1
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Fernandes Martins G, Horowitz JM. Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response. Phys Rev E 2023; 108:044113. [PMID: 37978593 DOI: 10.1103/physreve.108.044113] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2023] [Accepted: 08/07/2023] [Indexed: 11/19/2023]
Abstract
The limits on a system's response to external perturbations inform our understanding of how physical properties can be shaped by microscopic characteristics. Here, we derive constraints on the steady-state nonequilibrium response of physical observables in terms of the topology of the microscopic state space and the strength of thermodynamic driving. Notably, evaluation of these limits requires no kinetic information beyond the state-space structure. When applied to models of receptor binding, we find that sensitivity is bounded by the steepness of a Hill function with a Hill coefficient enhanced by the chemical driving beyond the structural equilibrium limit.
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Affiliation(s)
| | - Jordan M Horowitz
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Department of Biophysics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48104, USA
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2
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Owen JA, Talla P, Biddle JW, Gunawardena J. Thermodynamic bounds on ultrasensitivity in covalent switching. Biophys J 2023; 122:1833-1845. [PMID: 37081788 PMCID: PMC10209043 DOI: 10.1016/j.bpj.2023.04.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2022] [Revised: 03/29/2023] [Accepted: 04/13/2023] [Indexed: 04/22/2023] Open
Abstract
Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants and thereby reflect fundamental physical constraints on covalent switching.
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Affiliation(s)
- Jeremy A Owen
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
| | | | - John W Biddle
- Department of Systems Biology, Harvard Medical School, Boston, Massachusetts
| | - Jeremy Gunawardena
- Department of Systems Biology, Harvard Medical School, Boston, Massachusetts.
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3
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Jones RD, Jones AM. Model of ligand-triggered information transmission in G-protein coupled receptor complexes. Front Endocrinol (Lausanne) 2023; 14:1111594. [PMID: 37361529 PMCID: PMC10286511 DOI: 10.3389/fendo.2023.1111594] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 11/29/2022] [Accepted: 03/21/2023] [Indexed: 06/28/2023] Open
Abstract
We present a model for the effects of ligands on information transmission in G-Protein Coupled Receptor (GPCR) complexes. The model is built ab initio entirely on principles of statistical mechanics and tenets of information transmission theory and was validated in part using agonist-induced effector activity and signaling bias for the angiotensin- and adrenergic-mediated signaling pathways, with in vitro observations of phosphorylation sites on the C tail of the GPCR complex, and single-cell information-transmission experiments. The model extends traditional kinetic models that form the basis for many existing models of GPCR signaling. It is based on maximizing the rates of entropy production and information transmission through the GPCR complex. The model predicts that (1) phosphatase-catalyzed reactions, as opposed to kinase-catalyzed reactions, on the C-tail and internal loops of the GPCR are responsible for controlling the signaling activity, (2) signaling favors the statistical balance of the number of switches in the ON state and the number in the OFF state, and (3) biased-signaling response depends discontinuously on ligand concentration.
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Affiliation(s)
- Roger D. Jones
- Department of Biology, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States
- European Centre for Living Technology, Ca’ Foscari University of Venice, Venice, Italy
- Systems Engineering and Research Center, Stevens Institute of Technology, Hoboken, NJ, United States
| | - Alan M. Jones
- Department of Biology, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States
- Department of Pharmacology, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States
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4
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Chun HM, Horowitz JM. Trade-offs between number fluctuations and response in nonequilibrium chemical reaction networks. J Chem Phys 2023; 158:2888610. [PMID: 37144710 DOI: 10.1063/5.0148662] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2023] [Accepted: 04/17/2023] [Indexed: 05/06/2023] Open
Abstract
We study the response of chemical reaction networks driven far from equilibrium to logarithmic perturbations of reaction rates. The response of the mean number of a chemical species is observed to be quantitively limited by number fluctuations and the maximum thermodynamic driving force. We prove these trade-offs for linear chemical reaction networks and a class of nonlinear chemical reaction networks with a single chemical species. Numerical results for several model systems support the conclusion that these trade-offs continue to hold for a broad class of chemical reaction networks, though their precise form appears to sensitively depend on the deficiency of the network.
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Affiliation(s)
- Hyun-Myung Chun
- School of Physics, Korea Institute for Advanced Study, Seoul 02455, South Korea
| | - Jordan M Horowitz
- Department of Biophysics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48104, USA
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
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5
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Yang X, Chen Y, Zhou T, Zhang J. Exploring dissipative sources of non-Markovian biochemical reaction systems. Phys Rev E 2021; 103:052411. [PMID: 34134237 DOI: 10.1103/physreve.103.052411] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2020] [Accepted: 04/29/2021] [Indexed: 11/07/2022]
Abstract
Many biological processes including important intracellular processes are governed by biochemical reaction networks. Usually, these reaction systems operate far from thermodynamic equilibrium, implying free-energy dissipation. On the other hand, single reaction events happen often in a memory manner, leading to non-Markovian kinetics. A question then arises: how do we calculate free-energy dissipation (defined as the entropy production rate) in this physically real case? We derive an analytical formula for calculating the energy consumption of a general reaction system with molecular memory characterized by nonexponential waiting-time distributions. It shows that this dissipation is composed of two parts: one from broken detailed balance of an equivalent Markovian system with the same topology and substrates, and the other from the direction-time dependence of waiting-time distributions. But, if the system is in a detailed balance and the waiting-time distribution is direction-time independent, there is no energy dissipation even in the non-Markovian case. These general results provide insights into the physical mechanisms underlying nonequilibrium processes. A continuous-time random-walk model and a generalized model of stochastic gene expression are chosen to clearly show dissipative sources and the relationship between energy dissipation and molecular memory.
