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[MATH] , and the score is no longer the sufficient statistics. In the supplementary material we show further results and demonstrate how the improved likelihood ratio estimation leads to better inference results. |
Conclusions In this work we have presented a family of new inference techniques for the setting in which the likelihood is only implicitly defined through a stochastic generative simulator. The new methods estimate likelihoods or ratios of likelihoods with data available from the simulator. Most established inference m... |
While these additional quantities often require work to be extracted, they also prove to be very valuable as they can dramatically improve sample efficiency and quality of inference. Indeed, we have shown that this additional information lets us define loss functionals that are minimized by the likelihood or likelihood... |
We have demonstrated in three experiments that the new inference techniques let us precisely estimate likelihood ratios. In turn, this enables parameter measurements with a higher precision and less training data than with established methods. |
This approach is complementary to many recent advances in likelihood-free inference: the augmented data can be used to improve training for any neural density estimator or likelihood ratio estimator, as we have demonstrated for Masked Autoregressive Flows and Carl . It can also be used to define locally optimal summary... |
Finally, these results motivate the development of tools that provide a nonstandard interpretation of the simulator code and automatically generate the joint score and joint ratio, building on recent developments in probabilistic programming and automatic differentiation |
. We have provided a first proof-of-principle implementation of such a tool based on Pyro Acknowledgments We would like to thank Cyril Becot and Lukas Heinrich, who contributed to this project at an early stage. We are grateful to Jan-Matthis Lückmann for helping us automate the calculation of the joint likelihood rati... |
Pyro and to all participants of the Likelihood-free inference workshop at the Flatiron Institute for great discussions. We would like to thank Felix Kling, Tilman Plehn, and Peter Schichtel for providing the |
MadMax code and helping us use it, and to George Papamakarios for discussing the Masked Autoregressive Flow code with us. KC wants to thank CP3 at UC Louvain for their hospitality. Finally, we would like to thank Atılım Güneş Baydin, Lydia Brenner, Joan Bruna, Kyunghyun Cho, Michael Gill, Siavash Golkar, Ian Goodfellow... |
JB, KC, and GL are grateful for the support of the Moore-Sloan data science environment at NYU. KC and GL were supported through the NSF grants ACI-1450310 and PHY-1505463. JP was partially supported by the Scientific and Technological Center of Valparaíso (CCTVal) under Fondecyt grant BASAL FB0821. This work was suppo... |
# Source: arxiv 1805.12311 # Title: Reaction-diffusion kinetics on lattice at the microscopic scale # Sections: all # Downloaded: 2026-03-03T05:13:44.253116+00:00 |
Reaction-diffusion kinetics on lattice at the microscopic scale Abstract Lattice-based stochastic simulators are commonly used to study biological reaction-diffusion processes. Some of these schemes that are based on the reaction-diffusion master equation (RDME), can simulate for extended spatial and temporal scales bu... |
In addition, when a reaction compartment is populated with volume-excluding obstacles, MLM captures the non-classical reaction kinetics caused by anomalous diffusion of reacting molecules. |
missingum@section Introduction In the intracellular environment, macromolecules can be heterogeneously distributed in space and react stochastically at low concentrations. The conventional mass action-based approach is insufficient to describe the reaction-diffusion (RD) behavior of the macromolecules and thus, it is n... |
. Generally, we can represent space as a continuum (off-lattice) or a discretized lattice model. In the former, each molecule is represented as a point or a hard-body sphere that propagates via Brownian motion in continuous space |
. Bimolecular reaction is often modeled as a collision-based interaction according to the Smoluchowski model of diffusion-influenced reactions |
. In some models, the finite size of molecules is taken into account in the reaction and therefore, the effects of volume-exclusion by molecules can be reproduced |
. Although continuous space-time models are physically consistent, the cost of computation becomes significant when simulating non-dilute and crowded conditions in the cell |
On the other hand, in lattice approaches, the average diffusion behavior is adopted and the reactions follow either the simple first-order process, or the second-order process when two reactive molecules meet on the same lattice voxel. Such approaches reduce the computational cost even in crowded space and provide an e... |
, space is discretized into lattice voxels called subvolumes. In each subvolume, point-like molecules are assumed to be dilute and well-mixed. To obey the well-mixed condition, there is a limit to the size of the subvolume |
