Power Density Titration of Reversible Photoisomerization of a Fluorescent Protein Chromophore in the Presence of Thermally Driven Barrier Crossing Shown by Quantitative Millisecond Serial Synchrotron X-ray Crystallography

We present millisecond quantitative serial X-ray crystallography at 1.7 Å resolution demonstrating precise optical control of reversible population transfer from Trans–Cis and Cis–Trans photoisomerization of a reversibly switchable fluorescent protein, rsKiiro. Quantitative results from the analysis of electron density differences, extrapolated structure factors, and occupancy refinements are shown to correspond to optical measurements of photoinduced population transfer and have sensitivity to a few percent in concentration differences. Millisecond time-resolved concentration differences are precisely and reversibly controlled through intense continuous wave laser illuminations at 405 and 473 nm for the Trans-to-Cis and Cis-to-Trans reactions, respectively, while the X-ray crystallographic measurement and laser illumination of the metastable Trans chromophore conformation causes partial thermally driven reconversion across a 91.5 kJ/mol thermal barrier from which a temperature jump between 112 and 128 K is extracted.


S1. ESTIMATION OF ADIABATIC X-RAY INDUCED HEATING IN FIXED TARGET TR-SSX
Where is the maximum dosage in Gy calculated using RADDOSE3D 1,2 .
The input file used, using the values is

Beam
Type Gaussian FWHM 30 10 Collimation Rectangular 15 10 FLUX 1.9e12 ENERGY 12.7 The heat capacity of rsKiiro crystals is unknown.Estimates can be taken from experimental measurements of pure protein 3 and for lysozyme crystalsreported for varying degrees of hydration 14 At 300K temperature for pure protein C P = 5 x 10 2 J K -1 kg -1 , which would give rise to ΔT=124K.The measured C P values for lysozyme crystals, taken as a reference, very strongly depend on the hydration level, as shown by Miyazaki et al. 5 .At 300K, for 'wet' lysozyme crystals containing 45.7wt% Water C P was measured at 73.6 x 10 3 J K -1 mol -1 = 4.91 x 10 3 J K -1 kg -1 .For this value of C P , ΔT=13K At 300K, for 'dry' lysozyme crystals containing 7.4wt% Water C P was measured at 23.8 x 10 3 J K -1 mol -1 = 1.59 x 10 3 J K -1 kg -1 .For this value of C P , ΔT=39K.
In the presence of cooling, an estimate of a characteristic cooling time has been described 4 .For small particles submerged in liquids the characteristic cooling time τ c : Where c ps is the sample specific heat (J K -1 kg -1 ), ρ s the density (kg/m 3 ), L is the characteristic sample size (m), u is the velocity of the cooling medium, ν is the kinematic viscosity (10 -6 m 2 s -1 ) and κ s is the thermal conductance (W m -1 K -1 ) given by the Biot number,Bi:  = ℎ   Where h is the heat transport coefficient per unit area (J K -1 m -2 s -1 ).At 300K, Kriminski et al provide a value of κ s =0.61 W m -1 K -1. for liquid water and ν=η/ρ=1,002 x 10 -6 m 2 s -1 .The heat transport per unit area h (J K -1 m -2 s -1 ) where δt is the average thickness of the laminar thermal boundary layer (m) In the absence of flow of the coolant, the heat transfer is best approximated from heat convection using Newton's heat transfer law.An overall heat-transfer coefficient U is defined for two media from the additivity of thermal resistances Experimental measurement of thermal conductivity of lysozyme crystals were reported by Fujiwara et al. 6 with a value of 0.41 W m -1 K -1 .The effective heat-transfer coefficient U is calculated as U = 0.25 W m -1 K -1 .An estimate of the characteristic cooling time is made using this value   ≅

𝑈
The calculated values for the characteristic cooling time again depend strongly on the heat capacity of the crystal.For pure protein (C P = 5 x 10 2 J K -1 kg -1 ) τ=4.8 ms, for 'wet' lysozyme crystals containing 45.7wt% Water(4.91 x 10 3 J K -1 kg -1 .)τ=47 ms and for 'dry' lysozyme crystals containing 7.4wt% Water (23.8 x 10 3 J K -1 kg -1 ) τ=12 ms.The above example illustrates the calculation of potential heating profiles where crystals are embedded in stationary water reservoir with a characteristic dimension δ> 3L, in which case the final temperature is within less than 1 % of the reservoir initial temperature.However, for less surrounding water, the maximal temperature jump will be higher if the final temperature is consequently increased.

