1
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Zhang Z, Lu Z. Nonequilibrium Theoretical Framework and Universal Design Principles of Oscillation-Driven Catalysis. J Phys Chem Lett 2023; 14:7541-7548. [PMID: 37586077 DOI: 10.1021/acs.jpclett.3c01677] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/18/2023]
Abstract
At stationary environmental conditions, a catalyst's reaction kinetics may be restricted by its available designs and thermodynamic laws. Thus, its stationary performances may experience practical or theoretical restraints (e.g., catalysts cannot invert the spontaneous direction of a chemical reaction). However, many works have reported that if environments change rapidly, catalysts can be driven away from stationary states and exhibit anomalous performance. We present a general geometric nonequilibrium theory to explain anomalous catalytic behaviors driven by rapidly oscillating environments where stationary-environment restraints are broken. It leads to a universal design principle of novel catalysts with oscillation-pumped performances. Even though a single free energy landscape cannot describe catalytic kinetics at various environmental conditions, we propose a novel control-conjugate landscape to encode the reaction kinetics over a range of control parameters λ, inspired by the Arrhenius form. The control-conjugate landscape significantly simplifies the design principle applicable to large-amplitude environmental oscillations.
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Affiliation(s)
- Zhongmin Zhang
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, United States
| | - Zhiyue Lu
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, United States
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2
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Zhang Z, Du V, Lu Z. Energy landscape design principle for optimal energy harnessing by catalytic molecular machines. Phys Rev E 2023; 107:L012102. [PMID: 36797891 DOI: 10.1103/physreve.107.l012102] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/24/2022] [Accepted: 12/27/2022] [Indexed: 06/18/2023]
Abstract
Under temperature oscillation, cyclic molecular machines such as catalysts and enzymes could harness energy from the oscillatory bath and use it to drive other processes. Using an alternative geometrical approach, under fast temperature oscillation, we derive a general design principle for obtaining the optimal catalytic energy landscape that can harness energy from a temperature-oscillatory bath and use it to invert a spontaneous reaction. By driving the reaction against the spontaneous direction, the catalysts convert low free-energy product molecules to high free-energy reactant molecules. The design principle, derived for arbitrary cyclic catalysts, is expressed as a simple quadratic objective function that only depends on the reaction activation energies, and is independent of the temperature protocol. Since the reaction activation energies are directly accessible by experimental measurements, the objective function can be directly used to guide the search for optimal energy-harvesting catalysts.
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Affiliation(s)
- Zhongmin Zhang
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, USA
| | - Vincent Du
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, USA
| | - Zhiyue Lu
- Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, USA
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3
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Zhang D, Ouyang Q. Nonequilibrium Thermodynamics in Biochemical Systems and Its Application. ENTROPY (BASEL, SWITZERLAND) 2021; 23:271. [PMID: 33668768 PMCID: PMC7996154 DOI: 10.3390/e23030271] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/31/2021] [Revised: 02/16/2021] [Accepted: 02/17/2021] [Indexed: 12/05/2022]
Abstract
Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks' fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.
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Affiliation(s)
- Dongliang Zhang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China;
| | - Qi Ouyang
- The State Key Laboratory for Artificial Microstructures and Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China;
- Center for Quantitative Biology and Peking-Tsinghua Center for Life Sciences, AAIC, Peking University, Beijing 100871, China
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4
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Liu Q, Wang J. Quantifying the flux as the driving force for nonequilibrium dynamics and thermodynamics in non-Michaelis-Menten enzyme kinetics. Proc Natl Acad Sci U S A 2020; 117:923-930. [PMID: 31879351 PMCID: PMC6969527 DOI: 10.1073/pnas.1819572117] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The driving force for active physical and biological systems is determined by both the underlying landscape and nonequilibrium curl flux. While landscape can be experimentally quantified from the histograms of the collected real-time trajectories of the observables, quantifying the experimental flux remains challenging. In this work, we studied the single-molecule enzyme dynamics of horseradish peroxidase with dihydrorhodamine 123 and hydrogen peroxide (H2O2) as substrates. Surprisingly, significant deviations in the kinetics from the conventional Michaelis-Menten reaction rate were observed. Instead of a linear relationship between the inverse of the enzyme kinetic rate and the inverse of substrate concentration, a nonlinear relationship between the two emerged. We identified nonequilibrium flux as the origin of such non-Michaelis-Menten enzyme rate behavior. Furthermore, we quantified the nonequilibrium flux from experimentally obtained fluorescence correlation spectroscopy data and showed this flux to led to the deviations from the Michaelis-Menten kinetics. We also identified and quantified the nonequilibrium thermodynamic driving forces as the chemical potential and entropy production for such non-Michaelis-Menten kinetics. Moreover, through isothermal titration calorimetry measurements, we identified and quantified the origin of both nonequilibrium dynamic and thermodynamic driving forces as the heat absorbed (energy input) into the enzyme reaction system. Furthermore, we showed that the nonequilibrium driving forces led to time irreversibility through the difference between the forward and backward directions in time and high-order correlations were associated with the deviations from Michaelis-Menten kinetics. This study provided a general framework for experimentally quantifying the dynamic and thermodynamic driving forces for nonequilibrium systems.
