138]. A great benefit of the method is that multiple channels can be integrated on the same chip and at least one channel can be used as a reference channel.
The first GCI instrument, WAVE, was launched by Creoptix AG [139] (Wädenswil, Switzerland) and has shown its potential in the field of BIA even during its short presence in the market. Owing to a noise level < 0.01 pg mm -2 , the technique offers the measurement of surface mass below 1 pg mm -2 , making the WAVE perhaps the most sensitive label-free biosensor [20]. The sensitivity allows the detection of binding kinetics of analytes with a J o u r n a l P r e -p r o o f molecular weight even below 100 Da. Millisecond time resolution together with a fastflowing fluidic system is designed to resolve very fast binding kinetics characterized by high on-and off-rates (up to k a ≈ 3×10 9 M -1 s -1 and k d ≈ 10 s -1 ). Owing to this performance and high-throughput capability (a 4-channel sensor, able to handle a 2×384 well plate), the technique is especially suitable for fragment-based screening in drug discovery. The operation and data analysis software WAVEcontrol includes a versatile kinetic analysis functionality. GCI has found a broad range of applications in plant biology [140][141][142][143][144][145][146][147], cancer research [148], drug discovery [149] as well as the characterization of natural compound-protein interactions [132]. As recently presented, the top sensitivity of GCI allows for acquiring reliable kinetics of < 100 Da molecular weight (MW) analytes, making possible to reveal the interaction of Ni(II) ions (52 Da) with Ni-specific protein molecules [50].

Resonant waveguide grating (RWG) biosensor
Similar to OWLS and GCI, resonant waveguide grating (RWG) biosensors also use a diffraction grating embedded in a waveguide film to couple light into a waveguide. In the case of RWG the same grating is used for outcoupling the guided EM wave, practically meaning that the grating together with the waveguide serves as a resonant mirror (see Fig. 2 B). The criterion of constructive interference is only satisfied at a specific illuminating wavelength, called the resonant wavelength (λ R ). Any RI changes within the penetration depth of the evanescent field (ca. 150 nm) detunes the resonance and thus the waveguide modes can no longer be excited by the illuminating light with wavelength λ R . Waveguiding can be resumed at another resonant wavelength (λ R ' ≠ λ R ). The shift in resonant wavelength (Δλ = λ R ' -λ R , see the difference between resonance peaks in Fig. 2 B) is proportional to variations of N occurring in the waveguide-coating-medium system. An advantage of the RWG method is that light beams at a nominally normal incident angle can be applied to illuminate the biosensor [21,150]. This is significant for sampling large numbers of biosensors simultaneously, which is necessary for HTS [21]. Measuring samples in a typical highthroughput system often directly takes place in microplates [21]. A widely used RWG biosensor is the Epic benchtop (BT) system from Corning Inc.
[151] (Corning, NY, USA). A typical commercial Epic microplate consists of a glass substrate, a waveguide film (made of the biocompatible Nb 2 O 5 material) with the embedded grating structure as well as a biofunctional coating [152]. The Epic instrument monitors Δλ by sweeping the wavelength of broadband illuminating light over a 15 nm wide wavelength range with a 0.25 pm precision and records the resonant wavelength with a CCD detector. The Epic has a LOD of 2.2×10 -6 RIU (refractive index unit), which corresponds to 0.078 ng cm -2 in surface mass [129]. The great benefit of the technique is that the sensors are realized in the standard microplate format, where each well of a 96-or 384-well plate contains an independent sensor element in the center position, and all can be simultaneously interrogated. Moreover, Δλ of each well is measured in real-time with a resolution of 3 s, allowing the user to collect high-resolution kinetic data over long time intervals. The detection method and microplate-based construction make the Epic system an outstanding technique for performing high-throughput label-free biochemical and cellular assays in real-time [21,[153][154][155]. The Epic BT is provided with a controller and data collection software (LFAassay) which, however, does not include tools for kinetic analysis.
RWG technology has found a number of applications especially in cell biology and cellbased measurements for drug discovery. Recently, the technology was successfully applied to monitor the kinetics of cell adhesion [156][157][158] cell-surface and cell membrane receptorligand interactions [17,130], binding affinity [159], cytotoxicity [24], cellular signaling [160,161], and the functional state of surface-bound cells down to the single cell level [162]. Natural products were examined with this method as well on different cell lines [163], like hazardous herbs [164], and the popular green tea polyphenol epigallocatechin gallate (EGCG) [165,166]. Nanoparticle penetration into cells could also be monitored [167]. Experiments with extracellular vesicles [168] and an endothelial label-free biochip (EnLaB) assay [169] have been reported recently. Using a calibration methodology with robotic fluidic force microscopy (FluidFM), cell adhesion force kinetics of single cell populations have been monitored [162]. Clearly many diverse biological and biophysical process and phenomena can be examined thanks to the sensitivity and high-throughput capability of RWG biosensor.

Bio-layer interferometry (BLI)
Bio-layer interferometry (BLI) is an optical label-free technique for measuring biomolecular interactions (see Fig. 2 C). According to the maker of a commercial system, BLI platforms offer ease of use, high-throughput, reliability, and high precision measurements [170]. The technique analyzes the interference pattern of white light guided in an optical fiber reflected from two interfaces: an internal reference layer and a layer of immobilized protein on the biosensor tip. If the number of molecules bound to the biosensor tip changes, it causes a shift in the interference pattern. This alteration can be measured in real-time. Binding of an analyte in solution to the immobilized ligand on the biosensor tip surface increases the optical thickness at the biosensor tip. This results in a wavelength shift (Δλ), which is a direct measure of the alteration in thickness of the biological layer. Only molecules binding to or dissociating from the biosensor can change the interference pattern and create a response profile. The unbound molecules, changes in flow rate or in the RI of the medium do not significantly affect the interference pattern. This relative insensitivity to the medium extends the technique's capability to monitor crude samples without prepurification [171]. BLI has already been applied in vaccine research, for instance in clinical studies, epitope design, pathogen diversity and distribution studies, nucleic acid and molecular pathogen-host interaction studies as well as host immune response characterization [170,172].
The BLI technology is commercially available as the FortéBio Octet system (Sartorius AG, Göttingen, Germany), offering a sensitive analytical tool for measuring affinity and kinetics with a dynamic range of > 150 Da, affinities in the range of 10 pM -1 mM as well as association and dissociation rates in the range of k a = 10 1 -10 7 M -1 s -1 and k d = 10 -6 -10 -1 s -1 , respectively [171]. With this technique, fast results can be obtained; for instance, in the study of Mechaly et al., the workers carried out detection with limits of 10 pg/mL for ricin and 1×10 4 pfu/mL for F. tularensis within 17 min [173]. The FortéBio provides BLI appliances J o u r n a l P r e -p r o o f from 2-channeled (Octet K2, up to 2 assays in parallel) to high throughput ones working with 384-well tilted-bottom microplates and 96-well microplates (Octet HTX and Octet RED384) [171].

Interferometric Reflectance Imaging Sensor (IRIS)
A simple reflectance-based interferometric detection method has been devised by Özkumur and coworkers [174]. As an optical label free method, it owns all the advantageous points of those methods with an additional improvement for multiplexed measurements. The solid support of arrayed biomolecules is a silicon chip that has a thick uniform oxide layer on top. In its original form a tunable laser is incident on the array and reflection spectra are recorded for hundreds of spots simultaneously. The detection principle is that in case of a layered substrate, if light reflects from the substrate layer interfaces (Si-SiO2 and SiO2-air interface), characteristic spectral reflectance appears. Added mass on the SiO2 surface changes the optical path length difference generating a shift in the detected wavelength-dependent reflectivity curve. The next generation of the IRIS platform used multiple discrete LED sources making this technology more accessible, while it can achieve similar performance [175].
More recently single particle interferometric reflectance imaging sensor (SP-IRIS) technology was developed where an enhanced contrast in the scattering signal from nanoparticles enabled multiplexed detection of viruses in serum or whole blood. The SP-IRIS method was able to enumerate, estimate particle size, and phenotype exosomes from purified samples from cell culture, or directly from 20 µL of a clinical sample [176]. Tested with dynamic monitoring of antigen-antibody binding, a noise floor of 5.2 pg mm -2 was demonstrated.
On the market, the SP-IRIS technology has been realized as the ExoView R100 instrument of NanoView Biosciences (Brighton, MA, USA) targeting complex extracellular vesicle (EV) characterization [177]. The system applies both the label-free SP-IRIS method together with fluorescent detection to analyze EVs at the single particle level. SP-IRIS is used to measure the size distribution of EVs with counting and determining single EV size of as low as 50 nm. One chip accepts 35 uL of sample solution. Fluorescent detection is used to identify EV membrane proteins and other biomarkers.