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Affiliation(s)
- Xiyan Yang
- School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, People's Republic of China
| | - Yiren Chen
- College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, People's Republic of China
| | - Tianshou Zhou
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China.,Guangdong Province Key Laboratory of Computational Science, Guangzhou 510275, People's Republic of China
| | - Jiajun Zhang
- School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China.,Guangdong Province Key Laboratory of Computational Science, Guangzhou 510275, People's Republic of China
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6
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Shin G, Wang J. The role of energy cost on accuracy, sensitivity, specificity, speed and adaptation of T cell foreign and self recognition. Phys Chem Chem Phys 2021; 23:2860-2872. [PMID: 33471892 DOI: 10.1039/d0cp02422h] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
The critical role of energy consumption in biological systems including T cell discrimination process has been investigated in various ways. The kinetic proofreading (KPR) in T cell recognition involving different levels of energy dissipation influences functional outcomes such as error rates and specificity. In this work, we study quantitatively how the energy cost influences error fractions, sensitivity, specificity, kinetic speed in terms of Mean First Passage Time (MFPT) and adaption errors. These provide the background to adequately understand T cell dynamics. It is found that energy plays a central role in the system that aims to achieve minimum error fractions and maximum sensitivity and specificity with the fastest speed under our kinetic scheme for which numerical values of kinetic parameters are specially chosen, but such a condition can be broken with varying data. Starting with the application of steady state approximation (SSA) to the evaluation of the concentration of each complex produced associated with KPR, which is used to quantify various observables, we present both analytical and numerical results in detail.
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Affiliation(s)
- Gyubaek Shin
- Department of Chemistry, SUNY Stony Brook, 100 Nicolls Road, Stony Brook, NY 11794, USA. jin.wang.1.@stonybrook.edu
| | - Jin Wang
- Department of Chemistry, SUNY Stony Brook, 100 Nicolls Road, Stony Brook, NY 11794, USA. jin.wang.1.@stonybrook.edu and Department of Physics and Astronomy, SUNY Stony Brook, 100 Nicolls Road, Stony Brook, NY 11794, USA
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7
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Roy S, Bagchi B. Fluctuation theory of immune response: A statistical mechanical approach to understand pathogen induced T-cell population dynamics. J Chem Phys 2021; 153:045107. [PMID: 32752668 DOI: 10.1063/5.0009747] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
In this period of intense interest in human immunity, we attempt here to quantify the immune response against pathogen invasion through T-cell population dynamics. Borrowing concepts from equilibrium statistical mechanics, we introduce a new description of the immune response function (IMRF) in terms of fluctuations in the population number of relevant biological cells (effector and regulatory T-cells). We use a coarse-grained chemical reaction network model (CG-CRNM) to calculate the number fluctuations and show that the response function derived as such can, indeed, capture the crossover observed in a T-cell driven immune response. We employ the network model to learn the effect of vitamin-D as an immunomodulator. We solve our CG-CRNM using a stochastic Gillespie algorithm. Depending on the effector T-cell concentration, we can classify immune regulation regimes into three categories: weak, strong, and moderate. The IMRF is found to behave differently in these three regimes. A damped cross-regulatory behavior found in the dynamics of effector and regulatory T-cell concentration in the diseased states correlates well with the same found in a cohort of patients with specific malignancies and autoimmune diseases. Importantly, the crossover from the weakly regulated steady state to the other (the strongly regulated) is accompanied by a divergence-like growth in the fluctuation of both the effector and the regulatory T-cell concentration, characteristic of a dynamic phase transition. We believe such steady-state IMRF analyses could help not only to phase-separate different immune stages but also aid in the valuable connection between autoimmunity, optimal vitamin-D, and consequences of immunosuppressive stress and malignancy.
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Affiliation(s)
- Susmita Roy
- Department of Chemical Sciences, Indian Institute of Science Education and Research Kolkata, Campus Road, Mohanpur, West Bengal 741246, India
| | - Biman Bagchi
- Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India
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8
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Castellani G, Cooper LN, De Oliveira LR, Blais BS. Energy Consumption and Entropy Production in a Stochastic Formulation of BCM Learning. J Comput Biol 2020; 28:257-268. [PMID: 33370157 DOI: 10.1089/cmb.2020.0118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In a series of previous studies, we provided a stochastic description of a theory of synaptic plasticity. This theory, called BCM from the names of the three authors, has been formulated in two ways: the original formulation, where the plasticity threshold is defined as the square of the time-averaged neuronal activity, and a newer formulation, where the plasticity threshold is defined as the time average of the square of the neuronal activity. The newest formulation of the BCM rule of synaptic activity has interesting statistical properties, derived from a risk (or energy) function, the minimization of which leads to seeking of interesting projections in high-dimensional space. Moreover, these two rules, if implemented by a chemical master equation approach, show another interesting difference: the original rule satisfies the detailed balance, whereas the other not. Based on this different behavior, we found a continuous parameterization between these two rules. This parameterization shows a minimum that corresponds to maximum negative eigenvalues of the Jacobian matrix. In addition, the newest rule, due to the fact that it is in a nonequilibrium steady state (NESS), shows a higher level of plasticity than the original rule. This higher level of plasticity has to be interpreted in the framework of open thermodynamical systems and we show that entropy production and energy consumption in the newest rule are both less than in the original BCM rule.