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Affiliation(s)
- Qiong Liu
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, 130022 Jilin, China
| | - Jin Wang
- Department of Chemistry, Physics and Applied Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3400
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5
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Abstract
Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.
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Affiliation(s)
- Aidan I Brown
- Department of Physics , University of California, San Diego , La Jolla , California 92093 , United States
| | - David A Sivak
- Department of Physics , Simon Fraser University , Burnaby , British Columbia V5A 1S6 , Canada
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6
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Xie P, Guo SK, Chen H. ATP-Concentration- and Force-Dependent Chemomechanical Coupling of Kinesin Molecular Motors. J Chem Inf Model 2018; 59:360-372. [PMID: 30500195 DOI: 10.1021/acs.jcim.8b00577] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
A model is presented for the chemomechanical coupling of kinesin motors, which proposes that the rate constants of the chemical reaction are independent of the external force. On the basis of the model, we study theoretically the movement dynamics of the motors under varying external force and ATP concentration, such as the forward to backward stepping ratio, velocity, dwell time between two mechanical steps, stall force, and so on. The theoretical results reproduce quantitatively the diverse and even contradictory available single-molecule experimental data for different species of the motors. Furthermore, we study the dependence of the chemomechanical coupling ratio on ATP concentration and external force, with both ATP concentration and external force having large effects on the chemomechanical coupling.
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Affiliation(s)
- Ping Xie
- School of Materials Science and Energy Engineering , FoShan University , Guangdong , 528000 , China.,Key Laboratory of Soft Matter Physics and Beijing National Laboratory for Condensed Matter Physics , Institute of Physics, Chinese Academy of Sciences , Beijing 100190 , China
| | - Si-Kao Guo
- School of Materials Science and Energy Engineering , FoShan University , Guangdong , 528000 , China.,Key Laboratory of Soft Matter Physics and Beijing National Laboratory for Condensed Matter Physics , Institute of Physics, Chinese Academy of Sciences , Beijing 100190 , China
| | - Hong Chen
- School of Materials Science and Energy Engineering , FoShan University , Guangdong , 528000 , China
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7
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Brown AI, Sivak DA. Allocating and Splitting Free Energy to Maximize Molecular Machine Flux. J Phys Chem B 2018; 122:1387-1393. [PMID: 29290114 DOI: 10.1021/acs.jpcb.7b10621] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Biomolecular machines transduce between different forms of energy. These machines make directed progress and increase their speed by consuming free energy, typically in the form of nonequilibrium chemical concentrations. Machine dynamics are often modeled by transitions between a set of discrete metastable conformational states. In general, the free-energy change associated with each transition can increase the forward rate constant, decrease the reverse rate constant, or both. In contrast to previous optimizations, we find that in general flux is maximized neither by devoting all free-energy changes to increasing forward rate constants nor by solely decreasing reverse rate constants. Instead, the optimal free-energy splitting depends on the detailed dynamics. Extending our analysis to machines with vulnerable states (from which they can break down), in the strong driving corresponding to in vivo cellular conditions, processivity is maximized by reducing the occupation of the vulnerable state.
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Affiliation(s)
- Aidan I Brown
- Department of Physics, Simon Fraser University , Burnaby, British Columbia V5A1S6, Canada
| | - David A Sivak
- Department of Physics, Simon Fraser University , Burnaby, British Columbia V5A1S6, Canada
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8
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Ai G, Liu P, Ge H. Torque-coupled thermodynamic model for F_{o}F_{1}-ATPase. Phys Rev E 2017; 95:052413. [PMID: 28618520 DOI: 10.1103/physreve.95.052413] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2016] [Indexed: 01/23/2023]
Abstract
F_{o}F_{1}-ATPase is a motor protein complex that utilizes transmembrane ion flow to drive the synthesis of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and phosphate (Pi). While many theoretical models have been proposed to account for its rotary activity, most of them focus on the F_{o} or F_{1} portions separately rather than the complex as a whole. Here, we propose a simple but new torque-coupled thermodynamic model of F_{o}F_{1}-ATPase. Solving this model at steady state, we find that the monotonic variation of each portion's efficiency becomes much more robust over a wide range of parameters when the F_{o} and F_{1} portions are coupled together, as compared to cases when they are considered separately. Furthermore, the coupled model predicts the dependence of each portion's kinetic behavior on the parameters of the other. Specifically, the power and efficiency of the F_{1} portion are quite sensitive to the proton gradient across the membrane, while those of the F_{o} portion as well as the related Michaelis constants for proton concentrations respond insensitively to concentration changes in the reactants of ATP synthesis. The physiological proton gradient across the membrane in the F_{o} portion is also shown to be optimal for the Michaelis constants of ADP and phosphate in the F_{1} portion during ATP synthesis. Together, our coupled model is able to predict key dynamic and thermodynamic features of the F_{o}F_{1}-ATPase in vivo semiquantitatively, and suggests that such coupling approach could be further applied to other biophysical systems.
