Chemical diffusion master equation formulations of reactiondiffusion processes on the molecular level Mauricio J. del Razo1 2aStefanie Winkelmann3aRupert Klein1and Felix_2

2025-04-30 0 0 988.43KB 26 页 10玖币
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Chemical diffusion master equation: formulations of
reaction–diffusion processes on the molecular level
Mauricio J. del Razo,1, 2, a) Stefanie Winkelmann,3, a) Rupert Klein,1and Felix
Höfling1, 3
1)Freie Universität Berlin, Department of Mathematics and Computer Science, Berlin,
Germany
2)Dutch Institute for Emergent Phenomena, 1090GL Amsterdam, The Netherlands
3)Zuse Institut Berlin, Takustr. 7, 14195 Berlin, Germany
(Dated: 6 October 2022)
The chemical diffusion master equation (CDME) describes the probabilistic dy-
namics of reaction–diffusion systems at the molecular level [del Razo et al., Lett.
Math. Phys. 112:49, 2022]; it can be considered the master equation for reaction–
diffusion processes. The CDME consists of an infinite ordered family of Fokker–
Planck equations, where each level of the ordered family corresponds to a certain
number of particles and each particle represents a molecule. The equations at each
level describe the spatial diffusion of the corresponding set of particles, and they are
coupled to each other via reaction operators –linear operators representing chemical
reactions. These operators change the number of particles in the system, and thus
transport probability between different levels in the family. In this work, we present
three approaches to formulate the CDME and show the relations between them.
We further deduce the non-trivial combinatorial factors contained in the reaction
operators, and we elucidate the relation to the original formulation of the CDME,
which is based on creation and annihilation operators acting on many-particle prob-
ability density functions. Finally we discuss applications to multiscale simulations
of biochemical systems among other future prospects.
I. INTRODUCTION
It is a well-established paradigm to consider biochemical dynamics as an interplay between
the spatial transport (diffusion) of molecules and their chemical kinetics (reaction), both of
which are inherently stochastic. There exist different approaches for modeling and math-
ematically formalizing such reaction–diffusion processes, ranging from reaction–diffusion
master equations1–5, where spatial transport is modeled by diffusive jumps between local
compartments, to concentration-based approaches, such as deterministic6–9or stochastic
partial differential equations10 and partial integro-differential equations11. The preceding
modeling approaches may be regarded as approximations or limiting cases of particle-based
reaction–diffusion (PBRD) models, which explicitly resolve the diffusive trajectories of in-
dividual particles in space and time, as well as reactions between them. In the standard
PBRD models, particles move freely in space following Brownian motion, or any other form
of diffusion process12,13, and can undergo chemical reactions, which involve one, two or
more reactants in such a way that the reaction rate can depend on the positions or rela-
tive positions between the reactants14,15. Because of their high complexity, PBRD systems
are mostly studied numerically by means of Monte Carlo simulations of the underlying
stochastic reaction–diffusion process.
The mathematical formalization and analysis of PBRD models, however, is difficult be-
cause reactions constantly change the number of particles of each species, changing the
dimension and composition of the system. Recent work presents a probabilistic frame-
work and the characteristic evolution equation for PBRD termed chemical diffusion master
a)Corresponding authors: m.delrazo@fu-berlin.de, winkelmann@zib.de
arXiv:2210.02268v1 [cond-mat.stat-mech] 5 Oct 2022
2
equation (CDME)16. The CDME consists of an infinite ordered family of Fokker–Planck
equations (i.e., an enumerated collection), where each equation corresponds to a certain
number of particles n= 0,1,2, . . . . The equations, for each fixed n, describe the spatial
diffusion for the corresponding n-particle probability distribution, and they are coupled via
reaction operators that express the changes in the system’s state due to chemical reactions.
These operators change the number of particles in the system, and thus they can be con-
veniently expressed in terms of creation and annihilation operators16, following a classical
analogue of the quantum mechanical Fock space concept17,18. First steps towards solving
the CDME analytically by means of the Malliavin calculus were taken recently19. A more
comprehensive introduction on the topic can be found in ref. 16.
In this work, we explore the CDME from several perspectives and present three ap-
proaches to motivate and formulate it. This work not only improves our understanding of
how to formulate the CDME, but it also provides a more illustrative and accessible approach
to practitioners than the original work16. In general, the CDME is composed of a diffusion
operator and several reaction operators (one for each included reaction), all of them acting
on a symmetric many-particle distribution function. In analogy to the well-known chemical
master equation20–23, which characterizes spatially well-mixed stochastic reaction kinetics,
each reaction operator consists of a loss term describing the probabilistic outflow from a
given configuration state by the reaction, and a gain term that captures the probabilistic
inflow from other configuration states due to the reaction. The crucial part is to determine
these loss and gain operators for different types of reactions in the absence of a spatially
well-mixed setting; examples are binding and unbinding, creation and degradation, and
mutual annihilation. Here, non-trivial combinatorial factors enter for preserving symmetry
and normalization of the many-particle distribution functions under time evolution. The
local rate function, which defines the probability per unit of time for a reaction to take place
depending on the spatial positions of it’s reactants and products, has to be transformed into
an expression that takes the whole system state into account. This issue is addressed via
the following three approaches:
1. We use the local rate functions to specify also the loss and gain operators on a local
scale (acting on subsets of reactants and products), and then combine them into global
operators taking all combinations of reacting subgroups into account. The combina-
torial factors included in the operators are motivated by an inductive argument. The
CDME may then directly be expressed in terms of these global loss and gain operators
(section II).