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Affiliation(s)
- Gastone Castellani
- Department of Experimental, Diagnostic and Specialty Medicine, University of Bologna, Bologna, Italy
| | - Leon N Cooper
- Institute for Brain and Neural Systems, Brown University, Providence, Rhode Island, USA
| | - Luciana Renata De Oliveira
- Genetics and Molecular Cardiology/LIM 13, Heart Institute (InCor)/University of São Paulo Medical School, São Paulo, Brazil
| | - Brian S Blais
- Institute for Brain and Neural Systems, Brown University, Providence, Rhode Island, USA.,Department of Science and Technology, Bryant University, Smithfield, Rhode Island, USA
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9
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Fang X, Wang J. Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems. Annu Rev Biophys 2020; 49:227-246. [DOI: 10.1146/annurev-biophys-121219-081656] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.
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Affiliation(s)
- Xiaona Fang
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
| | - Jin Wang
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
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10
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Identifying feasible operating regimes for early T-cell recognition: The speed, energy, accuracy trade-off in kinetic proofreading and adaptive sorting. PLoS One 2018; 13:e0202331. [PMID: 30114236 PMCID: PMC6095552 DOI: 10.1371/journal.pone.0202331] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Accepted: 08/01/2018] [Indexed: 01/09/2023] Open
Abstract
In the immune system, T cells can quickly discriminate between foreign and self ligands with high accuracy. There is evidence that T-cells achieve this remarkable performance utilizing a network architecture based on a generalization of kinetic proofreading (KPR). KPR-based mechanisms actively consume energy to increase the specificity beyond what is possible in equilibrium. An important theoretical question that arises is to understand the trade-offs and fundamental limits on accuracy, speed, and dissipation (energy consumption) in KPR and its generalization. Here, we revisit this question through numerical simulations where we simultaneously measure the speed, accuracy, and energy consumption of the KPR and adaptive sorting networks for different parameter choices. Our simulations highlight the existence of a "feasible operating regime" in the speed-energy-accuracy plane where T-cells can quickly differentiate between foreign and self ligands at reasonable energy expenditure. We give general arguments for why we expect this feasible operating regime to be a generic property of all KPR-based biochemical networks and discuss implications for our understanding of the T cell receptor circuit.
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11
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Fan G, Bressloff PC. Population Model of Quorum Sensing with Multiple Parallel Pathways. Bull Math Biol 2017; 79:2599-2626. [PMID: 28887768 DOI: 10.1007/s11538-017-0343-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Accepted: 08/31/2017] [Indexed: 10/18/2022]
Abstract
Quorum sensing (QS) is a bacterial communication mechanism that uses signal-receptor binding to regulate gene expression based on cell density, resulting in group behaviors such as biofilm formation, bioluminescence and stress response. In certain bacterial species such as Vibrio harveyi, several parallel QS signaling pathways drive a single phosphorylation-dephosphorylation cycle, which in turn regulates QS target genes. In this paper, we investigate the possible role of parallel signaling pathways by developing a mathematical model of QS in V. harveyi at both the single-cell and population levels. First we explore how signal integration may be achieved at the single-cell level, and how different model parameters influence the process. We then consider two examples of signal integration at the population level: a one-population model responding to two environmental cues (cell density and mass transfer), and a two-population model with distinct cell densities. In each case, we use contraction analysis to reduce the population model to an effective single-cell model.
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Affiliation(s)
- Gaoyang Fan
- Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA
| | - Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA.
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12
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Roy U, Gopalakrishnan M. Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli. PLoS One 2017; 12:e0175309. [PMID: 28406996 PMCID: PMC5391091 DOI: 10.1371/journal.pone.0175309] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2016] [Accepted: 03/23/2017] [Indexed: 12/02/2022] Open
Abstract
A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance σa2 in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that σa2 peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.