Focal molography
Focal molography is the newest among the biosensor techniques introduced in this study. It is still under intensive development in a collaboration of Roche Diagnostics and ETH Zürich and has not been launched as a commercial instrument. It is worth presenting as a next-generation label-free optical biosensor that may open up new perspectives for kinetic measurements. The theoretical background of focal molography and application as a BIA technology was first described by Fattinger in 2014 [178].
In focal molography (a neologism coined from -focal molecular holography‖), detection is based on probing the mologram by laser light coupled into a monomode waveguide film J o u r n a l P r e -p r o o f that constitutes the sensor chip. The mologram is defined as a synthetic hologram, which is a precise nanopatterned assembly of receptor molecules. It is fabricated on a graft-copolymer layer attached to the waveguide film, where a submicrometer array of ridges (lines carrying the active sites) and inactive grooves are formed by a photolithographic process. The special arrangement of receptors is designed to diffract the probing coherent laser light and focus it into a diffraction limited focal spot. The spot is captured by a photodetector array where the molographic signal is recorded as the total light intensity of the focal spot. The unprecedented advantage of focal molography is that it can intrinsically discriminate specific binding from NSB, which otherwise always degrades biosensing signals [179]. This is realized by the special coherent assembly of the mologram, where the ridges and grooves are in effect the measuring and reference regions. As such, comprising hundreds of such separate regions, the technique can collect reference-corrected signal originating from a large number of tiny measuring and reference regions. Moreover, scattering by nonspecifically bound molecules that are randomly distributed in the copolymer layer do not contribute to the coherent signal [133,134]. NSB makes measurements in complex bodily fluids difficult by obscuring the signal of specific binding [179,180]; hence focal molography is expected to be a breakthrough in the selective detection of specific biomolecular interactions [181].
Based on measurements performed on a therapeutic antibody diluted in human plasma sample, the LOD was determined to be 200 ng ml -1 (1.3 nM). As an assay-independent measure, the upper limit bound of LOD expressed in surface load was found to be ~ 5 pg mm -2 . It is expected that further optimization in the lithography and readout method would improve the LOD by at least two orders of magnitude [133]. As recently demonstrated, the technique can also be applied in highly specific cell-based assays [182]. Due to the potential for multiplexing (several molograms on one single chip) and miniaturization, focal molography may be a future technology for point-of-care diagnostics [133,183].

Modeling adsorption kinetics
To quantitatively interpret experimental adsorption kinetics data, a kinetic model is needed that represents the adsorption mechanism and involves parameters that biophysicochemically characterize the system. When considering biomolecular interactions, the interaction of molecules with surfaces can be considered as specific or non-specific. In the case of specific binding the surface is usually assumed to be an array of distinct and independent binding sites, which strictly determine the positions where analytes can bind. In contrast, non-specific binding or adsorption defines the surface as a continuum everywhere potentially available for adsorbing species, at least initially as an empty surface; randomly adsorbed species create exclusion zones even for non-specific binding [184,185]. Hence, classification of a binding surface as a continuous or distinct array is somewhat arbitrary as it depends upon the size of the adsorbing ligand relative to the size of the binding sites constituting the adsorbent surface, with their arrangement also playing a role [37,184]. In many natural adsorption processes, the surface unambiguously presents a continuum to the adsorbing species, not an ordered array of independent, non-overlapping sites larger than the adsorbing ligand as assumed in the Langmuir model, and the sequential addition of molecules at randomly chosen positions engenders gaps between the adsorbed molecules which cannot J o u r n a l P r e -p r o o f later be filled. Similar reasoning applies to molecules adsorbing at randomly distributed discrete sites [186].
The most relevant field for the kinetic analysis of specific interactions is drug discovery, whereas non-specific adsorption kinetics find major interest in biosensor technologies, membrane and implant development, and maritime industries developing antifouling coatings. In such field not only the structures but also the kinetics may be complex, and the models devised to interpret the measured kinetics must appropriately match this complexity. It is important to recognize that many of the processes, including association, dissociation and many types of macromolecular conformational changes may operate on very different timescales, a proper understanding of which is essential for arriving at meaningful conclusions [184,187]. A practical summary of parameters that should be considered in the analysis of biomolecule adsorption data in terms of structure and kinetics was published by Migliorini et al. [188].

Fundamentals of adsorption kinetics. The Langmuir model
The Langmuir model can be considered as the fundamental theory of binding kinetics. It was devised for describing gas adsorption at discrete active sites on solid surfaces. A system may follow the Langmuir model if it fulfils specific restrictions as described elsewhere [189]. It is not prima facie obvious why a theory developed for ultraclean crystalline metal surfaces in high vacuum and at low temperature, in which every metal atom on the surface constitutes a binding site for a gas atom or small molecule, should be applicable to biomacromolecules adsorbing on a surface from solution, but for small antigens binding to immobilized antibodies it seems that the requisite conditions can be fulfilled.
Let us consider a solution of adsorptives in contact with a solid surface. Adsorptives or, more specifically, protein molecules or any particle P can adsorb onto binding sites S* of the surface resulting in adsorbed particles A , that can be also interpreted as P-S* complexes. The adsorption and desorption process can be described by the following reaction scheme: The rates of the forward (adsorption/association) and backward (desorption/dissociation) reactions are characterized by rate constants k a and k d , respectively. Considering that only monolayer formation is permitted (which is realistic for several proteins, e.g. bovine serum albumin, BSA) and the maximal coverage is defined by the surface density of the layer resulting from complete occupation of sites S*, the fractional coverage (Θ) can be expressed as: (13) J o u r n a l P r e -p r o o f where ν and M are the number per unit area and mass per unit area (also referred as surface mass or simply mass) of the adsorbed molecules, respectively. M max is the surface mass density that corresponds to the theoretically achievable maximal coverage (100% occupancy), a is the area occupied by one adsorbate bound to the binding site and m is the mass of one adsorbing species. The concentration of the available binding sites can be defined as the fraction of available sites (Φ), which is inversely related to Θ.
Based on the law of mass action, the net rate equation of the adsorption process written in terms of the time derivative of ν is: (14) where k a,num is the adsorption/association rate constant, c B,num is the bulk concentration of species per unit volume (both quantities are set to the number adsorbing species) and ν max = 1/a is the maximal number of analytes per unit area that can be bound to the surface representing complete occupancy. ν max is therefore the total number of binding sites per unit area. We can use mass M instead of ν if we multiply by m, obtaining: (15) Note, that we introduced here the kinetic rate parameter k a,mass = k a,num /a which will be later referred to as k a with typical units of cm s -1 . Also, c B,mass = mc B,num will be further simply used as c B The Langmuir model supposes the simplest case when all the binding sites that are not occupied by adsorbates are available for adsorption and a -monolayer‖ coverage (Φ = 0) can be reached: (16) One simple way is for checking whether the observed adsorption/binding process follows the Langmuir equation is plotting the measured M(t) data as dM/dt vs. M curve (see red line in J o u r n a l P r e -p r o o f Fig. 3 Characteristics of the Langmuir model compared to RSA. The curves are based on simulated data generated using the NBS-Kinetics software. Graph A shows simulated curves that demonstrate what we can obtain from a typical biosensor measurement. Graph B presents processed data where the mass derivative dM/dt was calculated and plotted against the surface mass M. The so-called pseudo-Langmuir data were simulated using the same conditions as for Langmuir (Φ = 1 -Θ), except that instead of allowing 100% coverage, a jamming limit of Θ j = 0.547 was set (see details in Section 3.3). This value is the same which was also used for simulating RSA data of adsorbing spherical objects. Although the pseudo-Langmuir data are closer to the RSA, the shape of the curve is clearly different. In affinity kinetic analysis, thus in the vast majority of reported SPR measurements, the measured response R (-resonance units‖, with unit of RU) is assumedunfortunately not always correctlyto be directly proportional to the number of bound analytes (i.e., R ν), and the following formula is widely used [190]: (17) which is essentially the same as Eq. (14) or (15). Here, R max is the maximal response (proportional to the amount of S* binding sites) and the typical units of k a,resp is M -1 s -1 (to get this, Eq. (14) is multiplied by the Avogadro number N Av = 6×10 23 mol -1 ). To be able to compare k a values obtained from Eq. (15) and Eq. (17) (k a,mass and k a,resp ) one can use the practical formula: (18) where k a,resp , k a,mass and a are given in their widely used M -1 s -1 , cm s -1 and nm 2 units, respectively. Using this conversion, one can see that the value of k a strongly depends on the mode of interaction. E.g., in the case of the non-specific adsorption of BSA from HEPES buffer onto silica [191], k a,mass = 1.1×10 -6 cm s -1 which is equivalent to k a,resp = 1.9×10 2 M -1 s -1 (a = 29 nm 2 ). As expected, this value is much below the k a,resp values characteristic to specific receptor-analyte interactions, and besides, non-specific adsorption would not be expected to fulfil the Langmuir criteria. Typical value of kinetic rate constants characterizing the adsorption of proteins are in the range of k a,mass = 10 -6 -10 -4 cm s -1 [192] and for desorption k d ≈ 10 -4 s -1 [186]. For the adsorption of microbial cells, a value of k a,mass ≈ 10 -5 cm s -1 seems to be typical [192]. Assuming a full -monolayer‖ coverage (Θ = 1) at saturation, the effective surface area occupied by a single receptor molecule can be calculated from the measured M max and the known m using Eq. (13).
Despite its restricting assumptions, which are seldom justified in biology, the Langmuir model is still widely used for kinetic analysis of specific interactions, perhaps because of its simplicity. It is offered to the user as the first approach in data evaluation, e.g. both the Biacore and Creoptix WAVE evaluation modules apply the Langmuir model, calling it -1:1 kinetics‖. However, the model usually fails to describe non-specific adsorption kinetics, such as the adsorption of protein molecules on sensor surfaces. The two main deviations from the Langmuir model are (i) the sensor surface (a metal or dielectric film) must be considered as a continuum, not an array of individual binding siteswhich even if they existed would be far smaller than the analyteand (ii) adsorbates can reside on the surface for several days or longer (k d < 10 -6 s -1 instead of the ≈ 10 -4 s -1 typical of ligands-receptor dissociation in homogeneous solution [186]), meaning that the adsorption was practically irreversible, contradicting Eq. (12)Error! Reference source not found.. Extended versions of the Langmuir theory have been discussed [193], but they cannot satisfactorily overcome its fundamental limitations. The only reported cases where the Langmuir model was found to be applicable to analytes binding to a continuum is where the ligands are mobile on the surface and can form clusters [194,195] These results were obtained with sensor chips coated with a lipid bilayer. In such cases a in Eq. (13) is the area occupied per analyte molecule and true monolayer coverage is achievable. Obviously to have clustering there must be interaction between the adsorbed species, hence the -no interactions‖ restriction of the Langmuir model is flouted, yet nevertheless the this model is successful.