, which in turn imposes a limit to the spatial resolution. Diffusion of molecules across subvolumes is modeled as a first-order reaction with a concentration dependent rate. Unimolecular and bimolecular reactions only occur within each subvolume with a rate defined by the propensity function |
. Compared to continuum-based schemes, RDME models RD from the mesoscopic to the macroscopic scale but not at the microscopic scale. However, there have been several efforts to overcome the well-mixed limit in RDME models and to bridge mesoscopic and microscopic scales |
Apart from the RDME lattice models, there is another class of schemes, which we refer as microscopic lattice method (MLM) that represents molecules at single particle resolution |
. In most of these schemes , the size of the voxel follows the molecule size, whereas in the small-voxel tracking algorithm (SVTA) |
, a particle can occupy multiple voxels, providing greater spatial resolution at the cost of higher computational complexity. In MLM, a molecule hops into a neighbor voxel at a constant rate such that normal diffusion is satisfied. Excluded volume arises naturally since the size of molecule is directly reflected by the... |
. In the collision-based reaction schemes, the steady-state reaction rate follows the macroscopic effective reaction rate when the reaction is activation-limited. However, the reaction accuracy of MLM has not been studied in detail when it is diffusion-influenced. In a recent work, Sturrock |
also reported several shortcomings in MLM, notably in the accuracy of Spatiocyte when estimating steady-state bimolecular reaction rates. |
Our focus in this work is to examine in detail the accuracy and consistency of MLM in solving diffusion-influenced reactions using theoretical analysis and numerical simulations. The theoretical framework here is constructed based on the hexagonal close-packed (HCP) lattice but is also applicable for any regular lattic... |
missingum@section Methods We begin by presenting the theoretical background of the Collins-Kimball approach in modeling irreversible bimolecular diffusion-influenced reaction. We highlight the particle-pair formalism for the reaction rate, which will be used in the MLM theory. We then briefly describe the Spatiocyte RD... |
Irreversible bimolecular diffusion-influenced reaction in continuum-based framework Consider an irreversible bimolecular reaction involving two distinct species: |
[EQUATION] where [MATH] and [MATH] are hard-sphere molecules with radii [MATH] and [MATH] , respectively. The molecules diffuse in three-dimensional (3D) space with diffusion coefficients [MATH] and [MATH] The time evolution of the species concentration is well-described by a time-dependent rate coefficient [MATH] |
[EQUATION] Smoluchowski derived the rate coefficient by relating the diffusion coefficient of the molecules with molecular collisions, leading to the product formation. In his work, [MATH] is made up of an immobile molecule and is surrounded by multiple diffusing [MATH] molecules. Collins and Kimball extended the Smolu... |
[EQUATION] Here, [MATH] is the collision rate, [MATH] and [MATH] The rate coefficient ( ) starts ( [MATH] ) at [MATH] but decays rapidly to |
[EQUATION] at long-time. [MATH] is the steady-state or the effective reaction rate constant given by [EQUATION] According to Noyes theory |
, the rate coefficient can be expressed equivalently using the particle-pair approach: [EQUATION] where [MATH] denotes the survival probability of an isolated reactant pair at time [MATH] given that they were initially in contact. Additionally, let [MATH] denote the rebinding-time probability distribution for a reactiv... |
[EQUATION] with [MATH] Note that the survival probability [MATH] is the same as the probability that the first rebinding event between an initially in-contact pair has not yet occurred at time [MATH] . Hence we can rewrite Eq. ( ) as |
[EQUATION] At long-time, we then have [EQUATION] where the integrated term gives the total rebinding probability: [EQUATION] Therefore, the effective rate constant ( ) can also be written in terms of the total rebinding probability: |
[EQUATION] The above relation was also described previously, but in the context of irreversible and reversible rate constants In subsequent sections, we use the relations described by Eqs. ( ) and ( 11 ) as the central concepts to derive the rate coefficient in MLM. |
Spatiocyte reaction-diffusion scheme In the Spatiocyte scheme (see Algorithm , space is discretized into HCP lattice because the arrangement allows the highest density of regular sphere voxels in 3D space. The voxel has a diameter [MATH] and can be occupied by at most, a single molecule. At each diffusion time step [MA... |
[EQUATION] The above relation is applicable when the reaction is activation-limited ( [MATH] ). For diffusion-influenced reactions ( [MATH] ), the collision rate [MATH] is reduced relative to the production rate [MATH] . The acceptance probability [MATH] would then have the issue of exceeding unity when [MATH] . The Sp... |
