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We have also examined the fluctuation of [MATH] at equilibrium, as depicted in Figure 6(c) . At [MATH] , the histogram of Spatiocyte matches the distribution curves of eGFRD and the well-mixed model (Gillespie method |
). At much reduced diffusion coefficient ( [MATH] ) however, both Spatiocyte and eGFRD shared similar distributions, with the width becoming narrower and the mean value shifting to the left. With the modified propensity function, RDMEm also exhibited similar distribution. The narrow width and the shifted mean are consi... |
It was reported that MLM would not be able to solve the first-order production-degradation reaction [MATH] accurately because of its spatial discretization scheme |
. When the number of total voxels in the compartment, [MATH] is less than [MATH] , the equilibrium concentration deviates from the well-mixed model. This deviation however, is a direct consequence of the volume exclusion property of MLM. Since each voxel can only occupy a single molecule, there would be an insufficient... |
. Just as in the cellular compartment, no more molecules can be added into the system when the number of generated molecules exceeds available free space. Moreover, since only about 34 % of the cell volume is occupied by macromolecules |
, it is also an unlikely scenario to fully occupy the voxels of HCP lattice with macromolecules. With the multi-algorithm implementation of Spatiocyte |
, we can use the Gillespie’s Next-Reaction method to simulate small molecules that are in large abundance and are homogeneously distributed. In this case, the equilibrium result is independent of spatial discretization since the method assumes the well-mixed condition. |
Dual phosphorylation-dephosphorylation cycle In mean-field models, the spatio-temporal correlation of microscopic rebinding events is not resolved explicitly because the correlation usually does not cause a significant impact on the dynamics at the macroscopic scale. One case where the correlation does influence the ma... |
, shown in Figure 7(a) . The substrate MAPK (K in Figure 7(a) ) is phosphorylated in a two-step process by the MAPK kinase (KK) and dephosphorylated by a phosphatase P. The phosphorylation and dephosphorylation processes proceed according to the Michaelis-Menten kinetics and exhibit distributive property |
, wherein the enzymes must unbind from the substrate before they can rebind and modify the second site. Upon phosphorylation or dephosphorylation, the respective enzymes are inactivated (denoted as KK* and P*), and reactivated (KK or P) after some time [MATH] . When the reactivation time is short and the enzyme-substra... |
using eGFRD. Processive behavior caused by rebindings of the same enzyme results in higher overall phosphorylation rate than the distributive case where the dissociated molecules can escape rebinding |
. Such microscopic spatio-temporal correlation has been shown to change the response sensitivity of the phosphorylation state, which can cause the subsequent removal of ultra-sensitivity or bi-stability in the system |
Rebinding events taking place within very short time scales are difficult to be captured by RDME because of the fine spatial resolution required. To test whether MLM can resolve such events faithfully, we use Spatiocyte to model the dual phosphorylation cycle with the same parameters from |
. Distributive and processive models are represented by Eqs. 1-5 of , and were solved using ODE solver. Figure 7(b) displays the steady-state response curves of Spatiocyte and reference theoretical models. Note that since the reactivation time [MATH] is equal to or less than the diffusion time step [MATH] (given in Fig... |
The parameter ranges examined so far have a stable steady-state as demonstrated by the response curves in Figure 7(b) . When the total concentration of the substrate is increased five-fold, the mean-field theory generates hysteresis, shown by the dotted and dash-dotted lines. The dotted line represents the response whe... |
As a side remark, in the original Spatiocyte scheme , the voxel adopts the size of the diffusing molecules. However, as we found in the Methods section, the voxel needs to be about 2% larger than the molecule size ( 27 ) for the total rebinding probability and the effective rate constant to be exactly the same as in th... |
Effects of excluded volume on bimolecular reaction Excluded volume in the cell arising from crowded obstacles such as macromolecules, Golgi apparatus or cytoskeletal elements can cause anomalous diffusion of reacting molecules |
