Investigating binding particles distribution effects on polymer translocation through nanopore
Introduction
Biopolymer translocation through nanopore is a ubiquitous process in both biology and biotechnology. Translocation of proteins through endoplasmic reticulum or organelles like mitochondria and RNA, through nanoporous membranes is of vital importance in cell metabolism [4], [24], [28]. The process is also important and noteworthy in drug delivery and gene therapy [22], [16], [20]. Moreover the application of biopolymer translocation in rapid and cheap polynucleic sequencing attracts many researchers in the field [25], [9], [11], [13], [10], [17]. After seminal experimental work of Kasianowicz et al. [16] on ssRNA translocation through α-hemolysin channel embedded in a bilayer, there has been flurry of theoretical works, experiments and simulations in the field of polymer translocation [22], [27], [33], [26].
Because of the great complexity in the process there isn't, yet, any universal consensus on the polymer translocation [26]. Experimentalists, in vitro, use strong electric field to bias translocation of the negatively charged biopolymers. In contrast there is not such a strong electric field in vivo and cell should find other mechanisms for its biopolymer translocation [34]. Brownian ratchet mechanism as a theoretical model has been introduced by Simon et al. in 1992 [31]. In this model a binding protein which binds to the polymer in the trans side will bias the translocation. After experimental work in the ratchet mechanism [21] Liebermeister et al. fit the data to estimate the transition probability per unit time in master equation approach [18]. Later Zandi et al. using a 2 dimensional Brownian molecular dynamic simulation on a stiff polymer calculate both the mean force and the mean passage time of the polymer in the presence of binding chaperones [38]. Afterwards in a series of work Ambjörnsson et al. calculate theoretically, from a master equation approach, the force and mean first passage time of an stiff polymer translocated through a nanopore in the presence of binding proteins [6], [5], [7].
Abdolvahab et al. then take the sequence into account for chaperone driving translocation and using a dynamic Monte Carlo simulation show how one can get information about the sequence by measuring the first and the second moments of the translocation times and discuss about their sequence dependency [3], [2]. They later approximate the whole probability density function for different sequences using a continuous advection–diffusion equation and discuss the importance of the Péclet number [1], [12]. The two dimensional flexible polymer is simulated using Langevin dynamics by Yu et al. [36]. They show that the translocation velocity has a maximum as a function of the chaperone concentration. Hereafter using 2 dimensional Langevin dynamics simulations they considered polymer's rigidity and its effect on translocation time [37]. They show that although the translocation time is an increasing monotonic function of the rigidity due to an increase in the radius of gyration and decreasing of the center of mass velocity in moderate chaperones concentrations, the dependency becomes more complex in the extremely low concentration.
Although many works has been done in chaperone assisted translocation and people study different factors in the field, to the best of my knowledge there is not any work on chaperones spatial distribution effects. Here we consider different spatial distributions of the binding proteins and how it will changed the translocation time of the polymer. Binding proteins may have different spatial distributions. As solving the Poisson–Boltzmann equation shows, exponential distribution is an important answer in the case of counterions [30], [15]. Here for binding proteins, specially Hsp70 [21], it may be different procedures leading to exponential distribution of chaperones in vicinity of the membrane. The cell can pinpoint protein synthesis, or transport them [8], [35], physically the membrane affinity could also be helpful. In the next section we will introduce our model and theory. The simulation is described in section 3. In the section 4 we will explain the sequence and how to consider and measure it. After addressing the different chaperones distributions in section 5, a theoretical estimation of how the translocation velocity should depend on spatial distribution parameter will be presented in section 6. Simulation results are depicted in section 7 and finally we will draw our conclusions in section 8.
Section snippets
Model and theory
As previous works [3], [2], [1], [5], [6], [7], in what follows we considered polymer as a one-dimensional entity. It provides both simplicity and allows us to focus on some special properties, here chaperone distributions, in more details and more clearly. Thus we have a stiff polymer with length L constructed of M monomers, each of length σ in which . In our model wall has not any width and we ignore the complexity of the interaction of wall and polymers [19]. Although its effect is
Simulation
Monte Carlo method is used for the simulation. We used Metropolis algorithm for constructing distribution which fulfill detailed balanced condition. Polymer can diffuse to the right and left with the same probably in the absence of chaperones. As there isn't any chaperone in the left side the probability of the polymer going to the right doesn't changed. But for going to the left there must not be any chaperone bound to the polymer near the wall. Chaperones will bind/unbind to/from the polymer
Sequence and adjacency probability
The polymer constructed of two different monomers, say, A and B. Their differences lay in the different binding energies between chaperones and the monomers. In our case the for monomers A and B, are 2 and −2 respectively. It means that chaperones don't like to bind to the monomer A. Pay attention that the not only depends on the binding energy but also on the local concentration of the chaperones (cf. Eq. (2)). These two different EBEs have been considered to clarify sequence
Chaperones spatial distribution
Binding particles, chaperones, may have different spatial distributions. In addition to uniform distribution in which considered before [3], [2], [1], [5], [6] we investigate exponential and step distribution functions and its results on the polymer translocation time distribution.
Exponential distribution: For including spatial distribution in the simulation we changed the chaperone's effective binding energy from to , where α is the exponential rate1
Theoretical estimation
Let's estimate what would be happen in this model with an exponentially distributed chaperones. For simplicity suppose that we have a polymer with length and with chaperone size . Because of the exponentially decreasing of the chaperone numbers the probability of chaperone bindings will become less and less as we go away from the cell membrane. Thus we can truncate the partition function of the system at some point. Bellow we write the partition function which ignore the chaperone
Simulation results
In this section we compare our results for different chaperones spatial distributions, introduced in section 5. There could be different mechanisms leading to exponential distribution of chaperones near the membrane. The cell can localize protein synthesis, or transport the chaperones [8], [35], physically also the membrane affinity could help.
The simulation results show that although by increasing the adjacency probability, , the translocation time of the polymer will be decreased in usual
Conclusions
Based on a dynamical Monte Carlo model we obtained the polymer translocation time probability density function for different chaperones spatial distributions. Simulation results show that the chaperones spatial distribution must be considered in calculating the translocation time of polymer driven by binding proteins. Ignoring this factor may lead to incorrect results even for the translocation regime. By increasing the exponential rate of the chaperone spatial distributions, the trend of
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