Bivalent model
Whereas the Langmuir model restricts one adsorbing species or analyte to bind at one surface binding site only, the bivalent model allows the bivalent analyte, that is equipped with two binding sites (e.g. an antibody), to bind first to one surface site and then consecutively to a second one (forming A1 and A2 complexes, respectively) [179,190,196]. The model has been extensively used to study antibody affinity profile [197] and cell signaling [198]. Model scheme and rate equations written in terms of R response units are as follows: where the total response R total = R A1 + R A2 and R A1 = R A2 = 0 at t = 0. The two-step binding kinetics is characterized by rate constants k a1 , k a2 , k d1 , k d2 (or their response-based versions).

The RSA model
Deviations from the Langmuir adsorption model are well known and discussed in textbooks [187,199]. Even if there is no affinity between adsorbed species, they cannot occupy J o u r n a l P r e -p r o o f each other's space and in this sense they do interact. Theoretically this short-range interaction can be taken into account by the pair interaction energy (potential). The deposition of proteins on the surface results in gaps between the adsorbed molecules small enough to exclude subsequent arrivals. The random sequential adsorption (RSA) model takes this into account along with other important features, namely that rigid particles are deposited on the surface at random positions selected sequentially, and that adsorption (deposition) is irreversible [200,201]. Experiments [202] verified the major predictions [200,201] of the RSA model: a short-and intermediate-time kinetics markedly different from that of the Langmuir model and a slow asymptotic approach to saturation. The original RSA model has been extended to take into account desorption and protein relaxation on the surface [203][204][205][206][207][208][209]. The astonishing success of the very simple basic RSA model was later shown to reside in the fact that the hydrodynamic friction experienced by particles as they approach the adsorbing surface perpendicular to it randomizes the motion such that it compensates for the highly spatially correlated redeposition attempts of a particle that has been rejected due to the prior presence of deposited particles [210,211]. with radius s is a = s 2 π. Each sphere has an exclusion zone (light green halo around spheres) and thus the total occupied area of one sphere is 4s 2 π. While overlapping of exclusion zones is permitted, the distance between the center positions of two spheres cannot be smaller than 2s. Besides the top view of a larger number of adsorbed spheres (A), three separate spheres both with side and top views are also shown providing a clear example of the RSA rules (B).
[Adapted with from Ref. [187], copyright Marcel Dekker Inc.; from Ref. [212] copyright American Physical Society] In kinetic equations, RSA yields a Φ whose value is always below the Langmuir 1-. The describing -dependence of Φ has been approximated theoretically and confirmed by systematic numerical simulations for objects with varying geometrical shapes. The RSA model satisfactorily accounts for the observation that the plateau of adsorption (the -jamming limit‖,  j ) of a protein onto a continuum is typically less than half the value expected for a packed monolayer. The coverage can be thus redefined as: (21) J o u r n a l P r e -p r o o f In the case of adsorbing disks,  j ≈ 55%. The physical meaning of  j is that although there is still plenty of unoccupied space, there is no free surface large enough to accommodate an additional adsorbate. Schaaf and Talbot found that for spherical adsorbates (subtending disks) the available surface function is [201]: (22) where the jamming limit is  j = 0.547. For typical spherocylindrical objects [213,214]: (23) with a jamming limit  j = 0.581; a range of aspect ratios was investigated. Three-dimensional expansion of the RSA model for spheroidal particles has been established by Adamczyk and Weroński [215]. Characteristics of a system following the RSA model is demonstrated in Fig. 3, especially in graph B. As compared to the Langmuir model, a nonlinear end tail appears in the RSA dM/dt vs M plot originating from the nonlinear Φ function. Such characteristic of an unknown system may suggest an RSA behavior. However, for note, this type of nonlinearity can also originate from more complex kinetic models involving more adsorbate states on the surface (shown e.g. in Section 4), even without calculating Φ according to the RSA model.
The Langmuir model defines a as a virtually occupied area that can be much larger than the actual area of the adsorbate in contact with the surface if the binding sites are sparsely distributed. The RSA model however provides a quantity that describes the adsorbate-surface interaction with a physically greater relevance. In this case a appearing in Eq. (21) can be interpreted as the actual area of a single adsorbate interacting with the surface, also termed as molecular footprint.

Multiple molecular states on the surface
The simplest kinetic model assumes only one state on the surface. This is doubtless valid for isotropic spheres of a metal or a simple binary compound such as CdTe but is an unrealistic description of a complex protein. In many cases, the complexity is manifested as multiple orientational and conformational states in which the adsorbates can be present on the surface. A biopolymer can undergo conformational changes that significantly increase the number of interfacial interactions [216]. As the probability of breaking up these interactions simultaneously is also increased, the probability of desorption becomes almost vanishingly small (k d < 10 -6 s -1 ) and the adsorption becomes practically irreversible [186]. In this case, a relatively simple approach is to separate the total surface adsorbed mass into reversibly and irreversibly adsorbed forms, viz. M = M r + M i . What makes the two states distinguishable on the surface is their different footprints a r and a i [207]. Eq.
Two-state RSA can encompass many variants, such as the adsorption of a single species that changes its conformation on the surface and a single species that can adsorb in two different orientations, and the kinetics of surface conformational change can be explicitly incorporated into the model [204][205][206]208]. If the data quality is good enough, as it should be from an OWLS or GCI measurement, the rate constants can all be determined by fitting [207].