. Anomalous diffusion has been shown to generate non-classical reaction kinetics on 2D and 3D lattices . Here, we use MLM on HCP lattice to examine the effects of volume exclusion on the bimolecular reaction [MATH] in the presence of uniformly distributed immobile obstacles. [MATH] and [MATH] have the radius [MATH] and... |
We first consider the effects of immobile obstacles on diffusing molecules. We calculate the time-dependent diffusion coefficient from the mean-squared displacement of simulated particle trajectories. The time-dependent diffusion coefficient in Figure 8(a) indicates that the diffusion is anomalous at short times and no... |
[EQUATION] where [MATH] is the percolation threshold for HCP lattice. We confirmed that the long-time diffusion coefficients obtained for [MATH] in Figure 8(a) (dashed lines) are consistent with [MATH] in Eq. ( 37 ). |
Figure 8(b) shows that the survival probability of [MATH] decays slower when the volume occupancy, [MATH] is increased. From the survival probability, we can calculate the rate coefficient according to Eq. ( 33 ) to obtain the kinetics. We replaced the constant concentration term [MATH] 33 ) with the time varying term ... |
As the volume occupancy approaches the percolation threshold (Figure 8(c) [MATH] ), the kinetics begins to deviate from the Collins-Kimball theory. The deviation is strongest at [MATH] , which is beyond the percolation threshold. Note that at lower volume occupany ( [MATH] ), the anomalous to normal diffusion crossover... |
Grima and Schnell have shown that reaction kinetics, either classical or non-classical, is not determined by the heterogeneity of the accessible space but rather by the reaction probability and the initial condition. In the Smoluchowski and Collins-Kimball framework, reaction follows classical kinetics when it is activ... |
. The Zip-Mandelbrot equation is valid for long-time kinetics whereas the Collins-Kimball rate ( ) describes the kinetics for all time ranges. |
Here, we have studied the kinetics of bimolecular reaction in the presence of immobile obstacles with MLM. When the total volume occupied by obstacles is much smaller than the percolation threshold and the observation time scale is longer than the anomalous to normal diffusion crossover time, the kinetics is still repr... |
or by extending the Smoluchowski and Collins-Kimball framework using a generalized diffusion equation missingum@section Conclusions |
In contrast to macroscopic and mesoscopic approaches, particle-based methods have the advantage to directly link microscopic parameters to the observed RD behavior, thus providing insights about the underlying mechanisms of the system. MLM shares this same advantage, but with reduced computational costs owing to its fi... |
but also track individual molecules on large eukaryotic cells and simulate membrane protein clustering in whole red blood cells Recently, Grima and colleagues developed a method called vRDME that incorporates volume exclusion into RDME |
. The method can approximate the continuum model very well by matching the steady-state rate constants of both models. We note that vRDME is a type of MLM since each voxel can occupy a molecule and bimolecular reactions occur by colliding reactants. In contrast to vRDME, our work here employs random walk theory and par... |
Contrary to the original assumption of Spatiocyte , the voxel should be larger than the molecule size (by about [MATH] for HCP lattice) to be quantitatively accurate. Numerical simulations showed that both the effective rate constant and the asymptotic time-dependent behavior have good agreements with the Collins-Kimba... |
Despite achieving the same total rebinding probability as the Collins-Kimball theory, the time-dependent behavior of MLM at time scales shorter than [MATH] is different than that theory (Figure ). One potential solution to obtaining the same behavior at such fine time scales is to make the voxel size smaller than the m... |
. This would reduce [MATH] but increase the cost of computation significantly because of the finer time steps and the higher number of collision checks required. |
MLM captures the effects of excluded volume naturally but comparing on-lattice behavior with continuum is not straightforward since the influence of volume exclusion and the resulting reaction kinetics vary according to the lattice arrangement |
. Moreover, since all diffusing species in this work have the same molecule size, it is not possible to replicate the effects of relative size of interacting molecules. To minimize such lattice artifacts and to better approximate off-lattice volume exclusion, we can improve the size representation of each molecule on l... |
