^{1,a)}and Michael E. Fisher

^{1,b)}

### Abstract

A broad class of chemical kinetic model for mechanoenzymes is analyzed theoretically in order to uncover structural aspects of the underlying free-energy landscape that determine the behavior under large resisting and assisting loads, specifically the turnover rate or, for a translocatory motor protein, the mean velocity, say, . A systematic graphical reduction algorithm is presented that provides explicit analytical expressions for mean occupation times in individual biomechanochemical states, for the splitting or backward/forward fractions, for the overall mean dwell time, and for the turnover rate. Application to the previously studied -state sequential and -parallel-chain models provides explicit structural criteria (independent of the zero-load transition rates) that determine whether diverges to large values *or*, conversely, exhibits extrema and converges to a vanishing value as the externally imposed load grows. Closed-form analytical extensions accommodate side-chain and looped side-chain reaction sequences in the enzymatic cycle. A general *divided-pathway* model is analyzed in detail.

The authors much appreciate correspondence and interactions with Professor A. B. Kolomeisky and Dr. Martin Lindén, and the interest of Professor D. Thirumalai.

I. INTRODUCTION

II. ENERGY LANDSCAPE STRUCTURE AND LARGE-LOAD BEHAVIOR

A. Basic sequential model

B. Parallel-chain models

C. Side-chain reactions

III. GENERAL KINETIC SCHEMES AND A SOLUTION ALGORITHM

A. Enzymatic cycles

B. Graphical reduction

C. Turnover rate

IV. APPLICATION TO SIMPLE CASES

A. -state sequential model

B. Parallel-chain models

C. Analysis of side-chain reactions

D. Looped side-chain reactions

V. DIVIDED-PATHWAY MODEL

### Key Topics

- Enzymes
- 17.0
- Free energy
- 8.0
- Motor proteins
- 7.0
- Reaction kinetics modeling
- 6.0
- Gibbs free energy
- 3.0

## Figures

Examples of well-known kinetic models for enzymes: (a) sequential -state model; (b) parallel-chain model with one sequential -state pathway and a second -state pathway. Extensions yielding tractable models: (c) a side chain attached to a state () of a kinetic scheme; (d) a looped side chain at a state ().

Examples of well-known kinetic models for enzymes: (a) sequential -state model; (b) parallel-chain model with one sequential -state pathway and a second -state pathway. Extensions yielding tractable models: (c) a side chain attached to a state () of a kinetic scheme; (d) a looped side chain at a state ().

Example of a general -state enzymatic cycle. The various dots and circles and the square labeled [0] represent the biochemical states () of the enzyme, while the bonds (or links) connect states that can be reversibly transformed into one another. Those states which have transitions to more than two other distinct states are *nodes* and are indicated by open circles, distinguishing them from the remaining states marked by solid dots. The waiting (or initial) state is [0].

Example of a general -state enzymatic cycle. The various dots and circles and the square labeled [0] represent the biochemical states () of the enzyme, while the bonds (or links) connect states that can be reversibly transformed into one another. Those states which have transitions to more than two other distinct states are *nodes* and are indicated by open circles, distinguishing them from the remaining states marked by solid dots. The waiting (or initial) state is [0].

The same enzymatic cycle depicted in Fig. 2 but now unfolded into a (potentially infinite) periodic chain of which only two unit cells, and , are shown. The absorbing boundaries introduced on the left and on the right at states and serve to cut the chain and generate a two-cell representation that then specifies a first-passage problem. This enables the efficient calculation of the average characteristics of the enzymatic cycles.

The same enzymatic cycle depicted in Fig. 2 but now unfolded into a (potentially infinite) periodic chain of which only two unit cells, and , are shown. The absorbing boundaries introduced on the left and on the right at states and serve to cut the chain and generate a two-cell representation that then specifies a first-passage problem. This enables the efficient calculation of the average characteristics of the enzymatic cycles.

Illustration of a bridge () connecting nodes () and () with associated bridge sites/states, , where the first and last bridge states are identified with the nodes () and (). The probability flow properties of the bridges are used in the algorithm to reduce the complexity of models of interest.

Illustration of a bridge () connecting nodes () and () with associated bridge sites/states, , where the first and last bridge states are identified with the nodes () and (). The probability flow properties of the bridges are used in the algorithm to reduce the complexity of models of interest.

Reduction of the basic sequential model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries and mean occupation times indicated; (c) reduced scheme with effective rates, and , see text. Note the individual occupation times, and , indicated.

Reduction of the basic sequential model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries and mean occupation times indicated; (c) reduced scheme with effective rates, and , see text. Note the individual occupation times, and , indicated.

Reduction of the parallel-chain model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries; (c) reduced scheme with the parallel effective rates: , , , and ; (d) final reduction to an -state model with effective rates and .

Reduction of the parallel-chain model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries; (c) reduced scheme with the parallel effective rates: , , , and ; (d) final reduction to an -state model with effective rates and .

A side chain extending from a state(). The integrated probability flow along a side chain vanishes.

A side chain extending from a state(). The integrated probability flow along a side chain vanishes.

A looped side chain attached to a state (). This motif like the side chain in Fig. 7 is an example of an additional structure that does not contribute to the integrated probability flow distribution.

A looped side chain attached to a state (). This motif like the side chain in Fig. 7 is an example of an additional structure that does not contribute to the integrated probability flow distribution.

The basic divided-pathway model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries; (c) a series of steps to reduce the original kinetic scheme to an sequential model and, thence, to the simplest model with appropriate effective rates, see text.

The basic divided-pathway model: (a) cyclic representation; (b) two-cell representation with absorbing boundaries; (c) a series of steps to reduce the original kinetic scheme to an sequential model and, thence, to the simplest model with appropriate effective rates, see text.

Examples of small enzymatic cycles belonging to the class of divided-pathway models; compare with Fig. 9. The mean properties of all these cycles can be extracted explicitly from the unified solution presented in Sec. V, see Eqs. (5.5)–(5.15).

Examples of small enzymatic cycles belonging to the class of divided-pathway models; compare with Fig. 9. The mean properties of all these cycles can be extracted explicitly from the unified solution presented in Sec. V, see Eqs. (5.5)–(5.15).

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