Dealing with mass transport and hydrodynamics
The adsorption process requires sufficient analytes close to the surface; their presence is maintained by transport from the bulk solution. The zeroth approximation is to say that the concentration close to the surface is always equal to the bulk concentration (c B ); i.e., the analyte is uniformly distributed within the entire volume of the flow cell and all parts of the biosensor surface are exposed to the same constant analyte concentration (one may refer to this as the -well-mixed‖ approach but in reality mixing can never be that good due to the friction at the walls). In this case, one can set c B as the concentration of analytes in the rate equation Eq. (15). Stable hydrodynamic conditions can be best obtained by applying laminar flow in the flow chamber, e.g., by a peristaltic pump or, even better (to avoid rhythmic pulsations) a syringe pump. The laminar flow is characterized by a low Reynolds number Re (which is, however, dependent on the flow geometry and only gives a rough estimation of the flow régime). Nevertheless, laminar flow normally obtains for Re < 2000 for a long, straight cylindrical tube, and in practical biosensors it can be as low as 1-10. On the other hand, even values larger than 2000 can maintain laminar flow if disturbance that would drive it into turbulence is avoided. Due to the characteristic parabolic velocity profile of laminar flow in a cylindrical tube, the number of analytes is inevitably depleted close to the surface and the transport through this vicinal region may be crucial both in terms of biosensor measurement outcome and in the kinetic analysis of the data. The presence of the depletion zone does not necessarily mean that transport is adsorption ratelimiting. If the rate of binding event is much slower than the transport, the kinetics are independent of transport and the flow cell can be treated as a well-mixed system. If the binding is fast, transport has a significant effect and can even dominate the kinetics [217][187] [218]. This case is termed mass transport limited (MTL), recognition of which plays a crucial role in kinetic analysis and in the design of flow chambers. To fully treat mass transport, we must know the spatiotemporal concentration distribution in the flow cell, which can be achieved by solving the full convective-diffusive partial differential equation (PDE). Although this approach assures high accuracy in principle, the high computational demands of the numerical solution make it unsuitable for end-users trying to fit the model to their data [219]. Furthermore, errors introduced by discretization do not guarantee reliable results. With rational approximations, however, the problem can be made more tractable. In practice, dealing with transport means that we should only know the J o u r n a l P r e -p r o o f concentration in the close vicinity of the surface (c S ). What considerations enter the calculation of c S ? Starting with the mathematical description of the convective-diffusive system, we go through the derivation of the framework that is used to deal with transport in adsorption kinetic analysis.

The PDE model
The fundamentals of concentration variations in streaming solutions confined in a chamber whose solid wall is a site of heterogeneous chemical reaction were originally discussed in terms of electrochemical problems. The PDE describing this system is closely related to the Navier-Stokes equations, which should be taken into account for a full hydrodynamic treatment of the system. In the field of biosensors, the PDE model was first applied for total internal reflection (TIR) measurement analysis [220][221][222]; the equations were adapted for the Biacore SPR flow cell [223], and further developed by Christensen [224] and others [217,225]. Based on these studies (in particular following the formalism of Myszka and co-workers), we introduce the model conditions and governing equations as follows. Let us assume a flow chamber of rectangular cross-section with length l, half-height h (full height b = 2h) and width w as shown in Fig. 5. The coordinates are given in the Cartesian system with x, y and z axes parallel to the sides l, w and h, respectively, where the origin is the inlet at the left bottom edge of the channel on the sensing surface (bottom plane). The solution enters with a flow direction parallel to x, linear flow velocity (flow rate) u x and free analyte concentration c B . We assume fully developed laminar flow along the whole length of the channel with its characteristic parabolic velocity profile on the vertical xz plane, described along the coordinate z by u x (z) = u max (z/b)(1-z/b). As such, the flow rate is zero at the bottom (z = 0) and top (z = b) channel boundaries and reaches its maximal value u max in the center (z = h). Considering a mean velocity value of u mean = (2/3)u max and cross-section area bw, the volumetric flow rate can be written as Q = bw(2/3)u max . For most channels, w >> h, hence concentration variations in the y direction can be simply neglected.  . (25). According to the model, the mass transport of analyte to the sensor surface is determined only by diffusion and convective flow, where the y component of diffusion as well as y and z components of the flow velocity are neglected. Other driving forces relevant for mass transport such as the presence of a substrate electric field [226,227] may also be included in the model. While at the sensor surface boundary (z = 0) the transported analyte mass flux density (j M ) should be equal to the binding/adsorption rate (reaction flux density, j R ) (26a), at the top boundary it should be zero due to the unreactive and impenetrable channel wall (26b). Note, j R is equal to the chosen cumulative equation for the mass derivative, e.g. Eq. (15). The concentration is constant with value of the injection concentration c inj (26c) at the inlet boundary, and as a reasonable approximation the analyte leaves the cell at the outlet due to convective flow only (26d). The system's initial conditions are determined by the association (at t = 0) (27a) and dissociation (at t = t d ) (27b). The PDE model can be solved only numerically. Numerical solutions of similar systems are extensively detailed in the literature, starting with the solution of equivalent electrochemical problems [228,229]. In the field of biosensing, the related papers cover mostly numerical simulations (using both finite difference (e.g., the Crank-Nicholson scheme) [220,224] or finite element methods [26,217,225,[230][231][232][233][234][235][236]) adapted to the Biacore SPR flow chamber with specific receptor-ligand binding according to simple Langmuir kinetics, which might be appropriate for receptors immobilized in a dextran hydrogel, but then the hydrodynamic profiles will be more complicated than those shown in Fig. 8. To the best of our knowledge, a numerical solution of the PDE model involving RSA-based kinetics at a planar surface of receptors for which Fig. 8 is applicable has not been published. Vijayendran and co-workers have shown that certain assumptions (e.g., neglecting convection but J o u r n a l P r e -p r o o f considering the diffusion equation for relevant dimensions) can make the PDE model more computationally tractable [237]. The full numerical solution is primarily used for simulations and, by retaining the complexity of the actual system, it can serve as an absolute standard for testing the validity of simpler models [219]. In applications, simplified versions of the PDE model, e.g., the two-compartment model, became widespread because analytical solutions to the equations for the proper convective-diffusive system are accessible through the steadystate approximation. Constantly increasing computational power may, however, soon enable the full set of equations to be numerically solved and simultaneously fitted to the data.