Realistic simulation of intracellular reaction-diffusion processes should also incorporate the influence of inter-molecular potentials such as van der Waals and hydrodynamic forces. By employing contact interactions on lattice as proposed by Fernando et al. |
or the SVTA approach with interaction potentials , it may be possible to incorporate the above forces in MLM. The theoretical framework presented in this work serves as a building block for further development and integration of MLM-based algorithms. |
missingum@section Author Contributions W.-X.C., K.K., K.T. and S.N.V.A. designed research; W.-X.C. performed research; W.-X.C., K.K., M.W., S.V.M., and S.N.V.A. analyzed data; and W.-X.C., K.K. and S.N.V.A wrote the manuscript. All authors read and commented on the manuscript. |
missingum@section Acknowledgements We thank Kozo Nishida for technical advice and support and Kylius Wilkins for critical reading of the manuscript. We thank Ramon Grima for providing the Matlab source of fast Brownian dynamics (point particle version) and Steven Andrews for his help with Smoldyn models used in the per... |
Appendix A Rebinding probability distribution The rebinding probability distribution is defined as (Eq. 3.10 in and Eq. S27 in [EQUATION] |
where [MATH] is the Green’s function in the diffusion equation: [EQUATION] subjected to initial condition [EQUATION] and boundary conditions such that |
[EQUATION] and [EQUATION] The latter condition is known as the radiation boundary condition. The Green’s function [MATH] has been solved in (p. 368 in |
) to be [EQUATION] where [MATH] For [MATH] , we thus have [EQUATION] Finally by substituting Eq. ( A.7 ) into Eq. ( A.1 ), we obtain the probability distribution |
[EQUATION] where [MATH] Appendix B Lattice initial rate Here we provide the derivation of the lattice initial rate, which was done previously in |
. Given two reacting species [MATH] and [MATH] , in which [MATH] are stationary and [MATH] are diffusing. The initial rate constant at time step [MATH] , can be estimated using the rate equation as |
[EQUATION] where [MATH] denotes the number of molecules of species [MATH] [MATH] denotes the change in [MATH] and [MATH] is the compartment volume. The number of successful reactions in a single step [MATH] can be crudely estimated as [MATH] where [MATH] is the average number of encounter, [MATH] is the total number of... |
For the activation-limited scheme, where [MATH] and [MATH] The initial reaction rate is then given by [EQUATION] Note that [MATH] is the sum of diffusion coefficients of the reacting pair, [MATH] |
Similary, for the diffusion-influenced scheme, where [MATH] and [MATH] , we have [EQUATION] Also note that the physical dimension of [MATH] satisfies [MATH] |
The above derivation for HCP lattice can be generalized to other lattice arrangements: [EQUATION] where [MATH] is the packing density of the lattice (e.g. [MATH] for the simple cubic lattice). |
Appendix C Voxel size As shown in main text, in order to match the MLM with the continuum-based model, the voxel size of HCP lattice has to be chosen such that |
[EQUATION] where [MATH] is the molecule size and [MATH] (p. 153 in ) is the total return probability on HCP lattice. More generally, the voxel length of any regular lattice arrangement follows that |
[EQUATION] For example, for the simple cubic lattice we have the voxel length: [EQUATION] about [MATH] larger than the molecule size ( [MATH] for the simple cubic lattice is given in p. 153 of |
). Appendix D First-passage time distribution on HCP lattice For [MATH] , we define [MATH] as the voxel occupation probability from [MATH] to [MATH] , that is, the probability of being at voxel [MATH] after [MATH] steps, given that the walk started at voxel [MATH] [MATH] as the first-passage time distribution from [MAT... |
The probability generating function of [MATH] and [MATH] is related through (Eq. I.18 in [EQUATION] where [MATH] is the lattice Green’s function for the face-centered cubic (FCC) lattice as defined in Eqs.(2.6)-(2.9) of |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] wherein [MATH] is the complete elliptic integral of the first kind. For the convenience of calculation, the voxel occupation probability is given as |
[EQUATION] where [EQUATION] The first-passage time distribution is related to the voxel occupation probability recursively via: [EQUATION] |