2. The global loss and gain operators are expressed in terms of many-particle propen-
sity functions, which define the probability per unit of time for a reaction to occur
as a function of the whole system state. We explicitly derive these many-particle
propensity functions from the given local rate functions using permutations and Dirac
δ-distributions. For the exemplary settings of decay and binding it will be shown that
the resulting CDME agrees with the one of the first approach (section III).
3. The operators in the CDME are expressed as expansions in terms of creation and
annihilation operators as in ref. 16. These expansions can be condensed in a com-
pact notation that allows us to write the CDME, for a given system of reactions, in a
simple, fast and straightforward manner. The combinatorial factors do not appear ex-
plicitly, instead they are naturally encoded in the creation and annihilation operators
(section IV). A dictionary specifying the relation between the compact notation for
the expansions and the concrete algebraic expressions in the classical representation
is provided in appendix A.
In all three approaches, we start with a simplified setting containing only one molecular
species, which drastically simplifies the notation, and then generalize to reactions involving
several species such as, complex formation and general association reactions.
3
II. THE CHEMICAL DIFFUSION MASTER EQUATION: AN INTUITIVE FORMULATION
We consider an open system of a varying number of diffusing particles of the same chem-
ical species in a finite space domain X⊂Rd. The diffusion process changes the spatial
configuration of the particles while the reaction process can change the number of particles
in the system. The configuration of the system is thus given by the numbers of particles
and their positions. The probability distribution of such a system is given as an ordered
family of probability density functions:
ρ= (ρ0, ρ1, ρ2, . . . , ρn, . . . ),(1)
where ρn(x(n))is the probability density of finding nparticles at the positions x(n)=
(x(n)
1, . . . , x(n)
n)for n≥1, while ρ0is the probability for no particles being present. As the
particles are statistically indistinguishable from each other, the densities must be symmetric
with respect to permutations of particle labels, e.g. ρ2(y, z) = ρ2(z, y)for all y, z ∈X, and
more generally
ρn(x(n)) = ρn(P x(n))for all P∈ Pn(2)
where Pnis the set of all permutations of an n-tuple. The normalization condition is
ρ0+
∞
X
n=1 ZXn
ρn(x(n))dx(n)= 1.(3)
In general, ρwill also depend on time, ρn=ρn(t, x(n)), but we will omit tfor simplicity.
As a remark, the distribution ρis an element of a linear function space similar to the Fock
space of quantum mechanics, see refs. 16–18 and section IV.
Given that there are M∈Nreactions, the CDME has the general form
∂ρ
∂t = D+
M
X
r=1
R(r)!ρ(4)
for a diffusion operator Dand reaction operators R(r). Each of the reaction operators
R(r)corresponds to one possible reaction, and it is conveniently split into loss and gain
operators1,
R(r)=G(r)− L(r).(5)
In the following, we will construct these loss and gain operators at first for reactions of a
single species and then for a multi-species scenario. In each case, we consider a system with
only one reaction, such that the index rcan be skipped. For systems with several reactions,
the results may simply be combined by summing up these operators as in eq. (4).
A. One species
To start with, we assume that there is only one chemical species A. The most general
reaction in this case is of the form
kA →lA (6)
for k, l ∈N0. The rate at which a reaction event occurs is given by λ(y(l);x(k))>0, and
it depends on the positions x(k)∈Xkof the reactants and the positions y(l)∈Xlof the
1Similarly, the reaction operator in ref. 16 was split into a particle conserving part (the loss operator) and
a non-conserving part (the gain operator).
4
FIG. 1. Diagram representing the loss of probability from the n-particle state due to the reaction
kA →lA (eq. (10)). The particle states are represented by a set of boxes, where each box correspond
to the index of a particle.
products. Note that the rate function λshould be symmetric with respect to pair exchanges
in both of its arguments.
We can now write the nth component of the CDME as
∂ρn
∂t =Dnρn+Gnρn+k−l− Lnρn(7)
for appropriate operators Dn,Gn,Lnreferring to diffusion, gain and loss, respectively2.
Reactions at the n-particle state produce a transition to the (n−k+l)-particle state. Thus,
the loss of probability for the n-particle state ρndepends only on itself. Similarly, reactions
at the (n+k−l)-particle state produce a transition to the ρnstate. Thus, the gain of
probability for the n-particle state depends on ρn+k−l.