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Affiliation(s)
- Guangkuo Ai
- Beijing International Center for Mathematical Research and School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
| | - Pengfei Liu
- Applied and Computational Mathematics, California Institute of Technology, Pasadena, California 91125, USA
| | - Hao Ge
- Beijing International Center for Mathematical Research and Biodynamic Optical Imaging Center, Peking University, Beijing 100871, People's Republic of China
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9
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Hwang W, Hyeon C. Quantifying the Heat Dissipation from a Molecular Motor's Transport Properties in Nonequilibrium Steady States. J Phys Chem Lett 2017; 8:250-256. [PMID: 27983853 DOI: 10.1021/acs.jpclett.6b02657] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Theoretical analysis, which maps single-molecule time trajectories of a molecular motor onto unicyclic Markov processes, allows us to evaluate the heat dissipated from the motor and to elucidate its dependence on the mean velocity and diffusivity. Unlike passive Brownian particles in equilibrium, the velocity and diffusion constant of molecular motors are closely inter-related. In particular, our study makes it clear that the increase of diffusivity with the heat production is a natural outcome of active particles, which is reminiscent of the recent experimental premise that the diffusion of an exothermic enzyme is enhanced by the heat released from its own catalytic turnover. Compared with freely diffusing exothermic enzymes, kinesin-1, whose dynamics is confined on one-dimensional tracks, is highly efficient in transforming conformational fluctuations into a locally directed motion, thus displaying a significantly higher enhancement in diffusivity with its turnover rate. Putting molecular motors and freely diffusing enzymes on an equal footing, our study offers a thermodynamic basis to understand the heat-enhanced self-diffusion of exothermic enzymes.
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Affiliation(s)
- Wonseok Hwang
- Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
| | - Changbong Hyeon
- Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
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10
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Qian H, Kjelstrup S, Kolomeisky AB, Bedeaux D. Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2016; 28:153004. [PMID: 26986039 DOI: 10.1088/0953-8984/28/15/153004] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Nonequilibrium thermodynamics (NET) investigates processes in systems out of global equilibrium. On a mesoscopic level, it provides a statistical dynamic description of various complex phenomena such as chemical reactions, ion transport, diffusion, thermochemical, thermomechanical and mechanochemical fluxes. In the present review, we introduce a mesoscopic stochastic formulation of NET by analyzing entropy production in several simple examples. The fundamental role of nonequilibrium steady-state cycle kinetics is emphasized. The statistical mechanics of Onsager's reciprocal relations in this context is elucidated. Chemomechanical, thermomechanical, and enzyme-catalyzed thermochemical energy transduction processes are discussed. It is argued that mesoscopic stochastic NET in phase space provides a rigorous mathematical basis of fundamental concepts needed for understanding complex processes in chemistry, physics and biology. This theory is also relevant for nanoscale technological advances.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
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11
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Cao Y, Gong Z, Quan HT. Thermodynamics of information processing based on enzyme kinetics: An exactly solvable model of an information pump. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062117. [PMID: 26172671 DOI: 10.1103/physreve.91.062117] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2015] [Indexed: 06/04/2023]
Abstract
Motivated by the recent proposed models of the information engine [Proc. Natl. Acad. Sci. USA 109, 11641 (2012)] and the information refrigerator [Phys. Rev. Lett. 111, 030602 (2013)], we propose a minimal model of the information pump and the information eraser based on enzyme kinetics. This device can either pump molecules against the chemical potential gradient by consuming the information to be encoded in the bit stream or (partially) erase the information initially encoded in the bit stream by consuming the Gibbs free energy. The dynamics of this model is solved exactly, and the "phase diagram" of the operation regimes is determined. The efficiency and the power of the information machine is analyzed. The validity of the second law of thermodynamics within our model is clarified. Our model offers a simple paradigm for the investigating of the thermodynamics of information processing involving the chemical potential in small systems.
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Affiliation(s)
- Yuansheng Cao
- School of Physics, Peking University, Beijing 100871, China
| | - Zongping Gong
- School of Physics, Peking University, Beijing 100871, China
| | - H T Quan
- School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
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12
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Zimmermann E, Seifert U. Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022709. [PMID: 25768533 DOI: 10.1103/physreve.91.022709] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2014] [Indexed: 06/04/2023]
Abstract
Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the F(1)-ATPase and the kinesin motor.
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Affiliation(s)
- Eva Zimmermann
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
| | - Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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13
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Lervik A, Kjelstrup S, Qian H. Michaelis–Menten kinetics under non-isothermal conditions. Phys Chem Chem Phys 2015; 17:1317-24. [DOI: 10.1039/c4cp04334k] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We extend the celebrated Michaelis–Menten kinetics description of an enzymatic reaction taking into consideration the presence of a thermal driving force.
