Enzyme Biophysics

The Eyring Equation

Transition state theory (TST) provides the fundamental framework for understanding reaction rates. The key assumption is that the reactants are in quasi-equilibrium with the transition state (activated complex). The rate constant is:

where $\Delta G^\ddagger$ is the free energy of activation, $k_BT/h \approx 6.2 \times 10^{12}$s$^{-1}$ at 25$^\circ$C is the universal attempt frequency, and$R = 8.314$ J/(mol$\cdot$K). The activation free energy can be decomposed:

The Eyring plot ($\ln(k/T)$ vs. $1/T$) gives slope $= -\Delta H^\ddagger/R$ and intercept containing $\Delta S^\ddagger$.

Catalytic Proficiency

The catalytic proficiency measures how much an enzyme accelerates a reaction compared to the uncatalyzed rate in solution:

where $\Delta\Delta G^\ddagger = \Delta G^\ddagger_{\text{uncat}} - \Delta G^\ddagger_{\text{cat}}$is the differential transition state stabilization. Typical values:

• • $\Delta\Delta G^\ddagger \approx 30{-}50$ kJ/mol: rate enhancement $\sim 10^5{-}10^9$
• • $\Delta\Delta G^\ddagger \approx 50{-}80$ kJ/mol: rate enhancement $\sim 10^9{-}10^{14}$
• • $\Delta\Delta G^\ddagger \approx 80{-}100$ kJ/mol: rate enhancement $\sim 10^{14}{-}10^{17}$ (OMP decarboxylase)

Enzymes achieve these enormous rate enhancements through transition state complementarity: the active site binds the transition state more tightly than the substrate by$\Delta\Delta G^\ddagger$. This is the basis for transition state analog drug design: molecules that mimic the transition state geometry bind with extraordinary affinity.

Competitive Inhibition

A competitive inhibitor $I$ binds to the free enzyme at the active site, competing with substrate. The modified scheme is:

The free enzyme is partitioned: $[E]_{\text{free}} = [E]_0 - [ES] - [EI]$. At steady state with $[EI] = [E][I]/K_i$:

$V_{\max}$ is unchanged (at saturating [S], all inhibitor is displaced). $K_M^{\text{app}} = K_M(1 + [I]/K_i)$ increases. On a Lineweaver-Burk plot, lines intersect on the y-axis (same $1/V_{\max}$, different slopes).

Uncompetitive Inhibition

An uncompetitive inhibitor binds only to the ES complex:$ES + I \rightleftharpoons ESI$ with $K_i' = [ES][I]/[ESI]$. The rate equation becomes:

Both $V_{\max}^{\text{app}} = V_{\max}/(1 + [I]/K_i')$ and$K_M^{\text{app}} = K_M/(1 + [I]/K_i')$ decrease by the same factor. Lineweaver-Burk: parallel lines (same slope, different intercepts).

Mixed (Noncompetitive) Inhibition

The inhibitor can bind both free enzyme (with $K_i$) and ES complex (with $K_i'$). When $K_i = K_i'$, this is pure noncompetitive inhibition:

The Lineweaver-Burk lines intersect to the left of the y-axis. When $K_i = K_i'$(pure noncompetitive), they intersect on the x-axis ($K_M$ unchanged,$V_{\max}$ decreases). Most real inhibitors show mixed inhibition with $K_i \neq K_i'$.

MWC (Concerted) Model

As derived in the cooperativity section, the MWC model with allosteric constant $L$and affinity ratio $c = K_R/K_T$ gives the fractional saturation:

An allosteric activator shifts the equilibrium toward R (decreases $L$), while an allosteric inhibitor shifts it toward T (increases $L$). This modulates the apparent affinity without directly competing with the substrate.