Activation-limited case ( [MATH] [MATH] For [MATH] , the rebinding-time probability distribution [MATH] is equivalent to the first-passage time distribution [MATH] as mentioned in the main text. |
Whereas for [MATH] , the rebinding-time probability distribution is given by [EQUATION] wherein [MATH] is the probability of reaching the origin for the [MATH] th time at [MATH] th step (I.1.9 in |
): [EQUATION] with [MATH] With Eq. ( D.10 ) we can obtain [MATH] recursively via [EQUATION] The generating function of [MATH] is related to the generating function of [MATH] |
[EQUATION] where in the last step we have [MATH] since for all [MATH] such that [MATH] , the return probability is zero. Using (Eq. I.20 in |
): [EQUATION] in Eq. ( D.12 ) we then have: [EQUATION] Finally the total rebinding probability of an in-contact pair on lattice is obtained by taking the limit [MATH] |
[EQUATION] where [MATH] (p. 153 in ) is the return probability on HCP lattice. Rebinding probability at long times The asymptotic behavior of the rebinding-time probability distribution [MATH] at large [MATH] can be estimated directly from the generating function. First we expand the generating function of the return p... |
and Eq. A.237 in [EQUATION] where [MATH] and [MATH] The corresponding expansion of the generating function of [MATH] is then [EQUATION] |
where we have ignored the term equal to or higher than [MATH] Recall that the generating function of the rebinding-time probability distribution for the activation-limited case: |
[EQUATION] By the expansion of the denominator we have [EQUATION] Substituting Eq. ( D.17 ) into [MATH] and collecting the leading terms gives |
[EQUATION] where [EQUATION] By means of singularity analysis of the generating function (see Eq. 2.3 of ), the corresponding asymptotic behavior of [MATH] as [MATH] is therefore |
[EQUATION] Rate coefficient at long times From the definition of rate coefficient on lattice using the particle-pair formalism, we have the [MATH] -step reaction rate coefficient: |
[EQUATION] which can be rewritten as [EQUATION] The first summation term is the total rebinding probability while the second term can be evaluated using the Euler-Maclaurin formula: |
[EQUATION] where we have used the definition [MATH] in the last step. Now we have the asymptotic reaction rate as [EQUATION] After rearrangement we have |
[EQUATION] Using the definition [MATH] , and applying the expressions for reaction acceptance probability in Eq. ( B.2 ) and voxel size in Eq. ( C.1 ), we obtain the long-time approximation as |
[EQUATION] which has the exact same form as the continuum case. Diffusion-influenced case ( [MATH] [MATH] ). The derivation of the effective rate coefficient in the diffusion-influenced case differs from the activation-limited case due to the difference in the simulation scheme (see Algorithm 1 in main text), namely in... |
The purpose of this section is to derive the long-time asymptotic behavior of the rate coefficient, which is independent of the transient time-dependent behavior. Hence, we parameterize the rebinding-time according to the eventful step [MATH] (which will be incremented after a physical movement or a reaction attempt), ... |
As shown in the main text, the rebinding-time probability distribution [MATH] is defined as [EQUATION] where [MATH] is the reaction probability and [MATH] is the in-contact probability of a reactive pair after [MATH] steps. |
The reaction probability is defined as [EQUATION] where the nominator term accounts for the probability of hopping to [MATH] from [MATH] and successfully reacting with the reactant located at [MATH] in one diffusion step, while the denominator term comes from the infinite sum representing the total probability of unsuc... |
Next, we derive the generating functions of two first-passage time distributions [MATH] and [MATH] that correspond to the current scheme. We start from |
[EQUATION] where the first term on the right-hand side relates to the failed reaction attempt [MATH] , the second term describes the hop from [MATH] , and the last term accounts for the trajectory [MATH] , which is continued by a series of [MATH] steps that have ended up in [MATH] again. |
From Eq. ( D.31 ), we obtain the generating function of [MATH] as [EQUATION] Thus we obtain [EQUATION] where [EQUATION] is given in terms of the generating function of [MATH] |
(the detailed derivation of Eq. ( D.34 ) is given in ). Now, we define the probability that a particle is in-contact after [MATH] -step as: |
[EQUATION] where the first term accounts for the trajectories [MATH] and [MATH] , the second term represents the trajectories [MATH] and the last term accounts for the initial condition. In detail, the coefficient |