Initialization: [MATH] 0, scheduler [MATH] for each species [MATH] do [MATH] max [MATH] , where [MATH] denotes the pair of reactive species [MATH] and [MATH] |
[MATH] , where [MATH] reaction acceptance probability [MATH] step acceptance probability [MATH] end for Main loop: while [MATH] and [MATH] |
do [MATH] next event in [MATH] get species identity [MATH] get current voxel location [MATH] reschedule next event, [MATH] for each molecule of species [MATH] |
do choose a random target voxel [MATH] [MATH] if [MATH] is vacant then draw [MATH] if [MATH] then walk succeeded, [MATH] else walk rejected, [MATH] |
end if else if [MATH] contains reactant species [MATH] then draw [MATH] if [MATH] then reaction [MATH] accepted, [MATH] else reaction failed and walk rejected, [MATH] |
end if else walk rejected, [MATH] end if end for end while Algorithm 1 Basic outline of the Spatiocyte algorithm for bimolecular reactions. [MATH] is the current simulation time, [MATH] [MATH] is the simulation duration, [MATH] is the reaction acceptance probability for the reactive pair of species [MATH] and [MATH] [M... |
Rebinding probability and reaction rate on HCP lattice As an alternative to the time-dependent reaction rate coefficient in Eq. ( ), we define a discrete-space version with a step-dependent rate coefficient on lattice as |
[EQUATION] where [MATH] is the simulation step, which is related to the simulation time by [MATH] [MATH] is the initial reaction rate constant on lattice (see Appendix ) and [MATH] is the lattice analogue of the rebinding-time probability function [MATH] in diffusion step [MATH] |
At long-time, the effective rate on lattice follows similarly to Eq. ( ): [EQUATION] where the summation term ( 14 ) corresponds to the total rebinding probability on lattice. |
To obtain the analytical expression for [MATH] , we consider again a reactive pair [MATH] and [MATH] , which are initially in-contact by occupying adjacent voxels on lattice. We are interested in the rebinding-time probability distribution as a function of the diffusion step [MATH] . Without losing generality, we can f... |
Activation-limited case ( [MATH] [MATH] We denote [MATH] as the voxel at origin, [MATH] as an element of the set of immediate neighbor voxels of [MATH] . We define [MATH] as the first-passage time distribution for a random walker to walk from voxel [MATH] to [MATH] , that is, the probability of arriving at voxel [MATH]... |
We first consider the rebinding-time probability distribution for the case [MATH] . Let [MATH] and [MATH] denote the first-passage time distributions to origin from origin and [MATH] , respectively. The two probabilities are related via |
[EQUATION] where [MATH] is the transition probability from [MATH] to [MATH] in a single step. This implies that the trajectory we are interested in, which is from an in-contact situation (e.g., [MATH] at [MATH] and [MATH] at [MATH] ) to the rebinding situation ( [MATH] hops to [MATH] ) in a single step, is equivalent t... |
Therefore, the rebinding-time probability distribution is fully described by [MATH] and is related to [MATH] . The latter can be obtained analytically from its probability generating function [MATH] |
(see Appendix ). As for [MATH] , the trajectories that have undergone failed reaction attempts before step [MATH] are included in the rebinding-time probability distribution: |
[EQUATION] where [MATH] is the probability to reach the origin for the [MATH] th time at the [MATH] th step (I.1.9 in ): [EQUATION] |
where [MATH] The generating function of [MATH] in terms of [MATH] is (see Appendix ): [EQUATION] By taking the limit [MATH] on [MATH] , we obtain the total rebinding probability on lattice as |
[EQUATION] where [MATH] . It was shown previously that the probability generating function of the HCP lattice is topologically equivalent to that of the face-centered cubic (FCC) lattice |
. Therefore, we have [MATH] , p. 153) for HCP lattice. Finally, if we set the initial rate [MATH] (see Appendix ) and substitute the total rebinding probability [MATH] from Eq. ( 19 ) into Eq. ( 14 ), we obtain the effective rate constant on lattice as |
[EQUATION] Diffusion-influenced case ( [MATH] [MATH] The rebinding-time probability distribution [MATH] of the diffusion-influenced scheme is defined as |
[EQUATION] where [EQUATION] is the probability for a successful reaction, [MATH] is the probability to select [MATH] given that the molecule is in [MATH] [MATH] for HCP lattice), and [MATH] is the probability that a particle is in contact after [MATH] -steps (see Appendix for more details). |
The probability generating function of [MATH] on HCP lattice is given by (see Appendix [EQUATION] where [MATH] is the probability generating function of [MATH] |
Taking the limit [MATH] , we get the total rebinding probability as (Appendix [EQUATION] which is identical to Eq. ( 19 ) in the activation-limited case. Similarly, by substituting the summation term in Eq. ( 14 ) with Eq. ( 24 ), we get the effective rate constant for the diffusion-influenced case as |