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Affiliation(s)
- Anders Lervik
- Department of Chemistry
- Norwegian University of Science and Technology
- Trondheim
- Norway
| | - Signe Kjelstrup
- Department of Chemistry
- Norwegian University of Science and Technology
- Trondheim
- Norway
- Process and Energy Laboratory
| | - Hong Qian
- Department of Applied Mathematics
- University of Washington
- Washington
- USA
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14
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Kolomeisky AB. Motor proteins and molecular motors: how to operate machines at the nanoscale. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2013; 25:463101. [PMID: 24100357 PMCID: PMC3858839 DOI: 10.1088/0953-8984/25/46/463101] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/07/2023]
Abstract
Several classes of biological molecules that transform chemical energy into mechanical work are known as motor proteins or molecular motors. These nanometer-sized machines operate in noisy stochastic isothermal environments, strongly supporting fundamental cellular processes such as the transfer of genetic information, transport, organization and functioning. In the past two decades motor proteins have become a subject of intense research efforts, aimed at uncovering the fundamental principles and mechanisms of molecular motor dynamics. In this review, we critically discuss recent progress in experimental and theoretical studies on motor proteins. Our focus is on analyzing fundamental concepts and ideas that have been utilized to explain the non-equilibrium nature and mechanisms of molecular motors.
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Affiliation(s)
- Anatoly B. Kolomeisky
- Rice University, Department of Chemistry, 6100 Main Street, Houston, TX 77005-1892, USA
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15
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Ge H, Qian H. Dissipation, generalized free energy, and a self-consistent nonequilibrium thermodynamics of chemically driven open subsystems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062125. [PMID: 23848645 DOI: 10.1103/physreve.87.062125] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2013] [Indexed: 05/15/2023]
Abstract
Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed "universe." A question remains with regard to what the minimally required description for the surrounding of such an open driven system is so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system or in a quasisteady state (QSS) with an excess amount of adenosine triphosphate (ATP), adenosine diphosphate (ADP), and inorganic phosphate (Pi). We introduce the key notion of minimal work that is needed, W(min), for the external regenerating system to sustain a NESS (e.g., maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the second law in terms of the same positive entropy production rate. The heat dissipation of a NESS without mechanical output is exactly the W(min). This provides a justification for introducing an ideal external regenerating system and yields a free-energy balance equation between the net free-energy input F(in) and total dissipation F(dis) in an NESS: F(in) consists of chemical input minus mechanical output; F(dis) consists of dissipative heat, i.e. the amount of useful energy becoming heat, which also equals the NESS entropy production. Furthermore, we show that for nonstationary systems, the F(dis) and F(in) correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics and identify a relative entropy H as a generalized free energy. We reach a new formulation of Markovian nonequilibrium thermodynamics based on only the internal kinetic equation without further reference to the intrinsic degree of freedom within each Markov state. It includes an extended free-energy balance and a second law which are valid for driven stochastic dynamics with an ideal external regenerating system. Our result suggests new ingredients for a generalized thermodynamics of self-organization in driven systems.
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Affiliation(s)
- Hao Ge
- Beijing International Center for Mathematical Research (BICMR) and Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, 100871, PRC.
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16
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Seifert U. Stochastic thermodynamics, fluctuation theorems and molecular machines. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2012; 75:126001. [PMID: 23168354 DOI: 10.1088/0034-4885/75/12/126001] [Citation(s) in RCA: 1171] [Impact Index Per Article: 97.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/18/2023]
Abstract
Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.
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Affiliation(s)
- Udo Seifert
- II. Institut für Theoretische Physik, Universität Stuttgart, 70550 Stuttgart, Germany
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17
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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.
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Affiliation(s)
- Huan-Xiang Zhou
- Department of Physics and Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306, USA.
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18
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Slanina F. Interacting molecular motors: efficiency and work fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:061135. [PMID: 20365146 DOI: 10.1103/physreve.80.061135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2009] [Revised: 11/05/2009] [Indexed: 05/29/2023]
Abstract
We investigate the model of "reversible ratchet" with interacting particles, presented by us earlier [F. Slanina, EPL 84, 50009 (2008)]. We further clarify the effect of efficiency enhancement due to interaction and show that it is of energetic origin, rather than a consequence of reduced fluctuations. We also show complicated structures emerging in the interaction and density dependence of the current and response function. The fluctuation properties of the work and input energy indicate in detail the far-from-equilibrium nature of the dynamics.
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Affiliation(s)
- Frantisek Slanina
- Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha, Czech Republic.