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Affiliation(s)
- Ushasi Roy
- Department of Physics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India
- * E-mail:
| | - Manoj Gopalakrishnan
- Department of Physics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India
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13
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Bressloff PC. Ultrasensitivity and noise amplification in a model of V. harveyi quorum sensing. Phys Rev E 2016; 93:062418. [PMID: 27415309 DOI: 10.1103/physreve.93.062418] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2016] [Indexed: 06/06/2023]
Abstract
We analyze ultrasensitivity in a model of Vibrio harveyi quorum sensing. We consider a feedforward model consisting of two biochemical networks per cell. The first represents the interchange of a signaling molecule (autoinducer) between the cell cytoplasm and an extracellular domain and the binding of intracellular autoinducer to cognate receptors. The unbound and bound receptors within each cell act as kinases and phosphotases, respectively, which then drive a second biochemical network consisting of a phosphorylation-dephosphorylation cycle. We ignore subsequent signaling pathways associated with gene regulation and the possible modification in the production rate of an autoinducer (positive feedback). We show how the resulting quorum sensing system exhibits ultrasensitivity with respect to changes in cell density. We also demonstrate how quorum sensing can protect against the noise amplification of fast environmental fluctuations in comparison to a single isolated cell.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
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14
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Iglesias PA. The Use of Rate Distortion Theory to Evaluate Biological Signaling Pathways. ACTA ACUST UNITED AC 2016. [DOI: 10.1109/tmbmc.2016.2623600] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
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15
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Affiliation(s)
- A. Subha Mahadevi
- Centre for Molecular Modelling, CSIR-Indian Institute of Chemical Technology, Tarnaka, Hyderabad, India 500607
| | - G. Narahari Sastry
- Centre for Molecular Modelling, CSIR-Indian Institute of Chemical Technology, Tarnaka, Hyderabad, India 500607
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16
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Roy S, Shrinivas K, Bagchi B. A stochastic chemical dynamic approach to correlate autoimmunity and optimal vitamin-D range. PLoS One 2014; 9:e100635. [PMID: 24971516 PMCID: PMC4074107 DOI: 10.1371/journal.pone.0100635] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2013] [Accepted: 05/29/2014] [Indexed: 01/26/2023] Open
Abstract
Motivated by several recent experimental observations that vitamin-D could interact with antigen presenting cells (APCs) and T-lymphocyte cells (T-cells) to promote and to regulate different stages of immune response, we developed a coarse grained but general kinetic model in an attempt to capture the role of vitamin-D in immunomodulatory responses. Our kinetic model, developed using the ideas of chemical network theory, leads to a system of nine coupled equations that we solve both by direct and by stochastic (Gillespie) methods. Both the analyses consistently provide detail information on the dependence of immune response to the variation of critical rate parameters. We find that although vitamin-D plays a negligible role in the initial immune response, it exerts a profound influence in the long term, especially in helping the system to achieve a new, stable steady state. The study explores the role of vitamin-D in preserving an observed bistability in the phase diagram (spanned by system parameters) of immune regulation, thus allowing the response to tolerate a wide range of pathogenic stimulation which could help in resisting autoimmune diseases. We also study how vitamin-D affects the time dependent population of dendritic cells that connect between innate and adaptive immune responses. Variations in dose dependent response of anti-inflammatory and pro-inflammatory T-cell populations to vitamin-D correlate well with recent experimental results. Our kinetic model allows for an estimation of the range of optimum level of vitamin-D required for smooth functioning of the immune system and for control of both hyper-regulation and inflammation. Most importantly, the present study reveals that an overdose or toxic level of vitamin-D or any steroid analogue could give rise to too large a tolerant response, leading to an inefficacy in adaptive immune function.
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Affiliation(s)
- Susmita Roy
- SSCU, Indian Institute of Science, Bangalore, Karnataka, India
| | | | - Biman Bagchi
- SSCU, Indian Institute of Science, Bangalore, Karnataka, India
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17
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Dasgupta T, Croll DH, Owen JA, Vander Heiden MG, Locasale JW, Alon U, Cantley LC, Gunawardena J. A fundamental trade-off in covalent switching and its circumvention by enzyme bifunctionality in glucose homeostasis. J Biol Chem 2014; 289:13010-25. [PMID: 24634222 DOI: 10.1074/jbc.m113.546515] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022] Open
Abstract
Covalent modification provides a mechanism for modulating molecular state and regulating physiology. A cycle of competing enzymes that add and remove a single modification can act as a molecular switch between "on" and "off" and has been widely studied as a core motif in systems biology. Here, we exploit the recently developed "linear framework" for time scale separation to determine the general principles of such switches. These methods are not limited to Michaelis-Menten assumptions, and our conclusions hold for enzymes whose mechanisms may be arbitrarily complicated. We show that switching efficiency improves with increasing irreversibility of the enzymes and that the on/off transition occurs when the ratio of enzyme levels reaches a value that depends only on the rate constants. Fluctuations in enzyme levels, which habitually occur due to cellular heterogeneity, can cause flipping back and forth between on and off, leading to incoherent mosaic behavior in tissues, that worsens as switching becomes sharper. This trade-off can be circumvented if enzyme levels are correlated. In particular, if the competing catalytic domains are on the same protein but do not influence each other, the resulting bifunctional enzyme can switch sharply while remaining coherent. In the mammalian liver, the switch between glycolysis and gluconeogenesis is regulated by the bifunctional 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase (PFK-2/FBPase-2). We suggest that bifunctionality of PFK-2/FBPase-2 complements the metabolic zonation of the liver by ensuring coherent switching in response to insulin and glucagon.
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Affiliation(s)
- Tathagata Dasgupta
- From the Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115
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18
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Zero-order ultrasensitivity: a study of criticality and fluctuations under the total quasi-steady state approximation in the linear noise regime. J Theor Biol 2013; 344:1-11. [PMID: 24309434 DOI: 10.1016/j.jtbi.2013.11.014] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2013] [Revised: 11/11/2013] [Accepted: 11/21/2013] [Indexed: 11/21/2022]
Abstract
Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a 'push-pull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker-Planck (drift-diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a 'critical point' for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm.