Two-compartment model
The two-compartment model is based on the analysis of the relative contribution of convection and diffusion to analyte transport. Far from the surface, well within the solution body, the concentration is maintained by convection at around c B . Although in most of the volume of the channel transport is governed by convection, its contribution to analyte transport vanishes in the close vicinity of the surface (Fig. 8), causing significant depletion in analyte concentration and molecular diffusion dominates transport. The relationship between convective and diffusive transport can be characterized by a dimensionless ratio called the Péclet number (Pe). Defined in direction x and taking l and h as characteristic channel distances, Pe is: If Pe >> 1 the depletion zone is much thinner than h and, in this case, convection is the dominant transport mechanism, if Pe << 1 molecular diffusion predominates across the channel. Various flow conditions have been discussed by Squires et al., showing that typical value of Pe can be in the order of 10 5 , but that low Pe values are essential for sensors designed to achieve the full collection of analytes from their solution on a sensor surface [26]. The region near the surface where the analyte is depleted is called the diffusion boundary layer (DBL) and has the characteristic thickness δ D . The DBL concept was first introduced by Nernst as the -diffusion layer‖ (stagnant film) model [238]. The model assumes that under a steady-state condition Fick's first law becomes simplified as the flux through the thin stagnant film towards the interface, depending on the constant concentration gradient measured at the two boundaries of the film: j M = D∂c/∂z ≈ D(c B -c S )/δ D . Here, D/δ D can be defined as a mass transfer coefficient (k m , typical units of cm s -1 ) determining the rate of the mass transport process. A simplified version of the (25) -(27) PDE model can be obtained if we divide the inside space of the flow cell into two plane-parallel compartments according to the DBL concept. Let us suppose that the concentration in the outer compartment is uniform both in space and time and equal to the injected bulk concentration c B . The inner compartment is the DBL itself, and the boundary between the two compartments is determined by the DBL thickness δ D . If the following assumptions are made: (ⅰ) the concentration is in a steady state (dc/dt = 0); (ⅱ) the velocity J o u r n a l P r e -p r o o f profile close to the surface is linear (in case of rectangular channel geometry, v(z) = 4zu max /h at z/h << 1); (ⅲ) diffusion in the direction of convective flow can be neglected (∂ 2 c/∂x 2 = 0); and (iv) the surface acts as a perfect sink (c S = 0, i.e. it can bind analytes infinitely fast and has infinite capacity), then the (25) convective-diffusive equation can be simplified [222] to ( ) (29) This system can be analytically solved as first shown by Lévêque [239] for an equivalent heat transport problem. The problem was extensively detailed in terms of mass transport by Levich, who delivered the solution for the case of laminar flow in tube and for various other systems [240]. The flux density governed by diffusion towards the cylindrical wall of a tube can be written as where r is the inner tube radius and x is the distance from the tube inlet to the position of interest (e.g., where the biosensing measurement is taking place) exposed to the mass flux. This is the Lévêque solution, which actually does not consider any interfacial binding kinetics. One main implication directly obtained from this formula is that it gives an analytical expression for the DBL thickness as δ D = (1/0.67)(Drx/u max ) 1/3 and thus for k m . Levich's related research mainly focused on electrochemistry, in particular to calculate the limiting electric current of electrochemical cells under fluid motion. Based on his landmark work, further efforts of Blaedel [241,242], Klatt [243] and Matsuda [244] provided the solution of (29) for various electrode designs including tubular and channel electrodes [245][246][247][248]. In case of tube geometry with circular cross section Eq. (30) applies, while for channels where the cross section is rectangular, the flux density is Here, h is the inner half thickness of the channel as shown in Fig. 5 B. Note that this expression is identical to Eq. (30) using the substitution of r = h. Regarding Eqs. (30) and (31), the extensive literature is rather confusing since many publications use different equation forms, which makes it hard to recognize that they are, in fact, identical. It can be seen that δ D depends on coordinate x with power 1/3 along the channel, providing its characteristic shape as shown in In the case of OWLS fluidics, Ramsden and Kurrat used the formula provided by Levich for tubes [187,249,250] written as: δ D = (1/0.67)[(2/3)DrxA/Q] 1/3 , where the rxA term can be combined in one parameter that depends only on cell geometry [27,251]. For the widely used Biacore SPR setups, the average <k m > is expressed by the following formula [252,253]: where b is the full thickness of the channel (b = 2h) as well as w and l are the channel width and length, respectively. This equation defines an average k m (and thus average flux) over the full length l of the channel, and it can be directly obtained from Eq. (31) by integrating from x = 0 to l according to x and dividing with l [224,254]. Another version of Eq. (32) can be found in refs [254] and [255], where the length of the sensing area and its location in the flow cell relative to the inlet are also taken into account. When the surface is assumed to be a perfect sink, i.e. all the analytes are continuously bound to the surface with an infinitely fast rate (k a → ∞ if there were no energy barrier [184]), the surface concentration is zero (c S = 0) and the flux is j M = k m c B . This is valid for electrons in electrochemistry but not for analyte binding in biosensing. When the binding process has an energy barrier (finite k a ), not all analytes transported to the surface will be immediately bound. Thus, the energy barrier retards adsorption and oppose depletion in the immediate vicinity of the surface resulting in c S > 0 (although still lower than c B ) and j M = k m (c B -c S ). This means that the binding process will be no longer fully transport-limited [184]. Note that the thickness of this vicinal region demarcated by the distance δ S from the surface is of the order of molecular dimensions (i.e., a few nm) as illustrated in Fig. 5 C. As shown by Ramsden, the change of analyte concentration within this layer can be written as [187] * + (33) This equation should be solved simultaneously together with the rate equations written in terms of adsorbed mass (see Eq. (15)). In this way, the PDE model could be reduced to a system of coupled ordinary differential equations (ODE). The main difficulty of solving these equations is that we do not accurately know the value of δ S . Another way for dealing with the influence of binding kinetics to the surface concentration is the projection of dc S /dt into the whole DBL, merging δ S and δ D [219,225,[256][257][258]: This latter approach became standard in the two-compartment analysis when dc S /dt is taken into account [225]. However, the accurate form is Eq. (33), as the volumetric concentration c S should be defined in a thin film close to the surface (Fig. 5 C). Being an undetermined parameter, the 1/δ S term often referred to be as the ratio of unit volume and surface area (S/V).
The thickness of the surface zone where dc S /dt should be considered can be eliminated from Eq. (33) and (34) if we make further assumptions. Let us assume that the flux j M and j R are balanced, which means that c S is in a steady state (dc S /dt = 0) [184,250,252]. If we also suppose that next to δ D , δ S can be neglected (δ D -δ S ≈ δ D ), this allows for the elimination of δ S . Reordering Eq. (33) one obtains an explicit formula for c S : (36) which is coupled with the equation describing the adsorption: The presence of makes it clear that not only the energy barrier but also the presence of previously adsorbed analytes prevents the surface from being a perfect sink. The approach of Eq. (36) is the standard method for calculating c S when mass transport is considered. The use of k m enables a simple interpretation of the coupled transport-adsorption process using the sequential scheme: where P B and P S represents analytes in the bulk phase and in the vicinity of the surface, respectively, and A r denotes the reversible adsorbed form of the analyte. This interpretation became widely used for explaining mass transport limitations in the SPR literature [218,252,253,259]. In the field of protein adsorption kinetics, the effect of hydrodynamics is mostly contained in δ D . Nevertheless, the two methods are actually equivalent, and they can be easily converted to each other using the substitution k m = D/δ D . As discussed by Karlsson et al. [252], using this k m based interpretation of transport limitation, distinguishing the transport-and binding-limited domain becomes quite straightforward. Substituting Eq. (36) into Eq. (37) one obtains (39) where the term k f = k a /(1+k a Φ/k m ) can be taken as the forward and k r = k d /(1+k a Φ/k m ) as the reverse rate constants of the whole process (38). It can be deduced from Eq. (39) that if the ratio k a Φ/k m << 1, then k f = k a and k r = k d , meaning that adsorption is far from the transportlimited domain and is binding-limited. In case of k a Φ/k m >> 1, then k f < k a and k r < k d , J o u r n a l P r e -p r o o f presenting a transport-limited system. It can be additionally inferred from Eq. (36) that in this case c S ≈ Mk d /k a (assuming c B k m /k a → 0). Eq. (39) also helps to understand how the association and dissociation phases of the entire adsorption curve can be distinguished in the data evaluation. In the initial linear phase of the association section, the adsorbed mass is small (M ≈ 0), thus k f c B Φ dominates and k r M can be neglected. Looking at the dissociation section, if it is assumed that c B = 0 from the starting time of washing (which does not always hold as shown in Section 3.5.3), then dM/dt = k r M and the kinetics are not affected by c S . As a result, for the binding-limited case k a and k r can be determined by analyzing the initial phase of association section and the entire dissociation section separately. The analysis of initial phase also enables k m to be determined. As Φ has its highest value at the starting point of the adsorption (Φ = 1 at t = 0) and decreases over time, transport limitation is most probable at the beginning of the process. Here, practically the ratio k a /k m determines whether a transport or binding limitation must be considered. When the kinetics is heavily transport-limited (initial phase), practically k f = k m and dM/dt = k m c B Φ. This relation allows k m to be determined by fitting the initial phase of the M vs. t dataset and taking the slope of the fitted line [260]. The strategy also allows for the determination of D: once k m is obtained, D can be calculated using Eq. (31) or (32) [225]. As seen from Eq. (39), within the intermediate domain ranging between the two limits, the simple linear formula k f -1 = k a -1 + k m -1 can be used for the determination of k a in the initial phase from the experimentally observed k f and calculated k m . Nevertheless, Déjardin and co-workers showed that this method can result in considerable error as δ D is assumed to be independent of k a and c S (Lévêque solution), however this is not true. Analyzing the exact solution of Eq. (29) when adsorption is also considered, they provided more accurate linear approximations for the two limits determining an intermediate domain in between [261,262]. As such, close to binding control, k f -1 = 0.827k a -1 + k m -1 , while close to transport control, k f -1 = k a -1 + 0.684k m -1 . Here, k a is related to a given position x on the surface. It was also pointed out that considering the average value of the rate parameters over the full length of the channel (<k f >, <k a >, <k m >), the error is smaller. Déjardin et al. developed a formula that describes the relationship of k f , k a as well as k m and allows for calculating k a from initial phase data. For further details about this model, see Ref. [262]. Nevertheless, under certain conditions the adsorption régime may strongly vary along the length of the channel. As demonstrated by a thorough analysis in the paper of Myszka and co-workers [225], the two-compartment model can provide excellent accuracy when fitting transport-limited data and it serves as a rather good approximation of the PDE model. The simplicity, implying meagre use of computational resources, and its simple interpretation has made the twocompartment approach the primary method in the kinetic analysis of measurements under transport limitation [219]. However, the model tends to fail if transient regimes influence the kinetics, e.g., when analyte solution is suddenly injected into the flow cell. With the motivation of refining the two-compartment model, three- [219] and multi-compartment [258] models have been developed. Future access to even larger computational power may allow these models to be further elaborated.

