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.