KNF (Sequential) Model

The Koshland-Nemethy-Filmer (KNF) model allows each subunit to change conformation independently upon ligand binding. For a dimer, the binding polynomial is:

where $K_t$ is the conformational equilibrium constant for a subunit,$K_s$ is the intrinsic binding constant, and $K_{ss}$ accounts for subunit-subunit interaction when both are in the ligand-bound conformation. The key differences from MWC:

• • MWC: Only positive cooperativity ($n_H \leq n$). Cannot explain negative cooperativity
• • KNF: Can explain both positive AND negative cooperativity
• • MWC: Predicts that unliganded T state is populated. Testable by structural methods
• • KNF: Conformational change is induced by binding. Half-of-sites reactivity

Modern single-molecule experiments on hemoglobin and other allosteric proteins suggest that reality lies between the two models, with elements of both concerted and sequential transitions depending on the system.

Hill Coefficient from MWC Parameters

The Hill coefficient at the midpoint ($\bar{Y} = 0.5$) can be derived from the MWC parameters. For the limit of large $L$ and $c \to 0$ (non-binding T state):

For hemoglobin ($n = 4$, $L \approx 10^5$, $c \approx 0.01$), this yields$n_H \approx 2.8$, matching experimental data. The Hill coefficient is always bounded:$1 \leq n_H \leq n$, and approaches $n$ only in the limit of infinitely strong cooperativity ($L \to \infty$, $c \to 0$).

Waiting Time Distributions

At the single-molecule level, enzyme turnover is a stochastic process. We observe the waiting times $\tau$ between successive catalytic events. For simple Michaelis-Menten kinetics with a single rate-limiting step, the waiting time distribution is exponential:

For an exponential distribution: $\langle\tau\rangle = 1/k_{\text{eff}}$ and the coefficient of variation $\text{CV} = \sigma/\langle\tau\rangle = 1$.

Dynamic Disorder

Landmark experiments on single cholesterol oxidase molecules (Lu et al. 1998) revealed that$\text{CV} > 1$ and waiting times are autocorrelated. This indicates dynamic disorder: the enzyme's catalytic rate fluctuates over time due to slow conformational dynamics.

The distinction between static and dynamic heterogeneity is crucial:

• • Static disorder: Different enzyme molecules have different rates, but each molecule's rate is constant. $\text{CV} > 1$ but no autocorrelation
• • Dynamic disorder: A single molecule's rate fluctuates over time. Both $\text{CV} > 1$ AND positive autocorrelation in successive waiting times

The waiting time autocorrelation function $C(m) = \frac{\langle \tau_i \tau_{i+m}\rangle - \langle\tau\rangle^2}{\langle\tau^2\rangle - \langle\tau\rangle^2}$decays with a timescale reflecting the conformational switching rate. This provides a unique window into protein dynamics that is invisible to ensemble measurements.

The Diffusion Limit

The maximum possible value of $k_{\text{cat}}/K_M$ is set by the rate at which enzyme and substrate can diffuse together. From the Smoluchowski equation:

Enzymes that have evolved to reach this limit are called "perfect" enzymes. The classic example is triosephosphate isomerase(TIM) with $k_{\text{cat}}/K_M \approx 4 \times 10^8$ M$^{-1}$s$^{-1}$.

The Albery-Knowles Model of Enzyme Evolution

Albery and Knowles (1976) proposed that enzyme evolution proceeds through three stages, progressively optimizing the free energy profile:

Stage 1: Uniform binding. The enzyme stabilizes all bound states equally (substrate, transition state, product) by $\Delta G_{\text{bind}}$. This lowers all barriers uniformly but does not change $k_{\text{cat}}/K_M$ because the substrate ground state and transition state are both stabilized.

Stage 2: Differential transition state stabilization. The enzyme evolves to bind the transition state more tightly than the ground state. This increases $k_{\text{cat}}$ while raising $K_M$ (weaker substrate binding relative to TS). The catalytic efficiency $k_{\text{cat}}/K_M$ increases.

Stage 3: Catalytic perfection. All internal barriers are optimized to be equal. The rate-limiting step becomes diffusion-controlled encounter. At this point:

Further evolution of the catalytic machinery cannot improve the rate. Only changes to the diffusion coefficient (electrostatic funneling, as in superoxide dismutase with$k_{\text{cat}}/K_M \approx 7 \times 10^9$ M$^{-1}$s$^{-1}$) can exceed the naive Smoluchowski limit.