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19
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Abstract
We develop a mechanochemical model for the dynamics of cellulase, a two-domain enzyme connected by a peptide linker, as it extracts and hydrolyzes a cellulose polymer from a crystalline substrate. We consider two random walkers, representing the catalytic domain (CD) and the carbohydrate binding module (CBM), whose rates for stepping are biased by the coupling through the linker and the energy required to lift the cellulose polymer from the crystalline surface. Our results show that the linker length and stiffness play a critical role in the cooperative action of the CD and CBM domains and that, for a given linker length, the steady-state hydrolysis shows a maximum at some intermediate linker stiffness. The maximum hydrolysis rate corresponds to a transition of the linker from a compressed to an extended conformation, where the system exhibits maximum fluctuation, as measured by the variance of the separation distance between the two domains and the dispersion around the mean hydrolysis speed. In the range of experimentally known values of the parameters of our model, improving the intrinsic hydrolytic activity of the CD leads to a proportional increase in the overall hydrolysis rate.
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Affiliation(s)
- Christina L. Ting
- Biochemistry and Molecular Biophysics, California Institute of Technology, Pasadena, CA 91125
| | - Dmitrii E. Makarov
- Department of Chemistry and Biochemistry and Institute for Theoretical Chemistry, University of Texas at Austin, Austin, Texas, 78712
| | - Zhen-Gang Wang
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125
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Qian H. Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations. J Phys Chem B 2007; 110:15063-74. [PMID: 16884217 DOI: 10.1021/jp061858z] [Citation(s) in RCA: 125] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Gibbsian equilibrium statistical thermodynamics is the theoretical foundation for isothermal, closed chemical, and biochemical reaction systems. This theory, however, is not applicable to most biochemical reactions in living cells, which exhibit a range of interesting phenomena such as free energy transduction, temporal and spatial complexity, and kinetic proofreading. In this article, a nonequilibrium statistical thermodynamic theory based on stochastic kinetics is introduced, mainly through a series of examples: single-molecule enzyme kinetics, nonlinear chemical oscillation, molecular motor, biochemical switch, and specificity amplification. The case studies illustrate an emerging theory for the isothermal nonequilibrium steady state of open systems.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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21
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Tsygankov D, Lindén M, Fisher ME. Back-stepping, hidden substeps, and conditional dwell times in molecular motors. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021909. [PMID: 17358369 DOI: 10.1103/physreve.75.021909] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2006] [Indexed: 05/14/2023]
Abstract
Processive molecular motors take more-or-less uniformly sized steps, along spatially periodic tracks, mostly forwards but increasingly backwards under loads. Experimentally, the major steps can be resolved clearly within the noise but one knows biochemically that one or more mechanochemical substeps remain hidden in each enzymatic cycle. In order to properly interpret experimental data for back-to-forward step ratios, mean conditional step-to-step dwell times, etc., a first-passage analysis has been developed that takes account of hidden substeps in N -state sequential models. The explicit, general results differ significantly from previous treatments that identify the observed steps with complete mechanochemical cycles; e.g., the mean dwell times tau(+) and tau(-) prior to forward and back steps, respectively, are normally unequal although the dwell times tau(++) and tau(--) between successive forward and back steps are equal. Illustrative (N=2) -state examples display a wide range of behavior. The formulation extends to the case of two or more detectable transitions in a multistate cycle with hidden substeps.
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Affiliation(s)
- Denis Tsygankov
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA.
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22
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Qian H. Cycle kinetics, steady state thermodynamics and motors-a paradigm for living matter physics. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2005; 17:S3783-94. [PMID: 21690724 DOI: 10.1088/0953-8984/17/47/010] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/07/2023]
Abstract
An integration of the stochastic mathematical models for motor proteins with Hill's steady state thermodynamics yields a rather comprehensive theory for molecular motors as open systems in the nonequilibrium steady state. This theory, a natural extension of Gibbs' approach to isothermal molecular systems in equilibrium, is compared with other existing theories with dissipative structures and dynamics. The theory of molecular motors might be considered as an archetype for studying more complex open biological systems such as biochemical reaction networks inside living cells.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
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23
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Möller W, Brown DM, Kreyling WG, Stone V. Ultrafine particles cause cytoskeletal dysfunctions in macrophages: role of intracellular calcium. Part Fibre Toxicol 2005; 2:7. [PMID: 16202162 PMCID: PMC1262770 DOI: 10.1186/1743-8977-2-7] [Citation(s) in RCA: 57] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2005] [Accepted: 10/04/2005] [Indexed: 11/25/2022] Open
Abstract
Background Particulate air pollution is reported to cause adverse health effects in susceptible individuals. Since most of these particles are derived form combustion processes, the primary composition product is carbon with a very small diameter (ultrafine, less than 100 nm in diameter). Besides the induction of reactive oxygen species and inflammation, ultrafine particles (UFP) can cause intracellular calcium transients and suppression of defense mechanisms of alveolar macrophages, such as impaired migration or phagocytosis. Methods In this study the role of intracellular calcium transients caused by UFP was studied on cytoskeleton related functions in J774A.1 macrophages. Different types of fine and ultrafine carbon black particles (CB and ufCB, respectively), such as elemental carbon (EC90), commercial carbon (Printex 90), diesel particulate matter (DEP) and urban dust (UD), were investigated. Phagosome transport mechanisms and mechanical cytoskeletal integrity were studied by cytomagnetometry and cell viability was studied by fluorescence microscopy. Macrophages were exposed in vitro with 100 and 320 μg UFP/ml/million cells for 4 hours in serum free medium. Calcium antagonists Verapamil, BAPTA-AM and W-7 were used to block calcium channels in the membrane, to chelate intracellular calcium or to inhibit the calmodulin signaling pathways, respectively. Results Impaired phagosome transport and increased cytoskeletal stiffness occurred at EC90 and P90 concentrations of 100 μg/ml/million cells and above, but not with DEP or UD. Verapamil and W-7, but not BAPTA-AM inhibited the cytoskeletal dysfunctions caused by EC90 or P90. Additionally the presence of 5% serum or 1% bovine serum albumin (BSA) suppressed the cytoskeletal dysfunctions. Cell viability showed similar results, where co-culture of ufCB together with Verapamil, W-7, FCS or BSA produced less cell dead compared to the particles only.