Flushing effect
Standard kinetic models suppose that the bulk concentration changes infinitely fast when, for example, the baseline phase of a biosensing run is switched to the analyte adsorption phase; i.e., concentration as a function of time follows a step function [184]. However, depending on the geometry of the flow cell, changing one solution to another (flushing the cell with the next solution) may take significant time. Changing water to an aqueous glycerol solution is an effective test to see how the bulk concentration actually develops under given hydrodynamic conditions. Glycerol is a suitable compound since it does not adsorb to the sensor surface and it diffuses relatively fast (D glycerol = 1.06×10 -5 cm 2 s -1 vs. D protein ≈ 10 -7 cm 2 s -1 ) and thus the obtained signal is practically not affected by diffusive transport.
The RI change in Fig. 6 A demonstrates such an experiment with OWLS employing an SH812-08 flow cell and 1 µL s -1 flow of, first, DI water, then 6% (w/w) aqueous glycerol solution, and lastly again DI water. This observed time-dependence in the variation of concentration can strongly affect the observed biosensor response. Note that in case of an OWLS or other optical biosensing experiment, where the adsorbed mass is calculated from the effective RIs using n C of the bulk solution, which is proportional to c B , the n C profile should be taken into account. A simple method of considering the concentration variations during flushing is to model its kinetics with the logistic function (Richard's curve), which has the form [263]: (40) where Y 0 and Y max are the initial and maximal values of the fitted curve and q and g are fitting parameters. As shown in Fig. 6 B-C, the change in n C during the two flushing phases could be fitted with this equation with excellent accuracy. The obtained fit parameters (given in the figure caption) can then be used when flushing affects the transport-limited section of the adsorption kinetic curve. The flushing curve can be incorporated if the steps in the original c B (t) curve are replaced by a continuous sigmoid-like shape determined by the fitted q and g parameters (Y max should be equal to c B as well as Y 0 is determined at the curve bottom). Further details about the inclusion of flushing effects in kinetic analysis will be given in Section 4.1.

Searching for the best fitting kinetic model. Effect of varying model parameters
What we can see on a typical adsorption curve is that the signal (commonly expressed as adsorbed mass) converges to saturation. In addition, a significant number of adsorbates may desorb when the solution is exchanged to baseline buffer, but a comparable number remains on the surface. The washing phase therefore plays a key role in analyzing adsorption curves, as it clearly shows that there is both a reversibly and a quasi-irreversibly adsorbed amount, indicating that the adsorbates may have at least two different states on the surface. A model that considers both reversibly and irreversibly adsorbed molecules on the surface should define two distinct states for the adsorbate characterized by separate mass values, M r and M i , respectively. In a label-free measurement, a total adsorbed mass, M = M r + M i , is observed. Two types of models that provides the simplest approach for describing such a system are the two-state parallel and consecutive models, schematically illustrated in Fig. 7.
Our example system shown in the figure is represented by the adsorption of wild-type flagellin (FliC) molecules on hydrophobized OWLS sensor surfaces, characterized in several studies in detail [251,264,265]. The bacterial filament monomer flagellin is an intensively studied protein owing to the ease of its modification by genetic engineering techniques, J o u r n a l P r e -p r o o f yielding various forms for a wide range of applications such as biorecognition element or antifouling coating (e.g., we showed that compact layers made of flagellin molecules can regulate bacterial adhesion [78]). Flagellins possess intrinsic anisotropy as they are asymmetric molecules with a capsule-like ellipsoidal shape. The anisotropy enables layers formed from the same molecules but different in surface mass, thickness and orientation under varying external conditions to be designed. The salt composition of the liquid medium can drastically affect the orientation of flagellin on the surface. Fig. 7 Schemes of the two-state parallel and consecutive adsorption kinetic models. The example schemes shown here represent flagellin adsorption on a hydrophobic surface. In the presence of kosmotropic salts [251], flagellin molecules adsorb on hydrophobic surfaces predominantly through their disordered hydrophobic D0 domain (shown in yellow; other domains are represented with different colors) and form a layer where molecules are oriented perpendicularly to the surface. As illustrated with blue, wider arrows, kosmotropic salts significantly increase k a .
[Adapted from Ref. [251], copyright 2018 American Chemical Society] The reaction schemes and equations corresponding to the two models are given below. The equations on the left and right represent the parallel and consecutive model, respectively.
Both models consist of a coupled ODE system involving two ODEs for describing mass changes and one for considering variations in c S (Eq. (47)-(48)). Here, dc S /dt calculation follows Eq. (35). In case of assuming dc S /dt = 0, this latter ODE can be eliminated and c S is calculated according to Eq. (49)-(50). The formula for expressing Φ depends on the chosen RSA model where the shape of the adsorbates must be considered (Eq. (22) or (23)). When calculating Θ with M r and M i , one can also take into account whether the two states are described with one footprint (a) (Eq. (21)) or whether they have two distinct footprints (a r and a i ) (Eq. (24)).
In order to bring these detailed kinetic models into a realistic context and show their power in practice, we demonstrate their application to the kinetic modeling of FliC adsorption, which represents a much more complex process than simple Langmuir-type binding. Fig. 8 shows the adsorption curves of FliC molecules added to the sensor surface in solutions containing different salts, NaF (chaotropic) or NaClO 4 (kosmotropic) [251]. The salts were selected from the Hofmeister series, ions of which can regulate hydrogen bonding interactions of proteins and surfaces [48,251]. As demonstrated in Fig. 8, the salt has a significant effect on the adsorption kinetics. Both the data corresponding to reversibly and irreversibly adsorbed states are shown and plotted in different formats. The dM/dt vs. t graph clearly shows the magnitude of rates over the measurement interval (Fig. 8 C, H) and the dM/dt vs. M graph indicates the relevant regimes and mechanism of adsorption ( Fig. 8 D, I).
In the latter graph, the slow increase in dM/dt shows that the adsorption is transport-limited and then, reaching the maximum rate, it becomes binding-limited, when the rate is determined by the area still available on the surface 139 . The nonlinear shape of the end tail in the descending phase suggests an RSA-type adsorption (see Fig. 3). Although the model considered diffusion, it failed in reproducing the initial phase. Calculating k m for the NaF measurement one can conclude that this system is far from remarkable transport limitation, as (k a + k i )/k m = 2.4×10 -2 << 1. This suggests that not the transport, but flushing of the flow cell is responsible for the discrepancy between the observations and fit (Fig. 8 F, G). In the following, we analyze the effect of considering dc S /dt ≠ 0 and including flushing in the variation of c B .
J o u r n a l P r e -p r o o f

Dependence of the kinetic fit on fit parameters and model conditions
The dependence of the fitted RSA kinetic model on the fit parameters in the case of protein adsorption has been analyzed by Kurrat et al. in depth[250]. The effect of hydrodynamic conditions on the observed kinetic curves has also been examined elsewhere, especially for SPR biosensors [225]. In this section we provide sets of simulated curves which are meaningful in demonstrating the effect of the kinetic regime and hydrodynamic conditions at which the system operates. The effect of systematically varied model parameters in the case of a Langmuir-type binding can be seen in Fig. 9 and Fig. 10. All the curves were simulated using our NBS-Kinetics software.
Similar tendencies can be observed for a more complex system than the Langmuir, as presented in Fig. 11 and Fig. 12. Here, varying kinetic curves are shown supposing the adsorption of flagellin molecules according to two-state parallel kinetics. It is obvious that if k a increases, c S will not follow c B and the measured mass curve will be affected in the 7 cm 2 s -1 , k d = 1×10 -3 s -1 , a r = 10 nm 2 , a i = 20 nm 2 ; for Q dependence: c B = 100 µg mL -1 , k a = 1×10 -4 cm s -1 , k i = 1×10 -8 cm s -1 ; c B dependence: Q = 1 µL s -1 , k a = 1×10 -5 cm s -1 , k i = 1×10 -8 cm s -1 .   Fig. 8, the overall fit has a very good quality (R 2 = 0.984-0.996), however, there are larger errors around the transient phases. The initial adsorption section was best fitted by the model supposing dc S /dt = 0 and involving flush effect (B). However, these conditions did not facilitate a better fitting of the washing section (C, D), and the best performing model was in fact the simplest one (dc S /dt = 0, no flush). This interesting discrepancy should be lying on conditions which were not implemented in the model, including hydrodynamic parameters that are neglected in the simplified equations.
Fitting the full PDE model is a good candidate to resolve such a minute effects, however as the best of our knowledge no attempt to fit the full PDE model has been published so far.

