Article Information

• PDF (154 kb)
• Full Text with Large Figures
• Cited by in Scopus (16)

PubMed

• Articles by Yu Zhou
• Articles by Anthony Auerbach

Related Articles

• Correction

  Correction Biophysical Journal, Volume 90, Issue 2, 15 January 2006, Pages 709 Full Text | PDF (38 kb)

• Thermodynamically Feasible Kinetic Models of Reaction Networ...

  Thermodynamically Feasible Kinetic Models of Reaction Networks Biophysical Journal, Volume 92, Issue 6, 15 March 2007, Pages 1846-1857 Michael Ederer and Ernst Dieter Gilles Abstract The dynamics of biological reaction networks are strongly constrained by thermodynamics. An holistic understanding of their behavior and regulation requires mathematical models that observe these constraints. However, kinetic models may easily violate the constraints imposed by the principle of detailed balance, if no special care is taken. Detailed balance demands that in thermodynamic equilibrium all fluxes vanish. We introduce a thermodynamic-kinetic modeling (TKM) formalism that adapts the concepts of potentials and forces from irreversible thermodynamics to kinetic modeling. In the proposed formalism, the thermokinetic potential of a compound is proportional to its concentration. The proportionality factor is a compound-specific parameter called capacity. The thermokinetic force of a reaction is a function of the potentials. Every reaction has a resistance that is the ratio of thermokinetic force and reaction rate. For mass-action type kinetics, the resistances are constant. Since it relies on the thermodynamic concept of potentials and forces, the TKM formalism structurally observes detailed balance for all values of capacities and resistances. Thus, it provides an easy way to formulate physically feasible, kinetic models of biological reaction networks. The TKM formalism is useful for modeling large biological networks that are subject to many detailed balance relations. Abstract | Full Text | PDF (281 kb)

• On Imposing Detailed Balance in Complex Reaction Mechanisms

  On Imposing Detailed Balance in Complex Reaction Mechanisms Biophysical Journal, Volume 91, Issue 3, 1 August 2006, Pages 1136-1141 Jin Yang, William J. Bruno, William S. Hlavacek and John E. Pearson Full Text | PDF (105 kb)

• …more

Copyright © 2005 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 89, Issue 6, 3680-3685, 1 December 2005

doi:10.1529/biophysj.105.067215

Biophysical Theory and Modeling

Φ-Value Analysis of a Linear, Sequential Reaction Mechanism: Theory and Application to Ion Channel Gating

Yu Zhou^*, John E. Pearson^† and Anthony Auerbach^*, ,  

^* Center for Single Molecule Biophysics and Department of Physiology & Biophysics, State University of New York at Buffalo, Buffalo, New York 14214
^† Theoretical Biology & Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545

Address reprint requests to A. Auerbach, Tel.: 716-829-2435.

Allosteric proteins are large, multimeric polymers that alternatively adopt more than one stable conformation. In experiments, such conformational changes often appear to be simple, two-state reactions. However, given the size and complexity of proteins, it is reasonable to assume that short-lived intermediate states exist during the course of a global isomerization.

One way to probe the intermediate states of an overall reaction (the transition-state ensemble; TSE) is by Φ-value analysis 1,2,3. A localized perturbation (e.g., a point mutation) that alters the equilibrium constant of a two-state reaction can do so by changing the forward rate constant, the backward rate constant, or both. The value Φ is the slope of the rate-equilibrium free energy relationship (REFER), which is a log-log plot of the forward rate constant versus the equilibrium constant for a perturbation series. The value Φ is a fraction that quantifies the relative extent to which the forward and backward rate constants were altered. A Φ-value of 1 indicates that only the forward rate constant changed and a Φ-value of 0 indicates that only the backward rate constant changed with the perturbation.