[EQUATION] which also follows Eq. ( 20 ). Henceforth, we adopt the same notations of the effective reaction rate and total rebinding probability for both the activation-limited and diffusion-influenced cases. |
Comparison with continuum-based theory Since the effective rate on lattice ( 25 ) has the same form of Eq. ( 11 ) in continuum, we can match them by equating the initial rate and total rebinding probability of the two: [MATH] and [MATH] . With the former relation, the reaction acceptance probability is connected to the... |
[EQUATION] Employing the [MATH] relation, the voxel size is found to be about [MATH] greater than the molecule size: [EQUATION] The Spatiocyte scheme is thus guaranteed to have the same effective rate and total rebinding probability as the continuum framework provided that Eqs. ( 26 ) and ( 27 ) are satisfied. In addit... |
[EQUATION] where [MATH] According to Eq. ( 27 ), accurate matching of both the effective rate and the total rebinding probability requires the voxel size to be larger than the molecule size. Nonetheless, during modeling we can fix the voxel size to be the same as the molecule size, [MATH] . In this case, it is still po... |
[EQUATION] However, this is done at the expense of losing accuracy in the total rebinding probability, since [MATH] For the reversible reaction, [MATH] , the local detailed balance on lattice is achieved by choosing a lattice dissociation rate constant [MATH] from the following equilibrium constant relation: |
[EQUATION] The MLM method can simulate the dissociation reaction as a first-order process with rate [MATH] and place the dissociated pair of molecules at an in-contact condition. |
Numerical simulations We verify the main theoretical results presented above with numerical simulations using Spatiocyte. Spatiocyte is included in E-Cell System version 4 |
, an open-source biochemical simulation environment that supports multiple algorithms, time scales and spatial representations. The Python notebooks used to generate the simulation results reported here are available at |
. The performance benchmark models for all tested methods are included in Spatiocyte package ( ) as examples. missingum@sectionI Results and Discussion |
We first validate the theory of total rebinding probability and its time-dependent behavior on lattice using numerical simulations. We examine the accuracy of the reaction rate coefficient and its time-dependent behavior on lattice. We then compare the diffusion and reaction performances of MLM and several other off-la... |
Numerical validation of MLM theory Rebinding probability We examine the rebinding probability distribution of a reactive pair, [MATH] and [MATH] that are initially in-contact. The theoretical rebinding-time probability distribution [MATH] and [MATH] are validated against numerical results. In the activation-limited cas... |
[EQUATION] Table ( ) shows the simulated and the expected theoretical values for [MATH] steps. The simulation results agree well with the expected values, with discrepancies never exceeding [MATH] . Since the theoretical rebinding-time probability distribution on lattice is validated by simulations, the analytical form... |
To illustrate the dependency of total rebinding probability on [MATH] , we obtained the probability at various [MATH] up to [MATH] . Table ( ) shows the simulated and the expected theoretical values for various [MATH] ratios. Both simulated and theoretical values coincide well, with discrepancies never exceeding [MATH]... |
We then evaluated the rebinding-time probability distribution by recording the time taken for [MATH] and [MATH] to associate immediately after a dissociation event. We performed the simulations for a large number of steps and independent runs. Figure shows the average number of rebinding events per unit time at [MATH] ... |
. We have corroborated this result with detailed asymptotic analysis that is provided in Appendix Note that in the diffusion-influenced case ( [MATH] ), finer step intervals generate rebindings at times smaller than the diffusion time step [MATH] , denoted by the vertical dashed line in Figure . In this temporal regime... |
Reaction rate We evaluated the accuracy of the effective reaction rate constant for irreversible bimolecular reactions ( ) over various [MATH] regimes on lattice. We considered an immobile species [MATH] and a diffusing species [MATH] that are uniformly distributed at initialization with concentrations [MATH] and [MATH... |
). From the survival probability, we calculated the time-dependent reaction rate coefficient using (Eq. 2.1 in [EQUATION] We adopted the following discretization scheme for the time derivative to get the discrete rate coefficient: |
[EQUATION] where [MATH] is the index of the discretized [MATH] and [MATH] . The boundary cases are computed as [EQUATION] where [MATH] denotes the final time step. The reaction rate coefficient obtained for various [MATH] ratios are shown in Figure 3(b) along with their corresponding theoretical curves from Eq. ( ). |
