^{1,a)}, Paul Inder

^{1,b)}and Raymond Kapral

^{1,c)}

### Abstract

A stochastic model for the dynamics of enzymatic catalysis in explicit, effective solvents under physiological conditions is presented. Analytically-computed first passage time densities of a diffusing particle in a spherical shell with absorbing boundaries are combined with densities obtained from explicit simulation to obtain the overall probability density for the total reaction cycle time of the enzymatic system. The method is used to investigate the catalytic transfer of a phosphoryl group in a phosphoglycerate kinase-ADP-bis phosphoglycerate system, one of the steps of glycolysis. The direct simulation of the enzyme-substrate binding and reaction is carried out using an elastic network model for the protein, and the solvent motions are described by multiparticle collision dynamics which incorporates hydrodynamic flow effects. Systems where solvent-enzyme coupling occurs through explicit intermolecular interactions, as well as systems where this coupling is taken into account by including the protein and substrate in the multiparticle collision step, are investigated and compared with simulations where hydrodynamic coupling is absent. It is demonstrated that the flow of solvent particles around the enzyme facilitates the large-scale hinge motion of the enzyme with bound substrates, and has a significant impact on the shape of the probability densities and average time scales of substrate binding for substrates near the enzyme, the closure of the enzyme after binding, and the overall time of completion of the cycle.

Computations were performed on the GPC supercomputer at the SciNet HPC Consortium, which is funded by the Canada Foundation for Innovation under the auspices of Compute Canada, the Government of Ontario, the Ontario Research Fund Research Excellence and the University of Toronto.

This work was supported in part by grants from the Natural Sciences and Engineering Council of Canada. The authors would like to Dr. Ramses van Zon for useful discussions.

I. INTRODUCTION

II. PROTEIN AND ITS CATALYTIC ACTIVITY IN SOLUTION

A. Network model of PGK and interactions with substrate

B. Solvent and its interactions with the protein and substrate

1. Penetrating solventmodel

2. Penetrating solvent without hydrodynamics

III. ENZYMATIC CYCLE DYNAMICS

A. Stochastic model for enzymedynamics

1. Fully stochastic model

IV. SIMULATION OF PGK ENZYME KINETICS

A. Diffusive dynamics

1. Explicit interaction model

2. Penetrating solventmodel

B. Substrate binding and reaction

C. Fully stochastic model

V. SUMMARY

### Key Topics

- Enzymes
- 115.0
- Solvents
- 90.0
- Proteins
- 37.0
- Enzyme kinetics
- 31.0
- Hydrodynamics
- 23.0

## Figures

The open conformation of phosphoglycerate kinase showing the (right) N- and (left) C-terminal domains of the protein. The N-terminal domain binds 3-phosphoglycerate and 1,3-bisphosphoglycerate, while the C-terminal domain binds MgATP and MgADP.

The open conformation of phosphoglycerate kinase showing the (right) N- and (left) C-terminal domains of the protein. The N-terminal domain binds 3-phosphoglycerate and 1,3-bisphosphoglycerate, while the C-terminal domain binds MgATP and MgADP.

(Left) Open conformation of the network model of PGK showing the approach of bPG to the binding pocket of the enzyme. (Right) Protein conformation after substrate binding has resulted in hinge closing to form the closed form.

(Left) Open conformation of the network model of PGK showing the approach of bPG to the binding pocket of the enzyme. (Right) Protein conformation after substrate binding has resulted in hinge closing to form the closed form.

Structure of the model used in the simulation of the diffusive encounters of the substrate with the enzyme and the full dynamics in the enzyme vicinity. The outer circle denotes the spherical volume with radius *r* _{2} containing a single enzyme molecule, while the inner-most circle with radius *r* _{1} denotes the spherical volume around the enzyme within which a full dynamical calculation is carried out. Within *r* _{1} the dynamical evolution is followed until the substrate leaves the spherical volume with radius *r* _{ i } > *r* _{1} or binds and reacts.

Structure of the model used in the simulation of the diffusive encounters of the substrate with the enzyme and the full dynamics in the enzyme vicinity. The outer circle denotes the spherical volume with radius *r* _{2} containing a single enzyme molecule, while the inner-most circle with radius *r* _{1} denotes the spherical volume around the enzyme within which a full dynamical calculation is carried out. Within *r* _{1} the dynamical evolution is followed until the substrate leaves the spherical volume with radius *r* _{ i } > *r* _{1} or binds and reacts.

The simulated value of the diffusion coefficient compared to the estimated time-dependent diffusion coefficient, *D*(*t*), in Eq. (6), versus *t* ^{−1/2} for an isolated Brownian particle with mass ratio = 10, ρ = 10, *k* _{ B } *T* = 1/3, σ = 0.5. From the fit of the data, the value of the diffusion coefficient is *D* = 0.063 in units of ℓ^{2}/τ.

