Axiomatic Motivation of Rate Equations

Most calculations that are presented in this work were done with rate equations. The concept of rate equations is presented together with the way of thinking that underlies the concept of cellular automata. This concept was presented as a general and basic theory that is suitable to predict the dynamics of any system that can be described by a mathematical expression that denotes information about the system by referring to its “state” and the probability for transitions between these states. In fact, rate equations are not only used to describe EET and ET processes but many systems, including optical transitions, particle reactions, complex chemical reactions and more general information processing in complex networks. Rate equations can even be used to describe complex systems such as sociological networks (Haken, 1990).

To understand the very basic concept that underlies the rate equation concept we present an axiomatic fundament that is suitable to motivate rate equations as a general mathematical framework that does not restrict the modeling with any constraints except basic axioms. We will speak about “systems” in the following. As systems we understand the objects that we want to investigate. These systems could be a piece of semiconductor, a cell, a plant, or the universe. It should be pointed out that the structure of the book allows for an omission of mathematical details to target the readers who are interested in biological background but not in the mathematical details. These readers are encouraged to skip the following parts and directly proceed with the next chapter.

As presented in (Schmitt, 2011), one can start from two very basic axiomatic prepositions to motivate rate equations:

  • 1) The assumption of “states” that define the system in form of information strings which could be numbers or any other type of coding. These strings contain all information that is experimentally accessible in a given system. One could talk about several states or substates of a system that define the “overall state”.
  • 2) The substates change over time. The time evolution can be described by differential equations. That means that we agree that there exists a transition in these systems that can be described by accurate mathematical treatment. The choice of differential equations is a specialization, however, it seems to be the most straightforward assumption that can define an equation for the “change” of the system in time.

Axiom 1) suggests that each state might be measurable in form of a mathematical object N{ with i forming a tupel of numbers or parameters that completely characterize the state. dN.

According to 2) 0 and therefore there must exist a function f dt

with dN.L = fi{xj, Nk, tj. f could depend on time t, any linear combination of the substates Nk themselves and other variables x,:

The so-called transfer matrix Tik contains all the factors for the linear combination of all substates k (taken together, they form the “state” of the system) and possible additional mathematical operators that describe

the time evolution of substate i. Even though f^xj,Nk, t) could be a nonlinear function in the substates, for the sake of simplicity we assume that f^Xj,Nk, tj is a linear function in the substates as a kind of first approximation. Since Xj are the variables that encode the information about the substates, these states (we might refrain from mentioning “substates” and call them just “states”) themselves should contain these variables as they contain all information that is experimentally accessible. As a consequence we can simplify eq. 1:

In a further simplification, we can assume that the dynamical behaviour may not change over time. The states themselves will of course change. Only the underlying dynamical concept is constant. In that case the proportionality factors Tik (t) do not depend on the time and we get

As we will see in chapter 1.4 (Rate Equations in Photosynthesis), the (protein) environment of a molecule can be assumed to introduce additional states (e.g. “relaxed” or “unrelaxed” states of the environment of the system). Therefore, the treatment of a closed system is in principle suitable to analyze even open systems if the states of the environment are known and incorporated into the theoretical description of the system.

From our simple axioms that there exist states and dynamics a kind of classical master equation formalism directly follows, as is visible in eq. 3. Eq. 3 is directly derived from the axiomatic postulate that ensembles are described by states and dynamic changes lead to transitions between these states. The number of transitions from one state to another is proportional to the population of this state and equal to the product of the transfer probability and the population of the states.

These axioms deliver a dynamical formulation of the temporal change of our states:

with the transfer matrix Tik=T . If we separate the supplying processes and the emptying processes (i.e. we separate the sum avoiding summation over k=i) we get:

Eq. 5 is a master equation for the evolution of population probabilities. For an excellent depiction of rate equations, the master equation and their applications see (Haken, 1990). The probability to find a certain state populated denotes to

If the set is complete, i.e. (t) = 0 then this formulation delivers the general master equation:

We have to keep in mind that eq. 5 is a simplified and linearized form of master equation. The full general ansatz is found in eq. 1. For example, eq. 5 does not describe biomolecular processes or processes that amplify or inhibit themselves (e.g. a cell division process or an allosteric protein regulation). But from eq. 1 a description for any process that is thinkable in the context of states and their change can be derived and solved numerically.

The time reversible/equilibrium case is equivalent to the existence of equilibrium probabilities Pk and P in such a manner that each term on the right side of eq. 7 vanishes separately, i.e. TjkPk= TkiPi. This means that the number of transfers from a substate Nk (which is found with probability Pk) to a substate N is the same as the backward reaction from state Ni to Nk .

In that case, the master equation (7) is called to be at a detailed balance. From eq. 7 we can see that due to this detailed balance condition the population probabilities are reciprocal to the transition matrix eleN T.

ments (determining the transition probabilities) for all times: - -l .

This means that the equilibrium is reached if there is no time dependency in the population densities of all substates (i.e. the population densities do not change over time).

As the number of transitions from a single state to another is same for forward and backward reaction in the equilibrium, the transition probabilities for forward and backward reaction are proportional to the relation between the populated states. In a microcanonical equilibrium, the Boltzmann distribution determines the population of states, i.e. forward and backward transfer of two coupled states are found to satisfy the Boltzmann relation

where Ei denotes the energy of the state i and kbT the thermal energy at temperature T. As we would expect from a single electron spreading along empty states, the overall equilibrium population is (see Haken, 1990).

For degenerated states eq. 9 has to be extended to (see Haken, 1990)

if Nk is gk -fold degenerated and Nt is g. -fold degenerated.

The formalism of rate equations as presented here has several advantages:

1) The theory contains all information about the population of the states in the system.

  • 2) Only two basic axioms and the linearity condition are sufficient to motivate the basic theory.
  • 3) The equations are rather simple (DGL 1st order).
  • 4) The theory allows the incorporation of further aspects by extending the space of states (see for example chapters 2.6 and 2.7 for concrete applications on EET and ET processes in plants).
  • 5) There exist common algorithms for solving the rate equations.
 
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