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Affiliation(s)
- Winfried Möller
- GSF National Research Center for Environment and Health, Clinical research group 'Inflammatory Lung Diseases', Robert Koch Allee 29, D-82131 Munich-Gauting, Germany
- GSF National Research Center for Environment and Health, Institute for Inhalation Biology, and Focus Network Aerosols and Health, Ingolstädter Landstr. 1, D-85746 Neuherberg/München, Germany
| | - David M Brown
- Napier University, School of Life Sciences, Edinburgh EH10 5DT, UK
| | - Wolfgang G Kreyling
- GSF National Research Center for Environment and Health, Institute for Inhalation Biology, and Focus Network Aerosols and Health, Ingolstädter Landstr. 1, D-85746 Neuherberg/München, Germany
| | - Vicki Stone
- Napier University, School of Life Sciences, Edinburgh EH10 5DT, UK
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Saffarian S, Collier IE, Marmer BL, Elson EL, Goldberg G. Interstitial collagenase is a Brownian ratchet driven by proteolysis of collagen. Science 2004; 306:108-11. [PMID: 15459390 DOI: 10.1126/science.1099179] [Citation(s) in RCA: 147] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
We show that activated collagenase (MMP-1) moves processively on the collagen fibril. The mechanism of movement is a biased diffusion with the bias component dependent on the proteolysis of its substrate, not adenosine triphosphate (ATP) hydrolysis. Inactivation of the enzyme by a single amino acid residue substitution in the active center eliminates the bias without noticeable effect on rate of diffusion. Monte Carlo simulations using a model similar to a "burnt bridge" Brownian ratchet accurately describe our experimental results and previous observations on kinetics of collagen digestion. The biological implications of MMP-1 acting as a molecular ratchet tethered to the cell surface suggest new mechanisms for its role in tissue remodeling and cell-matrix interaction.
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Affiliation(s)
- Saveez Saffarian
- Department of Biochemistry and Molecular Biophysics, Washington University School of Medicine, St. Louis, MO 63110, USA
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25
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Qian H, Elson EL. Fluorescence correlation spectroscopy with high-order and dual-color correlation to probe nonequilibrium steady states. Proc Natl Acad Sci U S A 2004; 101:2828-33. [PMID: 14970342 PMCID: PMC365705 DOI: 10.1073/pnas.0305962101] [Citation(s) in RCA: 60] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2003] [Accepted: 12/19/2003] [Indexed: 11/18/2022] Open
Abstract
In living cells, biochemical reaction networks often function in nonequilibrium steady states. Under these conditions, the networks necessarily have cyclic reaction kinetics that are maintained by sustained constant input and output, i.e., pumping. To differentiate this state from an equilibrium state without flux, we propose a microscopic method based on concentration fluctuation measurements, via fluorescence correlation spectroscopy, and statistical analyses of high-order correlations and cross correlations beyond the standard fluorescence correlation spectroscopy autocorrelation. We show that, for equilibrium systems with time reversibility, the correlation functions possess certain symmetries, the violation of which is a measure of steady-state fluxes in reaction cycles. This result demonstrates the theoretical basis for experimentally measuring reaction fluxes in a biochemical network in situ and the importance of single-molecule measurements in providing fundamental information on nonequilibrium steady-states in biochemistry.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
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26
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Qian H. Motor protein with nonequilibrium potential: Its thermodynamics and efficiency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:012901. [PMID: 14995657 DOI: 10.1103/physreve.69.012901] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2002] [Revised: 03/06/2003] [Indexed: 05/24/2023]
Abstract
A nonequilibrium potential function is introduced for a motor protein modeled by a rectified Brownian motion. This result provides a concrete case for a class of nonequilibrium systems in steady state with dissipation which possess a potential function. The potential micro is a natural generalization of the chemical potential for isothermal chemical species and micro=const if and only if the system is in an equilibrium. The steady-state flux J proportional, variant - nabla micro, and the total heat dissipation h(d) equals a surface integral integral microJ.dS, representing the energy input. In terms of micro and h(d) the thermodynamic energy conservation in the mesoscopic stochastic system can be rigorously established and various types of motor efficiency are elucidated.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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Qian H, Beard DA, Liang SD. Stoichiometric network theory for nonequilibrium biochemical systems. EUROPEAN JOURNAL OF BIOCHEMISTRY 2003; 270:415-21. [PMID: 12542691 DOI: 10.1046/j.1432-1033.2003.03357.x] [Citation(s) in RCA: 95] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Abstract