For physically non-interacting particles, the diffusion operator Dncan be expressed in
terms of the one-particle diffusion Dνapplied to the νth particle:
Dn=
n
X
ν=1
Dν,(8)
where Dνis the infinitesimal generator of the one-particle Fokker–Planck equation. For
example, one may think of Dνas something as simple as the d-dimensional Laplacian,
Dν=∇2
xν. Ignoring the reaction operators and assuming that there is no exchange of
particles with a reservoir outside of X24, all the resulting equations are uncoupled and
one obtains a family of uncoupled Fokker–Planck equations, unless there is an exchange
of particles with the world outside of X, in which case one ends up again with a similar
family of many-particle densities, albeit with a different structure of the coupling between
its levels25. For simplicity of the exposition, we assume reflecting boundaries for Xfrom
here on, i.e., a confinement by rigid walls.
The loss operator acting on the n-particle density will output the total rate of probability
loss of ρndue to all possible combinations of reactants. It is given in terms of the loss per
reaction Lν1,...,νk(local loss), which acts on kparticles at a time, with (ν1, . . . , νk)denoting
the indexes of the particles that it acts on. The loss per reaction quantifies how much
2In ref. 16, the loss operator was denoted by R(k), and the gain operator as R(k,l). We find the new
notation less cumbersome.
5
probability is lost to the current state due to one reaction, it is thus the integral over the
density and the rate function λover all the possible positions of the products:
(Lν1,...,νkρn) (x(n)) = ρn(x(n))ZXl
λ(y(l);x(n)
ν1,...,νk)dy(l),(9)
where x(n)
ν1,...,νk:= (x(n)
ν1, . . . , x(n)
νk). The total loss is then the sum of the loss per reaction
over all possible reactions,
Ln=X
1≤ν1<···<νk≤n
Lν1,...,νk.(10)
The form of the ordered sum guarantees that we count all the possible ways of picking up k
particles without double counting, see fig. 1for a diagram of the calculation. For the special
case of k= 0 we have
(Lnρn)(x(n)) = ρn(x(n))ZXl
λ(y(l))dy(l).(11)
Similarly, the gain operator acting on the n-particle density will output the total rate
of probability gain of ρn. It can be expressed in terms of the gain per reaction (local
gain) resulting from kreacting particles with indexes (ν1, . . . , νk)producing lproducts
with indexes (µ1, . . . , µl), termed Gµ1,...,µl. The gain per reaction quantifies how much
probability is gained by the current state due to one reaction, it is thus the integral over
the density and the rate function λover all the possible positions of the reactants:
(Gµ1,...,µlρn+k−l) (x(n)) = ZXk
λ(x(n)
µ1,...,µl;z(k))ρn+k−l(x(n)
\{µ1,...,µl}, z(k))dz(k),(12)
where the subscript \{µ1, . . . , µl}means that the entries with indices µ1, . . . , µlare ex-
cluded from the tuple x(n)of particle positions. Note that the indexes of the reacting
particles ν1, . . . , νkare not relevant for the gain since the reactants’ positions are integrated
out (and both the density and the rate function are symmetric). The total gain is then the
sum of the gain per reaction over all possible reactions,
Gn=(n−l)!
n!n+k−l
kn
X
µ1...µl=1
µi6=µj∀i,j
Gµ1,...,µl(13a)
=n
l−1n+k−l
kX
1≤µ1<···<µl≤n
Gµ1,...,µl,(13b)
where we used the symmetry of Gµ1,...,µlwith respect to the indices. The complicated form
of the gain operator is due to the fact that it needs to consider all the possible ways to pick
up kparticles from the n+k−l-particle state, just as the loss operator, but in addition,
it also needs to consider all the possible ways of incorporating lparticles into the current
state in a symmetry-preserving manner, see fig. 2for a diagram illustrating the calculation.
Note that the output of the loss and gain operators is also symmetric.
Let us use the preceding formulas for general reactions involving one species to derive the
CDME for some common reactions (for simplicity, we write ρn(xn, t)as ρn(xn)):
•Degradation A→ ∅: This case is recovered with k= 1, l = 0 using the rate function
λd(x) = λd(; x). The CDME reads
∂ρn
∂t (xn) =
n
X
ν=1
Dνρn(xn)+(n+ 1) ZX
λd(z)ρn+1(x(n), z)dz −ρn(x(n))
n
X
ν=1
λd(x(n)
ν).
(14)
摘要：

Chemicaldiusionmasterequation:formulationsofreactiondiusionprocessesonthemolecularlevelMauricioJ.delRazo,1,2,a)StefanieWinkelmann,3,a)RupertKlein,1andFelixHöing1,31)FreieUniversitätBerlin,DepartmentofMathematicsandComputerScience,Berlin,Germany2)DutchInstituteforEmergentPhenomena,1090GLAmsterdam...

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