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Santra M, Bagchi B. Kinetic proofreading at single molecular level: aminoacylation of tRNA(Ile) and the role of water as an editor. PLoS One 2013; 8:e66112. [PMID: 23840412 PMCID: PMC3688713 DOI: 10.1371/journal.pone.0066112] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2013] [Accepted: 05/02/2013] [Indexed: 11/25/2022] Open
Abstract
Proofreading/editing in protein synthesis is essential for accurate translation of information from the genetic code. In this article we present a theoretical investigation of efficiency of a kinetic proofreading mechanism that employs hydrolysis of the wrong substrate as the discriminatory step in enzyme catalytic reactions. We consider aminoacylation of tRNAIle which is a crucial step in protein synthesis and for which experimental results are now available. We present an augmented kinetic scheme and then employ methods of stochastic simulation algorithm to obtain time dependent concentrations of different substances involved in the reaction and their rates of formation. We obtain the rates of product formation and ATP hydrolysis for both correct and wrong substrates (isoleucine and valine in our case, respectively), in single molecular enzyme as well as ensemble enzyme kinetics. The present theoretical scheme correctly reproduces (i) the amplitude of the discrimination factor in the overall rates between isoleucine and valine which is obtained as (1.8×102).(4.33×102) = 7.8×104, (ii) the rates of ATP hydrolysis for both Ile and Val at different substrate concentrations in the aminoacylation of tRNAIle. The present study shows a non-michaelis type dependence of rate of reaction on tRNAIle concentration in case of valine. The overall editing in steady state is found to be independent of amino acid concentration. Interestingly, the computed ATP hydrolysis rate for valine at high substrate concentration is same as the rate of formation of Ile-tRNAIle whereas at intermediate substrate concentration the ATP hydrolysis rate is relatively low. We find that the presence of additional editing domain in class I editing enzyme makes the kinetic proofreading more efficient through enhanced hydrolysis of wrong product at the editing CP1 domain.
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Affiliation(s)
- Mantu Santra
- Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India
| | - Biman Bagchi
- Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India
- * E-mail:
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Bazzani A, Castellani GC, Giampieri E, Remondini D, Cooper LN. Bistability in the chemical master equation for dual phosphorylation cycles. J Chem Phys 2012; 136:235102. [PMID: 22779621 DOI: 10.1063/1.4725180] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.
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Affiliation(s)
- Armando Bazzani
- Physics Department of Bologna University, Bologna 40127, Italy
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Abstract
Cells often perform computations in order to respond to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds, it is expected that such computations require cells to consume energy. In particular, Landauer's principle states that energy must be consumed in order to erase the memory of past observations. Here, we explicitly calculate the energetic cost of steady-state computation of ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments, such as the spore germination networks of bacteria.
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Qian H, Roy S. An Information Theoretical Analysis of Kinase Activated Phosphorylation Dephosphorylation Cycle. IEEE Trans Nanobioscience 2012; 11:289-95. [DOI: 10.1109/tnb.2011.2182658] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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Realistic enzymology for post-translational modification: zero-order ultrasensitivity revisited. J Theor Biol 2012; 311:139-52. [PMID: 22828569 DOI: 10.1016/j.jtbi.2012.07.012] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2011] [Revised: 07/12/2012] [Accepted: 07/13/2012] [Indexed: 11/21/2022]
Abstract
Unlimited ultrasensitivity in a kinase/phosphatase "futile cycle" has been a paradigmatic example of collective behaviour in multi-enzyme systems. However, its analysis has relied on the Michaelis-Menten reaction mechanism, which remains widely used despite a century of new knowledge. Modifying and demodifying enzymes accomplish different biochemical tasks; the donor that contributes the modifying group is often ignored without the impact of this time-scale separation being taken into account; and new forms of reversible modification are now known. We exploit new algebraic methods of steady-state analysis to reconcile the analysis of multi-enzyme systems with single-enzyme biochemistry using zero-order ultrasensitivity as an example. We identify the property of "strong irreversibility", in which product re-binding is disallowed. We show that unlimited ultrasensitivity is preserved for a class of complex, strongly irreversible reaction mechanisms and determine the corresponding saturation conditions. We show further that unlimited ultrasensitivity arises from a singularity in a novel "invariant" that summarises the algebraic relationship between modified and unmodified substrate. We find that this singularity also underlies knife-edge behaviour in allocation of substrate between modification states, which has implications for the coherence of futile cycles within an integrated tissue. When the enzymes are irreversible, but not strongly so, the singularity disappears in the form found here and unlimited ultrasensitivity may no longer be preserved. The methods introduced here are widely applicable to other reversible modification systems.