The nicotinic acetylcholine receptor (AChR) is an allosteric membrane protein that regulates the flow of ions at the nerve-muscle and other cholinergic synapses. It has five subunits (total MW ∼290 kDa) and is roughly cylindrical in shape (∼170Å×70Å) 4. The ground states of the AChR gating isomerization are called closed (C; ions cannot readily pass through the channel) and open (O; monovalent cations flow rapidly). In single-channel patch-clamp recordings with a time resolution of ∼25μs, AChR gating appears to be a two-state process 5 and the transition from a nonconducting to a conducting conformation occurs within the time resolution of the instrumentation (≤3μs; 6). Nonetheless, both structural and mutagenesis studies suggest that many thousands of atoms move in the course of the gating conformational change.

Recently, numerical methods were used to show that Φ-values depend on the position of the perturbation within a linear reaction chain, with higher values indicating earlier positions in the sequence 7. In addition, the diffusion of energy across a flat TSE landscape was shown to account for many of the essential features of AChR gating. These include the two-state kinetic behavior in patch-clamp recordings, the presence of discrete domains within which residues all have the same Φ-value 8, the organization of these domains approximately along the long axis of the protein 9, and an apparent upper limit to the channel-opening rate constant of ∼1μs⁻¹10. The numerical analysis was limited because only the simplest reaction mechanism was examined (a single rate constant connecting all of the intermediate states), and because this mechanism was probed only by using simulations.

Here, we develop the analytical form of a REFER in the case of a bounded, linear chain of coupled reactions of arbitrary length and having arbitrary connecting rate constants. Although analytical solutions for the net rate-constant of a linear reaction chain have previously been described for specific cases 11,12, we are unaware of a general solution to this problem. We then apply the theory to the pattern of fractional Φ-values that has been observed for diliganded AChR gating.

The numerical calculations were done by using MAPLE 9.5 (Waterloo Associates, Waterloo, Ontario, Canada).

Consider a linear chain of reactions in which two stable end-states (C and O) are separated by an arbitrary number (n) of intermediate states (X):

Let k_f be the inverse of the mean first latency for a complete C→O passage under the condition that k₁, k_2n+2 ≪ all of the other rate constants. Specifically, we assume that the aggregate lifetime of the intermediates, [X₁…X_n], is vanishingly short compared to the lifetimes of the ground states, C and O.

Define r_j as the ratio of the exit rate constants (backward/forward) from X_j,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The denominator of Eq. (7) is the average number of times the C→X₁ transition occurs before the stable state O is reached. This number is a function only of the ratios of exit rate constants from the intermediate states. Hence, the inverse of the denominator of Eq. (7) gives κ as a function of the rate constants (the r-values) of an arbitrary, discrete, finite Markov chain:

(8)

(9)

(10)

We use the following reaction sequence as a numerical example:

Scheme 1The rate constant units are inverse time (e.g., ms⁻¹). For this reaction chain r₁=2, r₂=0.5, r₃=2, r₄=0.25, and r₅=1. We calculate from Eq. (7) that k_f=1.42ms⁻¹, which is in excellent agreement with the value calculated using Q-matrix methods. Increasing the magnitudes of the intermediate rates 10-fold did not alter k_f. In Scheme 2, the C→X₁ transition occurs, on average, approximately seven times before state O is occupied (κ_C→O ≈ 0.14).

Display large version of this figure

Display large version of this figure

The value Φ is the differential coefficient of log k_f/log K_eq, where K_eq is the equilibrium constant for the overall, C ↔ O reaction ([O]/[C] at equilibrium), which is the product of the intermediate equilibrium constants:

(11)

We see from Eq. (11) that K_eq will change equally for an equivalent change in any single r-value. However, Eq. (7) predicts that a change in r₁ will have the greatest effect on k_f, while a change in r_n will have the least effect on k_f. Thus, a change in r₁ will produce a high Φ-value, while an equivalent change in r_n will produce a low Φ-value. Qualitatively, the Φ-value for a perturbation series provides information about the position of the perturbed intermediate in the overall reaction sequence, with higher Φ-values indicating earlier positions in the chain.