Recall that the long-time asymptotic variant of the Collins-Kimball theory ( ) has the form [EQUATION] where [MATH] and [MATH] denote the steady-state rate constant and the time-dependent term, respectively. We fitted Eq. ( 35 ) to the numerical data, omitting early time points to avoid non-steady-state effects. The re... |
Performance Diffusion We compared the 3D diffusion performance of MLM using Spatiocyte (git 9757fb3) and three other off-lattice particle-based simulation methods, Smoldyn |
(version 2.55), eGFRD (in E-Cell System version 4.1.4) and fast Brownian dynamics (BD) (C++ program example in Spatiocyte git 9757fb3). When the molecules are represented as hard-spheres with volume exclusion, Spatiocyte required shorter run times than Smoldyn in all cases (Figure 4(a) ). Spatiocyte achieves comparable... |
If molecules are represented as dimensionless point particles, higher diffusion performance is expected since inter-molecular collisions can be ignored. Figure 4(b) shows the run times of Spatiocyte, Smoldyn and fast BD when diffusing point particles with the same simulation interval. eGFRD was not considered here sinc... |
Reaction Recently, Andrews benchmarked the performance of Smoldyn, MCell , eGFRD, SpringSaLaD and ReaDDy particle simulators when running the well-known Michaelis-Menten enzymatic reaction. Smoldyn required the least amount of time to complete the benchmark. Running the model on our hardware (see Figure for specificati... |
, we decreased the number of molecules, diffusion coefficients and reaction rates. The execution times of Spatiocyte, Smoldyn and eGFRD when running the model with the new parameters are presented in Figure . The simulators generated almost identical results. Spatiocyte and Smoldyn had similar run times, whereas eGFRD ... |
Application Examples We applied MLM to model two fundamental RD systems of intracellular signaling, the production-degradation process, previously studied using lattice-based methods |
, and the dual phosphorylation-dephosphorylation cycle of the mitogen-activated protein kinase (MAPK) cascade , a common motif found in signal transduction systems but with a response function that is highly sensitive to the binding kinetics. We also report the effects of excluded volume on the kinetics of a simple bim... |
Production-degradation process Consider the production and degradation processes of protein [MATH] represented by a zero-order production coupled with a second-order degradation: |
[EQUATION] The concentration of [MATH] will go through an initial transient state before settling down at a steady-state equilibrium, [MATH] that fluctuates according to the Poisson distribution |
. To confirm if MLM can recapitulate the production-degradation process correctly in 3D space, we have simulated the process with Spatiocyte and compared the outcomes with eGFRD and the well-mixed model. To generate the results of the well-mixed model, we solved the rate equation using an ordinary differential equation... |
Recently, the Spatiocyte scheme was reported to not only fail to reproduce the expected equilibrium value of [MATH] but also generate different values depending on the voxel size |
. In the report, the effective bimolecular rate [MATH] was used in the calculation of reaction acceptance probability instead of the intrinsic reaction rate constant [MATH] , which inevitably caused the deviation from the well-mixed model (see first row of Table ). As shown in Figure 6(a) and Table , there was no discr... |
The well-mixed model assumes the time scale of diffusion to be always shorter than that of the reactions. As a result, molecules are expected to be uniformly distributed at all times and reactions can take place independent of spatial localization. The well-mixed assumption is valid when describing activation-limited r... |
The reduction in equilibrium value when the diffusion coefficient is decreased was previously described by the microscopic theory of Agmon and Szabo |
. In contrast to the Collins-Kimball theory, Agmon and Szabo have considered the non-negligible effect of [MATH] concentration on the effective reaction rate, especially when the reaction is diffusion-influenced. The slow diffusion of molecules increases the effective contact radius, resulting in higher effective annih... |
On the other hand, RDME shows large deviation from the expected values at slow diffusion. The inability of conventional rate equation and RDME to correctly capture diffusion-influenced reactions has previously been noted and worked on before |
. By incorporating the diffusion coefficient into the bimolecular reaction propensity formula (Eq. 26 in ), the equilibrium concentration of RDME shows a better agreement with the expected values (see RDMEm, [MATH] in Figure 6(b) ). However, when the reaction is diffusion-limited ( [MATH] ), unlike MLM, the subvolume s... |
) that preserves the well-mixed condition. At [MATH] for example, the critical subvolume size is about 13 times the molecule diameter, any size smaller is invalid. |
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