Rate theories for biologists

Abstract

Some of the rate theories that are most useful for modeling biological processes are reviewed. By delving into some of the details and subtleties in the development of the theories, the review will hopefully help the reader gain a more than superficial perspective. Examples are presented to illustrate how rate theories can be used to generate insight at the microscopic level into biomolecular behaviors. An attempt is made to clear up a number of misconceptions in the literature regarding popular rate theories, including the appearance of Planck's constant in the transition-state theory and the Smoluchowski result as an upper limit for protein-protein and protein-DNA association rate constants. Future work in combining the implementation of rate theories through computer simulations with experimental probes of rate processes, and in modeling effects of intracellular environments so that theories can be used for generating rate constants for systems biology studies is particularly exciting.

Fig. 1

Thermodynamic control versus kinetic control. A protein in state A has two reaction pathways, leading to states B₁ and B₂, respectively. The forward and reverse rate constants of the two pathways are k_±1 and k_±2. Three of the constants are fixed: k₊₁ = 10 s⁻¹; k₊₂ = 0.1 s⁻¹; and k⁻¹= 0.01 s⁻¹. The ratio, k₊₁/k₊₂, of the two forward rate constants is thus fixed at 100. The fourth rate constant, k₋₂, is varied from 10⁻⁴ to 10⁻⁸ s⁻¹, yielding the five values for the ratio of the two equilibrium constants shown in the figure. The equilibrium concentration of B₁ is [B₁]_eq = C_tk₊₁k₋₂/(k₊₁k₋₂ + k₊₂k₋₁ + k₋₁k₋₂), where C_t is the total protein concentration; [B₂]_eq is obtained by reversing the indices 1 and 2. With the protein is initially in state A, the time dependence of the B₁ concentration is given by $$[{\mathrm{B}}_1]={[{\mathrm{B}}_1]}_{\text{eq}}-({[{\mathrm{B}}_1]}_{\text{eq}}({\lambda }_{+}-{\lambda }_{-}+k_{+1}+k_{-1}-k_{+2}-k_{-2})+2{[{\mathrm{B}}_2]}_{\text{eq}}{k}_{+1})e^{-{\lambda }_{+}t}/2({\lambda }_{+}-{\lambda }_{-})-({[{\mathrm{B}}_1]}_{\text{eq}}({\lambda }_{+}-{\lambda }_{-}-k_{+1}-k_{-1}+k_{+2}+k_{-2})-2{[{\mathrm{B}}_2]}_{\text{eq}}{k}_{+1})e^{-{\lambda }_{-}t}/2({\lambda }_{+}-{\lambda }_{-})$$ where λ_± = [k₊₁ + k₋₁ + k₊₂ + k₋₂ ± ((k₊₁ + k₋₁ − k₊₂ − k₋₂)² + 4k₊₁k₊₂)^1/2]/2. Again [B₂] is obtained by reversing the indices 1 and 2. Thermodynamic control means [B₁]/[B₂] → [B₁]_eq/[B₂]_eq, indicated by the arrows on the right, whereas kinetic control means [B₁]/[B₂] → k₊₁/k₊₂, indicated by the arrow at the top. Note that the two pathways can represent either unimolecular or bimolecular reactions. An example of the latter case is a protein binding with two different ligands; k₊₁ (or k₊₂) is then a pseudo-first-order rate constant given by the product of the ligand binding rate constant and the ligand concentration. The time interval in which kinetic control dominates is shaded in purple; the time interval in which thermodynamic control dominates is shaded in yellow.

Fig. 2

A simple model with intra-state equilibration and inter-state jump. (a) Illustration of the model. The two states, A and B, are represented by boxes; microstates within state A are represented by circles. (b) Time dependences of σ₁ and ρ₁/ρ_Aeq, representing intra-state equilibration, and of ρ_A, representing inter-state jump, for the following parameters values: k₁₂ = k₂₁ = k₁₃ = k₂₃ = 1, k₃₁ = k₃₂ = 2, and k_tr = 0.1. (c) The corresponding results when k₃₁ and k₃₂ are decreased to 0.125 and k_tr is increased to 1. The results are obtained from kinetic simulations of a single molecule. Briefly, the waiting time of the molecule in an initial microstate is generated from an exponential distribution function [see Eq. (2.10)], with the average waiting time equal to the inverse of the sum of the rate constants for all the pathways leaving that microstate. The probability for taking each of these pathways is proportional to the corresponding rate constant. The results shown are the average of 10⁶ repeat simulations; each simulation starts with the molecule in microstate 1. This simulation procedure is similar in spirit to the stochastic simulation algorithm of Gillespie (1977).

Fig. 3

A one-dimensional model for unimolecular reaction. (a) The potential energy function. (b) Results for the rate constant obtained from computer simulations (filled circles) and predicted by Melnikov and Meshkov [Eqs. (3.69) and (3.71); curve], for the potential energy function U(x) = (x² − 1)², with k_BT = 1/4 and m = 1. The simulations results are from Zhou (1989), by fitting the number correlation function [Eq. (3.20)] to an exponential function.

Fig. 4

Interaction energy functions. (a) Centrosymmetric model. The solid curve shows the energy function, with locations of the unbound state (i.e., P + L), transient complex (i.e., P*L), transition state, and native complex (i.e., PL) identified. The dashed curve is after smoothing out the transition state. (b) A smoothed energy function in relative translational (r) and rotational (Ω) space. The native complex is located in the deep well; the transient complex is located at the outer boundary of the energy well.

Fig. 5

Specification of the transient complex for the barnase-barstar protein pair (Alsallaq & Zhou, 2008). (a) Scatter plot of allowed (i.e., clash-free) configurations. Each scatter point represents a cluster of allowed configurations with the indicated contact number (N_c) and angle (χ) of relative rotation. The N_c level defining the transient complex is shown in dark color. (b) Transition of the standard deviation of χ, σ_χ, from the native complex (with high contact numbers) to the unbound state (with low contact numbers). The start of the sharp increase in, σ_χ, as indicated by an arrow, marks the transient complex.