We derive a closed-form expression for Φ for a perturbation series that changes a single r-value. The changes in log(k_f) and log(K_eq) consequent to a change in r₁ are

(12)

(13)

(14)

(15)

(16)

Typically, perturbations may change a single intermediate equilibrium constant, which will in turn alter two adjacent r-values. We can reformulate Eqs. (7) to reflect changes in the i^th microstate transition equilibrium constants, K_i,

(17)

(18)

By the application of the chain rule, the Φ-value for a perturbation of a single, intermediate equilibrium constant (K_m) for the m^th step of the reaction chain is

(19)

The Φ-values are not constants but are functions of the specific rate-constants of the model. Φ_m will vary from 0 to 1 as K_m decreases (∞→0). However, for any given value of K_m, the value of Φ_m always becomes smaller as m increases. Thus, for all values of K_eq, the relative value of Φ gives the relative position of the perturbation in the reaction chain.

We can estimate the curvature of a rate-equilibrium free-energy relationship for a bounded, linear sequence of reactions, i.e., the second derivative of log (k_f) versus log K_eq, as

(20)

To demonstrate the utility of this theory, we now apply Eq. (16) to the experimentally determined values for diliganded AChR gating. In AChRs, residues appear to be organized into discrete domains of constant Φ. So far, six such domains have been identified, with ∼Φ-values of 0.93, 0.80, 0.65, 0.54, 0.35, and 0.0. Moreover, these domains are spatially organized, according to Φ-value, approximately along the long axis of the protein.

For the purposes of this analysis, we assume that these six domains move sequentially during AChR diliganded gating: Because the lifetime of [X₁…X₅] is vanishingly brief, we have no information regarding whether the pore is conducting or nonconducting in these intermediate states. Given this reaction mechanism, we use Eq. (16) to solve for four of the five r-values for the intermediate state transitions. The results are r₂=1.15, r₃=0.73, r₄=1.72, and r₅=1.84. From Eq. (16), we can only conclude that r₁=0.13(k₁/k_f).

To solve for r₁, we use the additional experimental result that for perturbations of C↔ X₁, k_f=2ms⁻¹ when K_eq=1 5,16. Thus, from Eqs. (7),

(21)

(22)

These r-values can be incorporated into a kinetic model of diliganded AChR gating (K_eq=1):An energy landscape derived from this reaction sequence is shown in Fig. 1. Because of the overall tilt in the energy barriers of the TSE, according to this scheme a closed AChR makes ∼13 C→X₁ transitions before finally reaching the stable O state, but an open AChR makes only ∼2.8 O→X₅ transitions before reaching the stable C state. From Eqs. (8), we estimate that for diliganded AChR gating, κ_C→O≈0.078 and κ_O→C≈0.35.

Display large version of this figure

Figure 1

Energy landscape for diliganded AChR gating computed from experimental Φ-values (Eq. (16)) and an intrinsic gating rate constant of 2ms⁻¹. The corresponding kinetic model is Scheme 4, with all intermediate states (X₁–X₅) nonconducting and the forward rate constants from the intermediate states arbitrarily set to 10μs⁻¹. The solid line connects the barriers between the intermediate states. The depths of the wells along the reaction surface (open circles) are arbitrary. The transmission coefficients for the forward and reverse overall rate constants are ∼0.08 and ∼0.35, respectively. (Inset) The nonconducting interval duration distribution corresponding to the kinetic scheme. The arrow marks the approximate time resolution of the patch-clamp.

Equation (16) provides a formal relationship between Φ and the sequential position of a transition within a linear reaction chain flanked by absorbing boundaries, under the condition that the aggregate lifetime of the intermediate states is short relative to those of the ground states. Reactions that occur early in this sequence always produce higher Φ-values than those that occur later in the sequence. With this simple reaction mechanism, it is appropriate to classify sites of perturbation that yield higher Φ-values as moving in advance of those that produce lower Φ-values (“early” versus “late”), and sites that generate equivalent Φ-values as “synchronous”.

Our analysis indicates that with this model, Φ-values only provide information about the relative exit probabilities from the intermediate states in a linear series of coupled transitions (the r-values). That is, Φ-values reveal the relative heights of the energy barriers between microstates of the TSE, but provide no information about the absolute energies of the wells along the reaction surface. The assumption that the TSE lifetime is negligible renders the depths of these wells irrelevant.