We introduce the basic concepts and develop a theory for nonequilibrium steady-state biochemical systems applicable to analyzing large-scale complex isothermal reaction networks. In terms of the stoichiometric matrix, we demonstrate both Kirchhoff's flux law sigma(l)J(l)=0 over a biochemical species, and potential law sigma(l) mu(l)=0 over a reaction loop. They reflect mass and energy conservation, respectively. For each reaction, its steady-state flux J can be decomposed into forward and backward one-way fluxes J = J+ - J-, with chemical potential difference deltamu = RT ln(J-/J+). The product -Jdeltamu gives the isothermal heat dissipation rate, which is necessarily non-negative according to the second law of thermodynamics. The stoichiometric network theory (SNT) embodies all of the relevant fundamental physics. Knowing J and deltamu of a biochemical reaction, a conductance can be computed which directly reflects the level of gene expression for the particular enzyme. For sufficiently small flux a linear relationship between J and deltamu can be established as the linear flux-force relation in irreversible thermodynamics, analogous to Ohm's law in electrical circuits.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics and Bioengineering, University of Washington, Seattle, USA.
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Abstract
We provide a stochastic analysis of single-molecule enzymatic reactions that follow Michaelis-Menten kinetics. We show that this system can exhibit oscillatory behavior in the non-equilibrium steady-state at appropriate substrate concentrations. The stochastic model includes both enzyme dynamics and substrate turnover kinetics. The relationship between the probability of substrate survival and the time-correlation of enzyme conformation trajectories is discussed. Deterministic kinetics at large substrate concentrations are obtained as a limit of the stochastic model. We suggest that in addition to fluctuating enzyme conformation, the stochastic nature of substrate concentration fluctuations is another possible source of the complex behavior of single-molecule enzyme kinetics.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, WA Seattle 98195, USA
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Chen YD, Yan B, Rubin RJ. Fluctuations and randomness of movement of the bead powered by a single kinesin molecule in a force-clamped motility assay: Monte Carlo simulations. Biophys J 2002; 83:2360-9. [PMID: 12414673 PMCID: PMC1302325 DOI: 10.1016/s0006-3495(02)75250-8] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022] Open
Abstract
The motility assay of K. Visscher, M. J. Schnitzer, and S. M. Block (Nature, 400:184-189, 1999) in which the movement of a bead powered by a single kinesin motor can be measured is a very useful tool in characterizing the force-dependent steps of the mechanochemical cycle of kinesin motors, because in this assay the external force applied to the bead can be controlled (clamped) arbitrarily. However, because the bead is elastically attached to the motor and the response of the clamp is not fast enough to compensate the Brownian motion of the bead, interpretation or analysis of the data obtained from the assay is not trivial. In a recent paper (Y. Chen and B. Yan, Biophys. Chem. 91:79-91, 2001), we showed how to evaluate the mean velocity of the bead and the motor in the motility assay for a given mechanochemical cycle. In this paper we extend the study to the evaluation of the fluctuation or the randomness of the velocity using a Monte Carlo simulation method. Similar to the mean, we found that the randomness of the velocity of the motor is also influenced by the parameters that affect the dynamic behavior of the bead, such as the viscosity of the medium, the size of the bead, the stiffness of the elastic element connecting the bead and the motor, etc. The method presented in this paper should be useful in modeling the kinetic mechanism of any processive motor (such as conventional kinesin and myosin V) based on measured force-clamp motility data.
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Affiliation(s)
- Yi-der Chen
- Mathematical Research Branch, National Institute of Diabetes, Digestive and Kidney Diseases, National Institutes of Health, BSA Building Suite 350, 9000 Rockville Pike, Bethesda, MD 20892-2690, USA.
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Qian H. Equations for Stochastic Macromolecular Mechanics of Single Proteins: Equilibrium Fluctuations, Transient Kinetics, and Nonequilibrium Steady-State. J Phys Chem B 2002. [DOI: 10.1021/jp013143w] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195
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Abstract
In the past two years, several different RNA trafficking pathways have been characterized in oligodendrocytes; similar trafficking pathways have been discovered in neuronal and retroviral systems; co-assembly of multiple different RNAs into the same granules has been analyzed as a mechanism for coordinating gene expression; and a new hypothesis for RNA trafficking, based on the balance of power between kinesin and dynein in individual RNA granules, has been proposed.