S2. LASER HEATING
The measurements involved the use of two cw laser illuminations.The initial pre-conversion using the 473nm cw laser generates the metastable Trans state.Given that the characteristic cooling time is in the millisecond range, the rsKiiro crystals in the Trans state are assumed to be fully thermally equilibrated.The Power density titration series for the 405nm cw laser flash with 2ms duration, starts at a zero density.The crystallographic determination of remaining Cis state (Fig 2a, Fig 4, data point a) does not include the 405nm illumination and is used for the calculation of temperature jump caused by X-ray heating only (section S4.3).With subsequent increased power density up to 14.43 mJ/mm2 density and 2 ms duration, a saturating concentration of reversibly formed Cis state is measured (Figure 2, 4).An estimate of the laser heating can be made.rsKiiro is not a static absorber, and photochromic bleaching of the electronic absorption band at 400nm proceeds with a 18% quantum yield 7 .Taking into account an additional 5% fluorescence quantum yield, the maximum absorbed dose (J/kg) can be estimated: h   Where N A is Avogadros number (mol -1 ), ρ cryst is the density of the crystal (kg/l), the QY values for photoisomerization and fluorescence are given as fractions.[rsKiiro] in crystals is 48 mM This estimate provides an upper value of the absorbed dose D405nm-laser = 6.31 x 10 4 J kg -1 .This values is directly comparable to the calculated absorbed x-ray dose.
The estimate of the optical laser induced heating follows the same considerations as the x-ray heating above in section S1.This value of 6.31 x 10 4 J kg -1 is however applied to condition labelled f in figure 4d.In the titration series, from 0 to 14.4 mJ/mm 2 the absorbed laser dose will follow the concentration profile of the product, hence will follow the exponential growth profile as shown.For instance, condition e (6.74 mJ/mm 2 ) shows between 40% and 60% population increase relative to condition a, hence an absorbed dose of about half the absorbed x-ray dose is estimated to contribute to the temperature jump in that condition.As a result, a small deviation from a first-order behaviour is expected, but cannot be resolved with the obtained accuracy of the Cis population presented in Figure 4.

S3. ESTIMATED RSKIIRO PRECONVERSION EFFICIENCY
To calculate the preconversion efficiency, the relative rate of switching in rsKiiro at lower power and optical densities was scaled to the laser and crystal optical density used in the TR-SSX experiment.rsKiiro absorbs most strongly at 485 nm with a calculated absorption coefficient of 98,727 M -1 cm -1 .At the flash wavelength of 405 nm the absorption coefficient is calculated as 31,184 M -1 cm -1 .Steady state measurements at a power density of 0.0315 W/cm 2 (at 489 nm) gave a switching rate of 0.32 s -1 in rsKiiro at 298 K and pH 8.4.The power density of the 473 nm laser on the chip is 0.30 W/cm 2 which scales the rate constant to 2.50 s -1 when accounting for change in absorption between 473 nm and 488 nm (a ratio of 0.82:1).Over the one second illumination period this should convert 92% of the protein to the trans state.The average waiting time for a crystal once preconverted was 45 s during which time an estimated 4.3% of the population will thermally recover back to the cis state on average.The thermal recovery rate constant at room temperature was previously reported as 1.0x10 -3 s -1 7 (Data in Fig S2) This means the minimum expected cis population before X-ray probing of the crystal is calculated to be 12.5%.Because the calculated temperature jump is sensitive to the value used for the concentration of the trans state, we consider the possibility of experimental and fitting uncertainty of the laser profile and power density.This is estimated by assuming a difference in profile that could be the conversion of FWHM to the 1/e 2 profile.This would give an estimate of 0.22 W/cm 2 as a possible lower bound value for the power density.Consequently, a conversion rate constant of 2.2 s -1 results in the calculation of a minimum of 16.6% Cis population in addition to 4.3% thermal conversion, which equals 20.9%.Taking this value as an upper limit by assuming errors in the power density determination, the temperature jump calculations are done assuming a value of 12.5 +/-8.4 % for the Cis population.This uncertainty is propagated for the calculation of the temperature jump, section S4.3 below.

S4.1. Single Flash Yield and Error Propagation
The population change (N) due to the single 405 nm flash was calculated from the photodiode signal using the Beer-Lambert law: Where [Cis] is the concentration of the cis conformer and [CisTotal] the fully converted concentration (i.e. total sample concentration), I 0 is the initial intensity (or full trans), I [Cis] is the population due to the 405 nm flash and I [CisTotal] is the fully converted signal level of the cis state, as shown in figure S3.The standard error in the conversion, N was calculated as:

Where
, and are the standard error in the respective values, calculated as: where is the standard deviation in value X over n observations.

𝜎(𝑋)
The energy density in mJ/mm 2 was calculated as:  =  max × 10 - Where E max is the energy density with ND = 0 where ND is the optical density of the neutral density filter being used.The standard error in energy density, E can be calculated as: Figure S3: Plot of a typical flash photolysis trace.Showing the illumination scheme (bottom) and which points in the trace correspond to the values used to calculation the single flash yield.I Bck was the no light background value subtracted from all point to correct for the offset on the trace.