Scheme 3The overall shape of the barriers along the TSE for diliganded AChR gating (Fig. 1) was derived from two experimental observations, the map of Φ-values and the intrinsic gating rate constant. Previously, numerical analyses showed that a flat and isotropic TSE could account, approximately, for the kinetics of diliganded AChR gating 7. The results shown in Scheme 4 and Fig. 1 do not differ greatly from this simple model. The ratios of exit rate constants from each intermediate state are all <2, and the final barrier of the TS is only ∼1.5 k_BT higher than the initial barrier.

Display large version of this figure

Display large version of this figure

This landscape is provisional because the map of Φ-values is incomplete. The Φ-values that have been reported so far are nearly quantal (ΔΦ≈0.15), with the only deviation being the lack of an experimental Φ-value between Φ=0.35 and 0.0. Although one position (in the M4 segment of the β-subunit) has been found with Φ=0.17 19, we hesitate to add an intermediate state to the TSE based on only a single residue. If such an intermediate was confirmed, the computed landscape would become flatter still (r₁=1.85, r₂=1.15, r₃=0.73, r₄=1.73, r₅=0.95, r₆=0.94; κ_C→O=0.07, κ_O→C=0.17; and when K_eq=1, k₁=28.6ms⁻¹, k₁₄=11.8ms⁻¹).

With further assumptions, we can infer the depths of the wells of the TSE and a kinetic scheme for physiological gating. The conductances and forward rates of the intermediate states determine the first-passage rate between the first nonconducting intermediate (X₁) and the first conducting state, which is, perhaps, equal to the experimental channel-opening speed-limit 10. Further, if we assume that different agonists perturb only the C→X₁ rate constant, then we can generate a kinetic scheme for gating of AChRs having two bound molecules of the transmitter, ACh. Scheme 5, in which all of the X-states are nonconducting, predicts the apparent opening speed limit of ∼0.86μs⁻¹ and the wild-type diliganded gating parameters (with ACh) under standard patch-clamp recording conditions of k_opening ≈ 50,000s⁻¹, k_closing ≈ 2000s⁻¹, and K_eq ≈ 25 (rates are μs⁻¹):The mechanism of Scheme 1, in which the intermediate state transitions are completely coupled, accounts for many of the features of diliganded AChR gating. Moreover, the observation that for unliganded AChR gating, Φ ≈ 1 everywhere 17, is also consistent with the sequential mechanism which predicts that Φ→1 as K_eq→0. However, one significant deviation between theory with experiment is that the experimental REFERs appear to have less curvature than predicted using the simple linear chain mechanism. Although it is difficult to get accurate experimental estimates of curvature (because of a small range of K_eq and experimental scatter), we have not observed the trend in curvature predicted by Eq. (20) in the AChR kinetic data. Indeed, the REFER for position α418, for which Φ=0.5 over an ∼100-fold range of K_eq, has very little scatter and is remarkably linear 19. We do not yet know whether this discrepancy between model and experiment indicates that the linear reaction mechanism is fundamentally incorrect for AChR gating, or if this simple mechanism is simply incomplete and can be modified or extended to accommodate all of the features of AChR gating.

Display large version of this figure

Appendix

For a linear reaction scheme in which two stable end-states (C and O) are separated by n short-lived intermediate states (X)

(a)

and the mean first latency for a complete C→O passage is

(A1)

where

(A2)

and

(A3)

and

(a4)

let

(A5)

where

(A6)

The problem of solving the mean first latency L in Eq. (A1) is to find the solution x for

(A7)

From Eq. (A7),

(A8)

These can be rewritten as

(A9)

so that

(A10)

We define and solve Eq. (A9) by starting with the last equation for x_n+1 and then solving for the rest by using back substitution. This gives

(A11)

From Eqs. (A1), A3 and A5A1, A3, and A5,

(A12)

Note that each x_i ∝ 1/k_2i−1 so that k₁ ≪ k₃, k₅,…k_2n+1. Hence, x₁ ≫ x₂, x₃, …, x_n+1. Therefore,

(A13)

and

(A14)