[EQUATION] accounts for the total probability of arrival at [MATH] from a rejected reaction attempt (first sub-term) or from the adjacent neighbor [MATH] (second sub-term) given that there was no successful escape to [MATH] at the last simulation step [MATH] before the arrival, |
while the coefficient [EQUATION] accounts for the total probability of arriving at [MATH] from [MATH] given that there was no successful escape to [MATH] at the last simulation step [MATH] before the arrival, |
and finally [MATH] denotes the first-passage time distribution of the scheme with step-acceptance probability [MATH] (proof given in ). |
We then multiply Eq. ( D.35 ) with [MATH] [EQUATION] and take the sum to infinity to obtain [EQUATION] After collecting the terms, we obtain the generating function of [MATH] |
[EQUATION] Substituting Eq. ( D.30 ) and Eq. ( D.40 ) into Eq. ( D.29 ) then gives the rebinding-time probability distribution: [EQUATION] |
with the corresponding probability generating function [EQUATION] In the diffusion-influenced scheme of Spatiocyte, we have [MATH] [MATH] [MATH] and [MATH] . Using these parameters we then have the following quantities: |
[EQUATION] [EQUATION] [EQUATION] [EQUATION] where we have used definition Eq. ( D.58 ) in Eq. ( D.46 ). Using Eq. ( D.46 ), we obtain the limit of Eq. ( D.40 ) as: |
[EQUATION] Finally, we substitute Eq. ( D.47 ) into Eq. ( D.42 ) to obtain [EQUATION] Therefore, we have the total rebinding probability as: |
[EQUATION] Return probability [MATH] We denote [MATH] as the voxel occupation transition probability from [MATH] to [MATH] . It is related to [MATH] via the convolution relation ( |
, p. 121) [EQUATION] If a random walker started at [MATH] , it must go through [MATH] before reaching the destination voxel [MATH] . Then we have |
[EQUATION] Note that [MATH] Thus, with [MATH] , we have [EQUATION] Multiplying both sides with [MATH] gives [EQUATION] Then taking the sum of both sides from [MATH] to infinity gives |
[EQUATION] where [EQUATION] As such, we have [EQUATION] The total return probability to [MATH] from [MATH] is then [EQUATION] Using definition Eq. ( D.1 ) and [MATH] , finally we have |
[EQUATION] Return probability [MATH] If we increment the step count [MATH] for every successful step to a new voxel, then the first-passage time distribution from [MATH] to [MATH] at step [MATH] is given by |
[EQUATION] where [MATH] ,is the step acceptance probability. It can be shown that [EQUATION] Continuous time limit of the diffusion-influenced scheme |
In the diffusion-influenced scenario, Spatiocyte uses a different approach for hopping and reaction. Simulation progresses with a smaller time step [MATH] to resolve fast reaction events. We show that as [MATH] becomes smaller, the reaction and hopping events occur in a probabilistic manner that follows exponential tim... |
Hopping time distribution Consider a single particle hopping on a completely vacant lattice. Let [MATH] be the step acceptance probability for a particle heading to a vacant voxel. Then the probability of successful hopping after [MATH] trials is |
[EQUATION] The survival probability (no hopping) until [MATH] th trial is then [EQUATION] If we perform the trial every [MATH] sec such that [MATH] , where [MATH] is the average hopping rate per second. The survival probability becomes |
[EQUATION] where [MATH] Similarly, we have [EQUATION] Taking the limit of small [MATH] , we then have [EQUATION] Since [MATH] , when [MATH] is small enough, the hopping time distribution of a particle approximates the exponential distribution |
[EQUATION] with [MATH] Reaction time distribution Consider a reaction pair at an in-contact situation. The survival probability that they are still at the in-contact situation after [MATH] steps is |
[EQUATION] where [MATH] is the reaction probability and [MATH] is the escape probability. Let the simulation trial performed at infinitesimal time [MATH] , such that [MATH] |
The survival probability as a function of time is then [EQUATION] where [MATH] Note that the survival probability in this form includes both the probability of reaction and hopping events. Since the two events are independent of each other, the survival probability can be split into two separate terms: |
[EQUATION] where [MATH] is the average reaction rate and [MATH] is the average hopping rate. Therefore, the survival probability of the reaction also follows the exponential function |
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