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STOCHASTIC ELECTRON TRANSFER MODELS

I. Electron transfer model in shuttle molecule complexes localized in energy-conjugated mitochondrial membranes and photosynthesizing organelles - chloroplasts of plants and chromatophores of photosynthesizing bacteria.
Elementary transitions of an electron between individual states of a multi-enzymatic complex can be in the first approximation considered as Markovian process, that is, probabilities of transitions between individual states i and j of the multi-enzymatic complex in every moment of time can be characterized of certain numbers k_ij that do not depend on the prehistory of the system. Let S₁, S₂, …, S_n be states of the multi-enzymatic complex, and the probability that the complex is in the condition S_i in the moment of time t: p (S_i, t) = p_i(t), i=1,2,…, n.
For every moment of time , because the multi-enzymatic complex is necessarily in one of the introducing states, but cannot be in two different states at the same time.
If the complex is in the i-th state in the moment of time t, then the probability of its transfer into the j-th state during the period of time Δt can be expressed as a_ijΔt, i≠ j.. Here a_ij is the transition velocity constant, a_ij≥0, i≠ j. The probability that the complex will remain in the initial state after Δt is 1- a_iΔt, where the value a_i complies with the equality a_i= .

If in the moment of time t the system was in the i-th state, then in the moment of time (t + Δt) it will either remain in the initial state (with probability a_ijΔt), or will transfer in one of the n-1 remaining possible states (with probability a_ijΔt).

Sending Δt to zero, we will come to the following system of linear differential equations:

= - a_ip_i + p_j,  p_i(0) = c_i ≥0; =1.

II. The described model does not take into consideration presence of mobile shuttles of electrons in some regions of the electron-transport chain. In mathematical description of an electron transfer, differences in the organization of individual regions of electron transport must be taken into account. In the case when shuttles of electrons are mobile enough to interact with one another by encounters, reductive-oxidative states of individual shuttles can be considered as independent. In accordance with the mass action law, for the reaction of electron transfer between two one-electron shuttles C1 and C2 interacting by the scheme

k₁     k₂     k₃   
   D → C₁ → C₂ → A, (1)

kinetic equations describing the studied reductive-oxidative reactions can be written in the form

,

where D¹, C₁¹, C₂¹ are concentrations of the reduced form of the exogenous donor D and the electron shuttles C₁ and C₂; C₁⁰, C₂⁰, A⁰ are concentrations of the oxidated form of the shuttles C₁, C₂ and the exogenous acceptor A; k₁, k₂, k₃ are bimolecular constants of the velocities of the corresponding reactions.

III. Unlike the case of reductive-oxidative reactions in a solution, when every reduced molecule of the shuttle C₁ could pass an electron to any oxidated molecule C₂, in a complex the reduced shuttle C₁ can interact only with the oxidated shuttle C₂, which is in the same complex with C₁. To describe an electron transfer in a complex, all states, in which this complex can be, must be considered. The states of the complex can be defined as an ordered collection of states of shuttles, which constitute the complex. As this takes place, every shuttle in the complex can be either in the reduced, or the oxidated form, either in the protonated, or deprotonated state, and so on.
Let us designate the states of the complex as S₁, S₂, …, S_n and introduce the probability that the shuttle complex is in the state S_i in the moment of time t: p(S_i, t) = p_i(t), i = 1, 2, …, n. The events S₁, S₂, …, S_n are incompatible and form the full group of the events, that is why the normalization requirement is fulfilled. If transitions between the states S_i of the shuttle complex are viewed as Markovian process with a finite number of states and continuous time, then transitions of the complex from one state to another are described by a system of linear ordinary differential equations relative to probabilities:

; p_i(0) = b_i (i = 1, 2, ..., n), where b_i are initial values of the probabilities of the corresponding states. On examination particular electron-transport chains it is convenient to present a state of a complex in the form of a marked graph, in the vertices of which the states of the complex are placed, and arrows show possible transition between the states. In particular, for electron transfer in a complex of two shuttles on the chart (1) the marked graph of the states has the following form:

  (2)

Here upper indices 0 and 1 designate presence or absence of an electron on the corresponding shuttle. The numbers in brackets show the numbers of the complex states, k_i are velocity constants of the corresponding transitions, marked on the chart (1). In the general case velocity constants of transitions between the states of the complex can depend on states of the shuttles, which do not participate immediately in the reaction (cooperativity effect). The parameters α and β characterize the degree of cooperativity. For example, the velocity of electron inflow from the inner donor to the shuttle C₁ can be different, depending on the redox-state of the shuttle C₂ (α≠1). The following system of differential equations for the probabilities of the states of the complex C₁C₂ describes the functioning of the complex according to the chart (2), at α = β = 1:

         = k₃ p₂ - k₁ p₁;
         = k₂ p₃ - (k₁ + k₃) p₂;
         = k₁ p₁ + k₃ p₄ - k₂ p₃;
         = k₁ p₂ - k₃ p₄; p_i(0) = b_i.

After solving this system of equations relative to the probabilities of the states of the complex p_i, it is easy to find also the probabilities of the states of individual shuttles. For example, the probability that the first shuttle is in the oxidated state

p(C₁⁰) = p(C₁⁰C₂⁰) + p(C₁⁰C₂¹) = p₁ + p₂ = 1- p(C₁¹).

. The equations, which describe transitions between states of a multi-enzymatic complex, are linear. However, the number of these equations quickly increases with increasing of the number of shuttles. For a complex of n shuttles, where each of the shuttles can be either oxidated or reduced, the number of states is equal to 2ⁿ.