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Qian H. Cooperativity in Cellular Biochemical Processes: Noise-Enhanced Sensitivity, Fluctuating Enzyme, Bistability with Nonlinear Feedback, and Other Mechanisms for Sigmoidal Responses. Annu Rev Biophys 2012; 41:179-204. [DOI: 10.1146/annurev-biophys-050511-102240] [Citation(s) in RCA: 71] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195;
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Qian H. Cyclic conformational modification of an enzyme: serial engagement, energy relay, hysteretic enzyme, and Fischer's hypothesis. J Phys Chem B 2010; 114:16105-11. [PMID: 20863145 DOI: 10.1021/jp102400u] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
Reversible chemical modification of enzymes is one of the most important mechanisms in cellular signaling. We generalize this concept to include cyclic modification of enzyme conformations. The mechanism is fundamentally different from the ligand induced conformational change: It only requires a catalytic amount of ligand to activate an enzyme, but it does require an active chemical energy driving a "futile cycle" akin to the phosphorylation-dephosphorylation cycle. The mechanism covers several previously proposed models that include serial engagement for T-cell receptor activation, energy relay for proofreading in DNA replication and protein biosynthesis, the hysteretic enzyme, and Fischer's hypothesis on protein tyrosine phosphatase action. While for small proteins operating under a funnel-shaped energy landscape, multiple conformations with sufficiently long dwell times are not common, recent experiments suggest that for larger, multidomain proteins, cyclic conformational modification (CCM) is much more likely and evolution presumably finds a way to capitalize on this mode of regulation. CCM can be difficult to identify in cells; however, it is potentially an important, and yet overlooked, regulatory mechanism in cellular signal transduction. We suggest the serial engagement mechanism in T-cell activation to be a possible testing case for the CCM mechanism.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington Seattle, Washington 98195, USA
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Bishop LM, Qian H. Stochastic bistability and bifurcation in a mesoscopic signaling system with autocatalytic kinase. Biophys J 2010; 98:1-11. [PMID: 20074511 DOI: 10.1016/j.bpj.2009.09.055] [Citation(s) in RCA: 74] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2009] [Revised: 09/11/2009] [Accepted: 09/14/2009] [Indexed: 11/28/2022] Open
Abstract
Bistability is a nonlinear phenomenon widely observed in nature including in biochemical reaction networks. Deterministic chemical kinetics studied in the past has shown that bistability occurs in systems with strong (cubic) nonlinearity. For certain mesoscopic, weakly nonlinear (quadratic) biochemical reaction systems in a small volume, however, stochasticity can induce bistability and bifurcation that have no macroscopic counterpart. We report the simplest yet known reactions involving driven phosphorylation-dephosphorylation cycle kinetics with autocatalytic kinase. We show that the noise-induced phenomenon is correlated with free energy dissipation and thus conforms with the open-chemical system theory. A previous reported noise-induced bistability in futile cycles is found to have originated from the kinase synchronization in a bistable system with slow transitions, as reported here.
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Affiliation(s)
- Lisa M Bishop
- Department of Applied Mathematics, University of Washington, Seattle, Washington, USA.
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Huang Q, Qian H. Ultrasensitive dual phosphorylation dephosphorylation cycle kinetics exhibits canonical competition behavior. CHAOS (WOODBURY, N.Y.) 2009; 19:033109. [PMID: 19791989 DOI: 10.1063/1.3187790] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/28/2023]
Abstract
We establish a mathematical model for a cellular biochemical signaling module in terms of a planar differential equation system. The signaling process is carried out by two phosphorylation-dephosphorylation reaction steps that share common kinase and phosphatase with saturated enzyme kinetics. The pair of equations is particularly simple in the present mathematical formulation, but they are singular. A complete mathematical analysis is developed based on an elementary perturbation theory. The dynamics exhibits the canonical competition behavior in addition to bistability. Although widely understood in ecological context, we are not aware of a full range of biochemical competition in a simple signaling network. The competition dynamics has broad implications to cellular processes such as cell differentiation and cancer immunoediting. The concepts of homogeneous and heterogeneous multisite phosphorylation are introduced and their corresponding dynamics are compared: there is no bistability in a heterogeneous dual phosphorylation system. A stochastic interpretation is also provided that further gives intuitive understanding of the bistable behavior inside the cells.
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Affiliation(s)
- Qingdao Huang
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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Qian H, Shi PZ, Xing J. Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity. Phys Chem Chem Phys 2009; 11:4861-70. [PMID: 19506761 DOI: 10.1039/b900335p] [Citation(s) in RCA: 64] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present a simple, unifying theory for stochastic biochemical systems with multiple time-scale dynamics that exhibit noise-induced bistability in an open-chemical environment, while the corresponding macroscopic reaction is unistable. Nonlinear stochastic biochemical systems like these are fundamentally different from classical systems in equilibrium or near-equilibrium steady state whose fluctuations are unimodal following Einstein-Onsager-Lax-Keizer theory. We show that noise-induced bistability in general arises from slow fluctuations, and a pitchfork bifurcation occurs as the rate of fluctuations decreases. Since an equilibrium distribution, due to detailed balance, has to be independent of changes in time-scale, the bifurcation is necessarily a driven phenomenon. As examples, we analyze three biochemical networks of currently interest: self-regulating gene, stochastic binary decision, and phosphorylation-dephosphorylation cycle with fluctuating kinase. The implications of bistability to biochemical complexity are discussed.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
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Qian H, Shi PZ. Fluctuating Enzyme and Its Biological Functions: Positive Cooperativity without Multiple States. J Phys Chem B 2009; 113:2225-30. [DOI: 10.1021/jp810657j] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Hong Qian
- Departments of Applied Mathematics, University of Washington, Seattle, Washington 98195
| | - Pei-Zhe Shi
- Departments of Applied Mathematics, University of Washington, Seattle, Washington 98195
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Ge H, Qian M. Sensitivity amplification in the phosphorylation-dephosphorylation cycle: Nonequilibrium steady states, chemical master equation, and temporal cooperativity. J Chem Phys 2008; 129:015104. [DOI: 10.1063/1.2948965] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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32
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The effects of reversibility and noise on stochastic phosphorylation cycles and cascades. Biophys J 2008; 95:2183-92. [PMID: 18515389 DOI: 10.1529/biophysj.107.126185] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The phosphorylation-dephosphorylation cycle is a common motif in cellular signaling networks. Previous work has revealed that, when driven by a noisy input signal, these cycles may exhibit bistable behavior. Here, a recently introduced theorem on network bistability is applied to prove that the existence of bistability is dependent on the stochastic nature of the system. Furthermore, the thermodynamics of simple cycles and cascades is investigated in the stochastic setting. Because these cycles are driven by the ATP hydrolysis potential, they may operate far from equilibrium. It is shown that sufficient high ATP hydrolysis potential is necessary for the existence of a bistable steady state. For the single-cycle system, the ensemble average behavior follows the ultrasensitive response expected from analysis of the corresponding deterministic system, but with significant fluctuations. For the two-cycle cascade, the average behavior begins to deviate from the expected response of the deterministic system. Examination of a two-cycle cascade reveals that the bistable steady state may be either propagated or abolished along a cascade, depending on the parameters chosen. Likewise, the variance in the response can be maximized or minimized by tuning the number of enzymes in the second cycle.