J o u r n a l P r e -p r o o f
Although the effect is minute, it can result in significant deviations in the fitted kinetic data, as seen in Table 1.  As seen in Fig. 14

Surface energetics
The question of why certain types of molecules tend to adsorb on solid surfaces can be explained by energy conditions with energy contributors of the adsorbing molecules, the solvent molecules and the surface. In terms of proteins, the determinative energy-related parameters has been extensively studied by Norde and co-workers [266][267][268]. Information on the interaction of the adsorptive with the available surface area is contained in the rate coefficients. As the adsorption and desorption are generally considered as thermally activated processes, it is assumed that the rate coefficients can be written in the form (51) where U act is an activation energy for the process, and F an appropriately dimensioned frequency factor. In order to identify the forces involved, it can be extremely valuable to measure the adsorption kinetics for a given protein/surface combination under different solution conditions, i.e. pH, ionic strength [269], cosolute composition, and temperature. Electrostatics alone do not seem able to account completely for what is observed in many cases. Complementary techniques, including titration of adsorbed proteins, electrokinetic measurements of the surface potential of absorbed layers, and the direct measurement of the surface forces between adsorbed layers are making an important contribution towards resolving the current ambiguities. However, under physiological conditions, electrostatic forces may be negligible [48].
Kinetic analysis of biosensor data also allows for estimating the adsorbate-watersubstrate interfacial tension (γ PWS ) of the adsorbed species (e.g., a P protein molecule), which is defined as the specific work of adsorption during the adsorption process. To arrive at the determination of γ PWS based on kinetic data, we first write the Gibbs free energy change during the adsorption process (work of adsorption, ΔG a ) as [184,251] where ΔG PW , ΔG SW , ΔG PS as well as γ PW , γ SW , γ PS are the Gibbs free energy changes and interfacial tensions at the protein-water substrate-water and protein-substrate interfaces (water can be substituted by any solvent). Here we assumed that the area of adsorbed protein in J o u r n a l P r e -p r o o f contact with the surface can be given by the a footprint and that in the dissolved state this area is surrounded exclusively by solvent molecules. Upon adsorption the area a is disappearing at the PW and SW interfaces and it is appearing at the PS interface. Rearranging Eq. (52) one can write (53) Here, a new quantity γ PWS is introduced which is the protein-water-substrate interfacial tension defined as γ PWS = -(γ PW + γ SW + γ PS ). An attempt to link ΔG a to the quotient of the rate constants has been already made. Applying Eq. (51) for both k a and k d , the following equation can be obtained [251]: (54) where the factor δ S characterizes the thickness of surface adsorption zone potentially taking a value in the nanometer range. The reason for the appearance of δ S is to harmonize the dimensions in Eq. (54) and convert the k a dimension cm s -1 dimension to s -1 . Adsorbing species close to the surface than δ S are potentially in adsorbed state. In this sense, δ S can be used to convert the volume concentration into surface concentration in a straightforward way and we can define a surface related k a ' = k a /δ S , which can substitute ka in, e.g., Eq. (43) such as k a '(c S δ S )Φ. Rearranging Eq. (54) one can find that (55) Expressing ΔG a based on Eq. (53) and substituting into the above formula we arrive to calculate γ PWS using k a and k d determined from a kinetic analysis [251]: (56) In numerical calculations, δ S = 10 nm can be used, which is a reasonable approximation for the thickness of surface vicinal region. γ PWS is a complex quantity, depending on the waterexposed adsorbate surface, the nature of substrate surface as well as the quality and concentration of dissolved cosolutes. The significance of γ PWS is that it is a more precise characteristic parameter of the system than the previously studied γ PW which does not deal with the hydrophobic/hydrophilic properties of the substrate. Moreover, the accuracy of the determined γ PW was especially affected by the difficulties in determining interfacial tension in microscopic conditions. Based on these considerations, γ PWS is assumed to be a critical parameter in the phenomenological interpretation of protein adsorption processes, such as γ PW for conformation-related phenomena.
The application of the above theory in case of measured kinetic data was demonstrated in Kovacs et al., where the dependence of flagellin-related γ PWS was on the presence of different Hofmeister salts was investigated [251]. The calculated γ PWS values followed the expected J o u r n a l P r e -p r o o f Journal Pre-proof tendency (the presence of kosmotropes increased γ PWS while chaotropes decreased) as well as agreed with (the available few) literature values, confirming the validity of calculating γ PWS based on kinetic data. 5 Combined label-free measurements: unraveling structure and kinetics from biosensor data

Essence of combination. Hydration and viscoelasticity
In spite of the availability of various label-free techniques, each of which can monitor molecular interactions with good sensitivity and throughput, no single technique can achieve a full characterization of biofunctional coatings. Soft and hydrated interfacial layers present the greatest challenge due to their small optical contrast and conformational variability. To expand the range of analysis, combined label-free methods have been developed, primarily exploiting the advantages of optical and acoustic biosensors. In principle, optical methods are not sensitive to solvent molecules associated with the adlayer, since compared to the solvent molecules already present in the bulk phase, they do not add a significant shift to the RI. As a result, when data are referenced to the baseline, the associated solvent is in effect subtracted, which means that only the signal of deposited analytes themselves (so-called -dry‖ mass) is recorded. Contrary to optical methods, QCM measures both the dry analyte and any associated solvent, because it is sensitive to any added mass within the penetration depth that oscillates together with the crystal. Although the solvated mass in itself does not allow for the determination of solvent content, by using the -dry‖ OWLS mass measured on the same layer one can separate the solvated mass (M QCM ) into dry analyte (M optical ) and solvent masses (M QCM -M optical ) and determine the solvation factor (H A ) [270][271][272]: (57) Evaluation of QCM data by the Kelvin-Voigt model requires knowledge of the mass density of the adlayer. Hitherto, when the layer is assumed to be strongly hydrated, ρ A = 1000 kg m -3 is commonly used, with little understanding of the validity of the assumption. Now, the value of ρ A can be determined from the measured optical and QCM mass data (the 3 rd overtone QCM Sauerbrey mass (M QCM,S ) is recommended) if the density of adsorbing and solvent molecules are known [67,107]: (58) where ρ analyte and ρ solvent are the mass density of surface bound analyte and solvent molecules, respectively. Due to the ability to obtain information about the amount of coupled water inside the adlayer complemented by additional measured properties such as dominant chain conformation and viscoelasticity (Fig. 3), the combination of QCM and OWLS data results offers an excellent tool for biolayer research [107,273].
J o u r n a l P r e -p r o o f Fig. 15 Principle of QCM sensitivity to adlayer structure. The inset drawings on the top correspond to the bottom resonance curves and represent QCM chips in different conditions. See the colors for matching the drawings and curves. Inset image A represents an unloaded QCM chip, where only a thin part of the bulk solution oscillates together with the crystal. The shear wave generated by the chip oscillation is also shown. B. If the QCM chip is loaded by a thin and rigid adlayer with a low water content, the decay of the penetrating shear wave is similar to the unloaded case, only a frequency shift is detected, but Γ is almost unchanged. C. In case of a thick, viscoelastic adlayer potentially possessing a large water content, the layer does not oscillate perfectly together with the crystal resulting in an increase in dissipation and thus a marked widening in Γ.
The combination of QCM with optical techniques has been realized in three ways: (ⅰ) combination in one device probing exactly the same surface simultaneously by the two techniques; (ⅱ) one device but the measurement is performed on separate surfaces using separate flow cells; and (ⅲ) measurement by separate instruments on different surfaces albeit at the same time. Although the first approach is obviously the best it is a great challenge for instrument developers; nevertheless SPR-QCM [274] and SE-QCM [106,[275][276][277] (ellipsometry-compatible QCM module from Biolin Scientific AB) setups have been already demonstrated, but despite the benefits offered by waveguide-based methods, the combination with QCM has still not been achieved. Currently (ii) is available (OWLS QCM 3000 Biosensor system from MicroVacuum Ltd. [278]), allowing the same reagent solution to be used and maintaining the same temperature of the surfaces. The combination of QCM and OWLS data measured in parallel OWLS-QCM experiments have already shown its potential in hydrated nanolayer characterization [67,131,271,272]. Another interesting venture is the combination of GCI with spectroscopic ellipsometry (GCI-SE) in one instrument, which may offer the simultaneous exploitation of the high sensitivity provided by GCI and the spectroscopic capabilities of SE, allowing complex multilayer structures to be analyzed in detail [279]. SE-SPR is a (i) type combination realizing the generation of surface plasmons and simultaneous SE measurement in the proper configuration J o u r n a l P r e -p r o o f [280][281][282][283][284]. Whether the data are obtained from (i) or (ii), the post-measurement combination of QCM and optical data are present complex data analysis challenges. Several comprehensive studies have already combined data obtained from separate SPR-QCM [104,107,112,[285][286][287], SE-QCM [107,270], OWLS-QCM [67,131,271,272], DPI-QCM [91,288,289] as well as RWG-OWLS [129,290] experiments. Of note, our recent work explored the possibility to calibrate the label-free optical biosensor signals with direct adhesion-force measuring technologies [162]. Such combinations will open up novel research direction in adhesion research and could deepen our understanding in terms of structure-kinetic relations.