The simulated value of the diffusion coefficient compared to the estimated time-dependent diffusion coefficient, *D*(*t*), in Eq. (6), versus *t* ^{−1/2} for an isolated Brownian particle with mass ratio = 10, ρ = 10, *k* _{ B } *T* = 1/3, σ = 0.5. From the fit of the data, the value of the diffusion coefficient is *D* = 0.063 in units of ℓ^{2}/τ.

Plot of Eq. (13) for the self-diffusion coefficient *D* for a tagged particle in the penetrating solvent model as a function of the mass ratio μ. From this figure we see that if the mass ratio is selected to be μ ≈ 28.5 the value of the diffusion coefficient in the penetrating solvent model, *D* ≈ 0.063, matches that in the explicit interaction model.

Plot of Eq. (13) for the self-diffusion coefficient *D* for a tagged particle in the penetrating solvent model as a function of the mass ratio μ. From this figure we see that if the mass ratio is selected to be μ ≈ 28.5 the value of the diffusion coefficient in the penetrating solvent model, *D* ≈ 0.063, matches that in the explicit interaction model.

Probability densities for the time of substrate binding (black), closing time of the protein (red), and the overall cycle time (blue). These probability densities were constructed from analytical fits to the simulation data for the full solvent model as a function of time. The results are for simulation conditions μ = 10, *k* _{ B } *T* = 1/3, ρ = 10, with a solvent-bead interaction σ = 0.5 ℓ, corresponding to σ = 2.5. Points on the curves are chosen to indicate statistical uncertainties in the construction of the densities.

Probability densities for the time of substrate binding (black), closing time of the protein (red), and the overall cycle time (blue). These probability densities were constructed from analytical fits to the simulation data for the full solvent model as a function of time. The results are for simulation conditions μ = 10, *k* _{ B } *T* = 1/3, ρ = 10, with a solvent-bead interaction σ = 0.5 ℓ, corresponding to σ = 2.5. Points on the curves are chosen to indicate statistical uncertainties in the construction of the densities.

Time series showing the reduction in the number of solvent particles in the vicinity of the bPG substrate as it binds to the enzyme. The red curves show the number of solvent particles in the cell containing the bPG substrate as a function of time, while the black curves denote the distance of the substrate to the enzyme binding site. (Top) σ = 0.5, (bottom) σ = 0.7.

Time series showing the reduction in the number of solvent particles in the vicinity of the bPG substrate as it binds to the enzyme. The red curves show the number of solvent particles in the cell containing the bPG substrate as a function of time, while the black curves denote the distance of the substrate to the enzyme binding site. (Top) σ = 0.5, (bottom) σ = 0.7.

Probability densities *P*(*t*) for the time of substrate binding (top panel), enzyme closing time (middle panel), and total reaction cycle time (bottom panel). The black curves correspond to results for the interacting solvent model, the red curves correspond to the results for the penetrating solvent model with hydrodynamics, and the blue curves are the results for the penetrating solvent model without hydrodynamics.

Probability densities *P*(*t*) for the time of substrate binding (top panel), enzyme closing time (middle panel), and total reaction cycle time (bottom panel). The black curves correspond to results for the interacting solvent model, the red curves correspond to the results for the penetrating solvent model with hydrodynamics, and the blue curves are the results for the penetrating solvent model without hydrodynamics.

Probability density *P* _{conv}(*t*) of the substrate conversion time to products versus time expressed in milliseconds for the explicit solvent model. The other models yield essentially identical results since the substrate conversion is determined primarily by diffusion when the substrate is at physiological concentrations.

Probability density *P* _{conv}(*t*) of the substrate conversion time to products versus time expressed in milliseconds for the explicit solvent model. The other models yield essentially identical results since the substrate conversion is determined primarily by diffusion when the substrate is at physiological concentrations.

Absorption time probability density versus time. The top panel is the absorption time for the absorption onto an inner sphere at *r* _{1} = 7 starting from a radial distance *r* = 10 in length units ℓ. The bottom panel shows the absorption time density (top) and cumulative distribution (bottom) for the outer sphere, where the outer absorbing sphere radius is set to be *r* _{2} = 31.6 and *r* = 10.

Absorption time probability density versus time. The top panel is the absorption time for the absorption onto an inner sphere at *r* _{1} = 7 starting from a radial distance *r* = 10 in length units ℓ. The bottom panel shows the absorption time density (top) and cumulative distribution (bottom) for the outer sphere, where the outer absorbing sphere radius is set to be *r* _{2} = 31.6 and *r* = 10.

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