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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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34
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Abstract
Biochemical systems and processes in living cells generally operate far from equilibrium. This review presents an overview of a statistical thermodynamic treatment for such systems, with examples from several key components in cellular signal transduction. Open-system nonequilibrium steady-state (NESS) models are introduced. The models account quantitatively for the energetics and thermodynamics in phosphorylation-dephosphorylation switches, GTPase timers, and specificity amplification through kinetic proofreading. The chemical energy derived from ATP and GTP hydrolysis establishes the NESS of a cell and makes the cell--a mesoscopic-biochemical reaction system that consists of a collection of thermally driven fluctuating macromolecules--a genetically programmed chemical machine.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
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35
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Andrews BW, Iglesias PA. An information-theoretic characterization of the optimal gradient sensing response of cells. PLoS Comput Biol 2007; 3:e153. [PMID: 17676949 PMCID: PMC1937015 DOI: 10.1371/journal.pcbi.0030153] [Citation(s) in RCA: 64] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2007] [Accepted: 06/18/2007] [Indexed: 12/27/2022] Open
Abstract
Many cellular systems rely on the ability to interpret spatial heterogeneities in chemoattractant concentration to direct cell migration. The accuracy of this process is limited by stochastic fluctuations in the concentration of the external signal and in the internal signaling components. Here we use information theory to determine the optimal scheme to detect the location of an external chemoattractant source in the presence of noise. We compute the minimum amount of mutual information needed between the chemoattractant gradient and the internal signal to achieve a prespecified chemotactic accuracy. We show that more accurate chemotaxis requires greater mutual information. We also demonstrate that a priori information can improve chemotaxis efficiency. We compare the optimal signaling schemes with existing experimental measurements and models of eukaryotic gradient sensing. Remarkably, there is good quantitative agreement between the optimal response when no a priori assumption is made about the location of the existing source, and the observed experimental response of unpolarized Dictyostelium discoideum cells. In contrast, the measured response of polarized D. discoideum cells matches closely the optimal scheme, assuming prior knowledge of the external gradient—for example, through prolonged chemotaxis in a given direction. Our results demonstrate that different observed classes of responses in cells (polarized and unpolarized) are optimal under varying information assumptions. For many cell types, the direction of migration is determined in response to spatial differences in the concentration of chemoattractant, a process known as chemotaxis. Precise chemotaxis—that is, motility with low directional distortion—requires that cells make accurate decisions based on the stochastic fluctuations inherent in cell-surface receptor occupancy. Here, we use rate distortion theory, a branch of information theory, to determine chemotaxis strategies for cells based on this imperfect information about their environment. In engineering, rate distortion theory provides the information processing capabilities required to achieve a desired accuracy. We demonstrate that more accurate chemotaxis requires greater information. We also show that a priori information can improve chemotaxis efficiency. We compare the optimal signaling schemes to existing experimental measurements and models of eukaryotic gradient sensing and demonstrate that different observed types of cellular responses (polarized and unpolarized) are optimal under varying information assumptions. Our results also highlight the constraints that noise places on the performance of cellular systems.
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Affiliation(s)
- Burton W Andrews
- Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, Maryland, United States of America
| | - Pablo A Iglesias
- Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, Maryland, United States of America
- * To whom correspondence should be addressed. E-mail:
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Han Z, Vondriska TM, Yang L, MacLellan WR, Weiss JN, Qu Z. Signal transduction network motifs and biological memory. J Theor Biol 2007; 246:755-61. [PMID: 17374382 PMCID: PMC2701969 DOI: 10.1016/j.jtbi.2007.01.022] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2006] [Accepted: 01/25/2007] [Indexed: 11/19/2022]
Abstract
Memory is a ubiquitous phenomenon in biological systems, yet the mechanisms responsible for memory, and how to manipulate it at the subcellular level, remain poorly understood. Subjected to transient stimuli, biological systems can exhibit short early responses and/or prolonged (or permanent) late responses. Experimental evidence suggests that early responses (short-term memory) involve post-translational modification of existing proteins and/or their intracellular relocalization, whereas late responses (long-term memory) depend on new protein synthesis. Although this provides an intuitive explanation at the basic molecular level, it does little to clarify the important dynamics that actually maintain memory at the systems level. In this study, we use mathematical modeling to study dynamical mechanisms of biological memory. We first examined the response of four fundamental motifs (positive/negative feedforward and feedback) to external stimuli. Because motifs do not exist in isolation within the cell, we then combined these motifs to form signaling modules to understand how they confer biological memory. These motifs, and different combinations thereof, displayed distinct behavior in response to external stimuli. The principles described in this study have important implications for experimental approaches to identify the mechanisms for biological memory and for the development of therapeutic strategies to modulate signaling network responses in the setting of human disease.