S4.2. Flash Photolysis calculation
To fit the flash-photolysis data a model was derived based on first order reaction kinetics.On millisecond times scales the cis-trans isomerization can be considered a single-step process:

Cis
The switching rate constant depends on the photon flux q, the proportion of absorbed photons χ in a sample of absorbance, A, the quantum yield of Trans/Cis switching,Φ, and the thermal rate, k TH : For the case where OD « 1 a Taylor expansion can be used to approximate the absorbed fraction as: 1 -10 - ≈  ln(10)

And 𝜒≅ln (10)𝜖[𝑇𝑟𝑎𝑛𝑠]𝑙
Where ε is the molar extinction coefficient (M -1 cm -1 ) and l the path length (cm) Using this approximation and the Beer-Lambert law, the switching rate can be rewritten as: Where k 1 is a first order rate constant The rate of thermal recovery of the trans state during the flash-photolysis measurement is much smaller than the rate of photo-induced switching, so can be neglected in the 2ms flash duration.For a Fluence (mJ mm -2 ) delivered in the 2ms flash to drive the Trans-to-cis reaction Where k is a fluence-based first-order rate constant in the units of mm 2 /mJ 8 .

𝑘 = ln (10)𝜙𝜀 10
The data were fitted (Figure S4) using the orthogonal distance regression package in scipy. 57This minimizes residuals by accounting for errors in both the dependent and independent variables when fitting, which was appropriate due to the significant errors in both the energy density and the measured intensity.

S4.3. X-Ray Crystallography Photolysis Modelling
The crystallography rate measurements can be considered a three step process:

Cis + Trans
Preconvert Cis + Trans Flash Cis + Trans X -ray Probe

Cis + Trans
As mentioned in the section S3 above, the initial preconversion step and thermal recovery before measurement should lead to an initial cis population of 12.5% which was constant through the experiment.
The temperature jump ΔT is calculated from the Arrhenius equation, using E a =91 +/-5 KJ mol -1 and k 1 =4.08 x 10 -4 s -1 at a temperature of 293 K.
Where k 2 is calculated from the concentration [Cis] measured at 0 mJ/mm 2 power density (Figure 4) that provides the converted fraction X The RMS error for the Ea dominates the error on the calculation of ΔT, more strongly than the errors in the determination of the cis concentration from spectroscopy (Section S3) and the uncertainly of the crystallographic quantification.The uncertainty of +/-0.05 fraction of the [Cis] determined from the crystallographic methods at 0mJ/mm 2 adds to the uncertainty typically between 2 and 5 degrees K.The resulting average and error is reported in Table 1 in the main manuscript.
To illustrate, the uncertainties in the values for Ea, the Cis concentration at zero power density for the 405nm laser from spectroscopy, and the uncertainty in the crystallographic quantification methods propagate as follows, as shown in tables S1,S2 and S3 below.S3.Calculated temperature jump values for the maximal estimated value of 0.209 determined for the Cis concentration from spectroscopy, for the mean (91.5 KJ/mol), minimum (81.5 KJ/mol) and maximum (101.5 KJ/mol) values for the activation energy Ea.

S5. KNIFE EDGE LASER FOCAL SPOT SIZE DETERMINATION
To ensure accurate determination of power density of the 405nm laser, both in the home lab and at the synchrotron beamline, knife edge scans were performed to ascertain the spot sizes.Traces were fitted as described in 9 using SciPy in python to extract the FWHM of the focus profile.
Angstroms cubed.The temperature change ΔT as a result of deposition of this absorbed dose in the absence of cooling depends on the heat capacity Cp Δ =

Fig S1 .
Fig S1.Illustration of temperature jump temporal profile assuming an effective heating time constant of τ= 5 ms and characteristic cooling times of 4.8 ms (blue), 12 ms (green) and 47 ms (red).The calculation assumes an infinite Volume for surrounding water using convection heat transfer.The temperature jump is plotted as a fraction of ΔTmax, which is the value in the absence of cooling ΔT=D/cp

Figure S2 .
Figure S2.Kinetic trace of the absorption at 489nm measuring the thermal recovery rate constant for the Trans-to-Cis conversion at 298 K and pH 8.4, yielding a value of 1.0x10 -3 s -1 .

Figure S4 :
Figure S4: Population change plotted against energy density obtained from optical flash photolysis data collection.
01 s time point that measures [Cis].

Figure S5 :
Figure S5: Knife edge scans of the laser focus at the home lab (left) and P14.2 (T-REXX) at PETRA III (right).

S7.
SSX ENERGY TITRATION OCCUPANCY DETERMINATION BY MINIMISATION OF R FREE AND R WORK .

Figure S7 :
Figure S7: Plots of R Free (purple) and R Work (green) against the cis (1-trans) occupancy level for all illumination powers.

Table S2 .
Calculated temperature jump values for the minimal estimated value of 0.041 determined for the Cis concentration from spectroscopy, for the mean (91.5 KJ/mol), minimum (81.5 KJ/mol) and maximum (101.5 KJ/mol) values for the activation energy Ea.