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Affiliation(s)
- J H Carson
- Department of Biochemistry, University of Connecticut Health Center, Farmington, CT 06030, USA.
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32
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Qian H. Nonequilibrium steady-state circulation and heat dissipation functional. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:022101. [PMID: 11497631 DOI: 10.1103/physreve.64.022101] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2000] [Indexed: 05/23/2023]
Abstract
A nonequilibrium steady-state (NESS), different from an equilibrium, is sustained by circular balance rather than detailed balance. The circular fluxes are driven by energy input and heat dissipation, accompanied by a positive entropy production. Based on a Master equation formalism for NESS, we show the circulation is intimately related to the recently studied Gallavotti-Cohen symmetry of heat dissipation functional, which in turn suggests a Boltzmann's formulalike relation between rate constants and energy in NESS. Expanding this unifying view on NESS to diffusion is discussed.
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Affiliation(s)
- H Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA.
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Kuthan H. Self-organisation and orderly processes by individual protein complexes in the bacterial cell. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2001; 75:1-17. [PMID: 11311713 DOI: 10.1016/s0079-6107(00)00023-7] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In the bacterial cell, individual multimeric proteins and multiprotein assemblies perform and control orderly processes. Individual motor enzyme complexes accomplish highly complex functions, such as nucleic acid and protein syntheses, with impressive efficiency and fidelity. Lac operon repression by the lac repressor is effectively controlled via a single molecular switch. There are only few copies of, for example, DNA polymerase holoenzyme and lac repressor and few specific target molecules/sites, with which these protein complexes interact, present in a single E. coli cell. These interactive processes take place in submicron-sized spaces characterised by extreme crowding (volume exclusion) of macromolecules and small molecules, heterogeneity and non-ideality. Recent evidence reinforces the fundamental difference of the cytoplasmic as compared with in vitro ("test tube") reaction conditions. This is reflected in the breakdown of the applicability of "bulk phase" thermodynamic, macroscopic chemical kinetic and diffusion laws to interactions of individual macromolecules and target sites in a single cell. Stochastic kinetic models and stochastic simulations enable the statistical description and analysis of biochemical reactions and binding processes which involve small numbers of reactants. New unifying concepts and models are required for the quantitative understanding of the microscopic self-organisation of multi-protein complexes and the dynamic order at the single-protein assembly and single-switch level in the living cell.
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Chen YD, Yan B. Theoretical formalism for bead movement powered by single two-headed motors in a motility assay. Biophys Chem 2001; 91:79-91. [PMID: 11403886 DOI: 10.1016/s0301-4622(01)00153-3] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Kinesins and dyneins are protein motors that can use the free energy of ATP hydrolysis to carry a cargo and move uni-directionally along a microtubule filament. The purpose of this paper is to derive the formalism connecting the ATP-driven translocation reactions of these motors on microtubule filaments and the movement of the bead carried by the motor in a motility assay in which the bead is clamped at an arbitrary constant force. The formalism is thus useful in elucidating the load-dependent kinetic mechanism of the free-energy transduction of the motor using the mechanical data obtained from the motility assay. The formalism is also useful in assessing the effect on the measured motility data of various physical and hydrodynamic parameters of the assay, such as the size of the bead, the viscosity of the medium, the stiffness of the elastic element connecting the motor and the bead, etc. In a previous paper [Biophys. J. 67 (2000) 313] (hereafter referred to as paper I), we have derived the formalism for the case that the motor in the assay has only one head. In this paper we extend the derivation to the case that the motor is two-headed. The formalism is derived based on a simple two-state hand-over-hand model for the movement of the motor on microtubule, but can be easily extended to more complicated kinetic models. Effects of various hydrodynamic parameters on the velocity of the bead are studied with numerical calculations of the model. The difference between the formalism presented in this paper and the widely used "chemical" formalism, in which the movement of the kinesin and the bead is described by pure chemical reactions, is discussed.
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Affiliation(s)
- Y D Chen
- Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, MD 20892-2690, USA.
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Qian H, Qian M. Pumped biochemical reactions, nonequilibrium circulation, and stochastic resonance. PHYSICAL REVIEW LETTERS 2000; 84:2271-2274. [PMID: 11017261 DOI: 10.1103/physrevlett.84.2271] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/1999] [Indexed: 05/23/2023]
Abstract
Based on a master equation formalism for mesoscopic, unimolecular biochemical reactions, we show the periodic oscillation arising from severe nonequilibrium pumping is intimately related to the periodic motion in recently studied stochastic resonance (SR). The white noise in SR is naturally identified with the temperature in the biochemical reactions; the drift in the SR is associated with the circular flux in nonequilibrium steady state (NESS). As in SR, an optimal temperature for biochemical oscillation is shown to exist. A unifying framework for Hill's theory of NESS and the SR without periodic forcing is presented. The new formalism provides an analytically solvable model for SR.
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Affiliation(s)
- H Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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