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Affiliation(s)
- Zhangang Han
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
| | - Thomas M. Vondriska
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
- Departments of Anesthesiology, David Geffen School of Medicine, University of California, Los Angeles, California 90095
| | - Ling Yang
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
| | - W. Robb MacLellan
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
- Departments of Physiology, David Geffen School of Medicine, University of California, Los Angeles, California 90095
| | - James N. Weiss
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
- Departments of Physiology, David Geffen School of Medicine, University of California, Los Angeles, California 90095
| | - Zhilin Qu
- Departments of Medicine (Cardiology), David Geffen School of Medicine, University of California, Los Angeles, California 90095
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Qian H. Reducing intrinsic biochemical noise in cells and its thermodynamic limit. J Mol Biol 2006; 362:387-92. [PMID: 16934833 DOI: 10.1016/j.jmb.2006.07.068] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2006] [Revised: 07/21/2006] [Accepted: 07/26/2006] [Indexed: 11/16/2022]
Abstract
In living cells, the specificity of biomolecular recognition can be amplified and the noise from non-specific interactions can be reduced at the expense of cellular free energy. This is the seminal idea in the Hopfield-Ninio theory of kinetic proofreading: The specificity is increased via cyclic network kinetics without altering molecular structures and equilibrium affinites. We show a thermodynamic limit of the specificity amplification with a given amount of available free energy. For a normal cell under physiological condition with sustained phosphorylation potential, this gives a factor of 10(10) as the upper bound in specificity amplification. We also study an optimal kinetic network design that is capable of approaching the thermodynamic limit.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
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Qian H. Nonequilibrium potential function of chemically driven single macromolecules via Jarzynski-type Log-Mean-Exponential Heat. J Phys Chem B 2006; 109:23624-8. [PMID: 16375340 DOI: 10.1021/jp0545391] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Applying the method from recently developed fluctuation theorems to the stochastic dynamics of single macromolecules in ambient fluid at constant temperature, we establish two Jarzynski-type equalities: (1) between the log-mean-exponential (LME) of the irreversible heat dissiption of a driven molecule in nonequilibrium steady-state (NESS) and ln P(ness)(x) and (2) between the LME of the work done by the internal force of the molecule and nonequilibrium chemical potential function mu(ness)(x) identical with U(x) + k(B)T ln P(ness)(x), where P(ness)(x) is the NESS probability density in the phase space of the macromolecule and U(x) is its internal potential function. Psi = integral mu(ness)(x) P(ness)(x) dx is shown to be a nonequilibrium generalization of the Helmholtz free energy and DeltaPsi = DeltaU - TDeltaS for nonequilibrium processes, where S = - kB integralP(x) ln P(x) dx is the Gibbs entropy associated with P(x). LME of heat dissipation generalizes the concept of entropy, and the equalities define thermodynamic potential functions for open systems far from equilibrium.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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39
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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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Bassingthwaighte JB, Chizeck HJ, Atlas LE, Qian H. Multiscale modeling of cardiac cellular energetics. Ann N Y Acad Sci 2005; 1047:395-424. [PMID: 16093514 PMCID: PMC2864600 DOI: 10.1196/annals.1341.035] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher-level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the "eternal cell," which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter-mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long-term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large-scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.
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Qian H, Reluga TC. Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch. PHYSICAL REVIEW LETTERS 2005; 94:028101. [PMID: 15698232 DOI: 10.1103/physrevlett.94.028101] [Citation(s) in RCA: 67] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2003] [Indexed: 05/24/2023]
Abstract
We develop a rigorous nonequilibrium thermodynamics for an open system of nonlinear biochemical reactions responsible for cell signal processing. We show that the quality of the biological switch consisting of a phosphorylation-dephosphorylation cycle, such as those in protein kinase cascade, is controlled by the available intracellular free energy from the adenosine triphosphate (ATP) hydrolysis in vivo: DeltaG=k(B)Tln(([ATP]/K(eq)[ADP]), where K(eq) is the equilibrium constant. The model reveals the correlation between the performance of the switch and the level of DeltaG. The result demonstrates the importance of nonequilibrium thermodynamics in analyzing biological information processing, provides its energetic cost, establishes an interplay between signal transduction and energy metabolism in cells, and suggests a biological function for phosphoenergetics in the ubiquitous phosphorylation signaling.
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Affiliation(s)
- Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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Oster G. Clocks and patterns in myxobacteria: a remembrance of Art Winfree. J Theor Biol 2004; 230:451-8. [PMID: 15363668 DOI: 10.1016/j.jtbi.2004.04.038] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2004] [Accepted: 04/27/2004] [Indexed: 11/16/2022]
Abstract
At the beginning of their aggregation phase waves of cell density sweep across the surface of myxobacteria colonies. These waves are unlike any other in biology. Waves can be linear, concentric or spiral and when they collide, instead of annihilating one another they appear to pass through each other unchanged. Moreover, the wavelength determines the spacing and pattern of fruiting bodies that will rise up presaging sporulation. The explanation for these waves was suggested by the work of Art Winfree on cellular clocks, and confirmed by a mathematical model that explains all of the observed wave behavior. The story of how this model evolved illustrates the roles of chance and scientific networking in the search for the explanation of a new phenomenon.
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Affiliation(s)
- George Oster
- Department of Molecular and Cell Biology, College of Natural Resources, University of California, Berkeley, CA 94720-3112, USA.
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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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