Chain conformation
OWLS is uniquely sensitive to the orientation of molecules in the adsorbed layer, owing to the fact that both TE and TM polarizations of the excited modes propagate in the waveguide layer. The field intensity of TE and TM waves oscillate in planes perpendicular to each other, thus they are differently sensitive to oriented molecules on the surface [264] (see Fig. 17Error! Reference source not found.). An adlayer consisting of oriented molecules (ordered structure) is optically anisotropic, manifesting optical birefringence (or double reflection). In general, the RI of a birefringent layer can be decomposed into ordinary (n A,o ) and extraordinary (n A,e ) RIs and the sign of birefringence is defined by the ratio of n A,o and n A,e . Although the usually measured two waveguide modes do not allow the simultaneous determination of all the three optogeometrical parameters (d A , n A,o , n A,e ), the averaged adlayer RI ñ A calculated by the 4-layer homogeneous and isotropic mode equations can indicate the presence and extent of birefringence. Alternatively, d A should be independently measured, e.g. by X-ray diffraction [80]. Only a few techniques (e.g., polarized visible attenuated total reflection spectroscopy [291]) have the capability of measuring n A,o and n A,e simultaneously. If one measures a thin adlayer made of molecules with predominantly parallel orientation with J o u r n a l P r e -p r o o f the surface, an overestimated ñ A is observed; for an adlayer consisting of molecules having their conformation predominantly perpendicular to the surface, an underestimated ñ A is observed. As a result, the sign of ñ A is an indication of the type of optical anisotropy and thus the adlayer nanostructure. Once M is calculated, the NBS-OWLS analysis software plots the ñ A -M curve, which timeimplicitly shows the change in ñ A as a result of each unit of mass added to the surface (see Error! Reference source not found.) As ñ A is sensitive to optical anisotropy in the adlayer, the observed deviation in ñ A compared to the expected value suggests the sign and magnitude of adlayer birefringence. Fig. 17 Principle of OWLS sensitivity to adlayer structure and its exploitation on obtaining structural information. Two distinctive situations are shown where negative birefringence is produced by only polymer chains lying down and positive birefringence is produced by only end-grafted chains. The refractive index ellipsoids (left) demonstrates the relative magnitude of n A,o and n A,e in the case of the two structures. The drawings highlight both the adlayer structures and the orientational sensitivity of the TE and TM waveguide modes by indicating the electric field directions of the sensing modes. The data shown in the ñ A -M graph represents the formation of carboxymethyl dextran (CMD) layer grafted to OWLS chip surface using different surface chemistries as indicated in the legend. The expected RI value of CMD (ñ A = 1.45) given on the ñ A axis can be used as a reference point on the right birefringence axis. The direction of the experiment in time and the separate experimental phases are indicated by the arrows and numbers. While using aminosilane-modified surface at pH 7.0 resulted in mostly side-on grafting of the CMD chains, on epoxysilane-modified surface at pH 2.3 the end-on grafting dominated over a considerable part of the grafting. However, at the end of the experiment, the layer structure was found to be rather lying down. The data were adapted from Ref [30].

Complex structural and kinetic analysis with the combination of OWLS-QCM data
It has become a common strategy that an adlayer is examined both by an optical technique, such as SPR or OWLS, and by QCM (characterization of biomolecule adsorption and interactions [108,131,292] and living cell adhesion [293,294]) and the obtained data analyzed together. Error! Reference source not found. outlines the workflow that can be applied when QCM and OWLS data are both available. Such analysis is especially relevant in case of nanolayers with high water content and ordered structure. The combination of QCM J o u r n a l P r e -p r o o f (whence hydrated mass, viscoelasticity) and OWLS (whence dry mass, optical anisotropy) data enables complex kinetic-structural analysis revealing conformational and hydrationrelated rearrangements of (bio)polymer chains. Further details of such evaluation can be found in Saftics et al. [272]. It should be emphasized that this type of analysis requires the adlayer under investigation to be formed on the same type of surface; i.e., with the same surface chemistry on both the OWLS and QCM sensing surfaces.

Fig. 18
Workflow for complex data analysis to reveal the dynamics of structural changes in a thick, hydrated adlayer based on combination of QCM and OWLS measurement data. The figure directs through the key stages of data analysis starting with the measured kinetic data. The second stage demonstrates the model applied for evaluating the listed input data with providing the resulted important outputs which are then further analyzed. These analysis stages involve both simple data visualization strategies (combined analysis) and fitting of kinetic models (kinetic fit). The complex data post-processing and analysis emphasize that the quantities obtained in the first evaluation round offer deeper insight into the structure and dynamic behavior of hydrated nanolayers. The figure contains measurement data adapted from Ref. [272].

Introduction to software packages developed for label-free biosensor data analysis
We publish three software packages along with this paper: (ⅰ) NBS-OWLSanalysis and (ⅱ) NBS-QCManalysis that allows for the evaluation of QCM and OWLS measurement data, respectively; and (ⅲ) NBS-Kinetics, which provides a kinetic analysis platform for evaluating any biosensor kinetic data, especially for characterizing adsorption kinetics. All the programs have a graphical interface where the different analysis steps are separated in tabs. The programs can be used both to load the measurement data, prepare them for the specific analysis (baseline-and offset-corrections), perform the evaluation based on the corresponding optical, mechanical or binding kinetic model, and finally to visualize and save data. Fig. 19 presents an overview about the graphical interface of the three programs. Further details about the software package including summary of main features and hints of some specific J o u r n a l P r e -p r o o f functions can be found in the Supporting Information. Note: the software is under continuous development and improvements may be provided in later versions. In this study, we have presented an extensive review on label-free biosensor techniques capable of revealing the nanostructure and formation kinetics of nanolayers with especial relevance for biomedical research. Our principal intention was to demonstrate the variety of available label-free methods, whose ongoing development and expanding field enables one to study fast and complex interfacial biophysicochemical phenomena on the nanoscale with unprecedented throughput and sensitivity. The fundamentally different measurement principles provide broad insight, obtaining both the optical and mechanical characteristics of interfacial layers, determining their mass and thickness as well as deducing their nanostructure and dynamic alterations. Exceptional measurement data can be obtained if the combination of such methods can be established, making it possible to resolve the structure and formation kinetics of ultrathin layers in aqueous conditions. The combination of OWLS (or DPI) and QCM data is especially powerful as it allows for the simultaneous determination of chain orientation and hydration degree through detecting anisotropy as well as dry and hydrated mass. The kinetics of adsorption processes on the solid/liquid interface has great importance, e.g., in drug discovery or in the development of biofunctional and antifouling coatings for biosensors. With particular focus on hydrodynamic conditions and the mass transport limitation problem, we discussed the experimental and theoretical considerations that should be addressed when kinetic data are fitted. Starting from the PDE model, we devised the widely used twocompartment model in detail. To provide representative examples, we presented kinetic curves simulated with our developed kinetic analysis tool. Taking advantage from these simulations we showed how the kinetic curves depend on variations in parameters (kinetic rate constants, footprints, concentration, flow rate) involved in the model equations. We demonstrated that using the two-compartment model with modeling surface vicinal concentration differently, even at exceptional fit quality, small discrepancies in the fit may result in relatively large variations in the fitted data. As a significant addition to this review, we introduced our software packages developed for the evaluation of OWLS and QCM measurement data and for kinetic analysis, providing useful analysis tools for the scientific community. k a,num k a,resp k d k a1,resp , k a2,resp k d1,resp , k d2,resp

Software availability
The data analysis tools presented in this review are available upon request at the email address of robert.horvath‖at‖energia.mta.hu or r74horvath‖at‖gmail.com. The software packages will also be made available on the website of the Nanobiosensorics Laboratory (www.nanobiosensorics.com).