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Thermodynamics and its relevance to self-replication and cell growth
Thermodynamic and Stoichiometric Laws Ruling the Fates of Growing Systems
Research article with Y. Sughiyama and T. J. Kobayashi
Physical Review Research 6, 023173 (2024); Arxiv:2312.14435 (2023)
We delve into growing open chemical reaction systems (CRSs) characterized by autocatalytic reactions within a variable volume, which changes in response to these reactions. Understanding the thermodynamics of such systems is crucial for comprehending biological cells and constructing protocells, as it sheds light on the physical conditions necessary for their self-replication. Building on our recent work, where we developed a thermodynamic theory for growing CRSs featuring basic autocatalytic motifs with regular stoichiometric matrices, we now expand this theory to include scenarios where the stoichiometric matrix has a nontrivial left kernel space. This extension introduces conservation laws, which limit the variations in chemical species due to reactions, thereby confining the system's possible states to those compatible with its initial conditions. By considering both thermodynamic and stoichiometric constraints, we clarify the environmental and initial conditions that dictate the CRSs' fate - whether they grow, shrink, or reach equilibrium. We also find that the conserved quantities significantly influence the equilibrium state achieved by a growing CRS. These results are derived independently of specific thermodynamic potentials or reaction kinetics, therefore underscoring the fundamental impact of conservation laws on the growth of the system.
Information geometry of dynamics on graphs and hypergraphs
Research article with T. J. Kobayashi, D. Louthcko, S. A. Horiguchi and Y. Sughiyama
Information Geometry 7(1), 97-166 (2024): Arxiv:2211.14455 (2022)
We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz–Hodge decomposition and the Otto structure in L2- Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can broaden the applicability of information geometry to various problems of linear and nonlinear dynamics.
Research article with T. J. Kobayashi, D. Louthcko, and Y. Sughiyama
Physical Review Research 4, 033208 (2022): Arxiv:2206.00863 (2022)
We derive the Hessian geometric structure of nonequilibrium chemical reaction networks (CRN) on the flux and force spaces induced by the Legendre duality of convex dissipation functions and characterize their dynamics as a generalized flow. With this structure, we can extend theories of nonequilibrium systems with quadratic dissipation functions to more general ones with nonquadratic ones, which are pivotal for studying chemical reaction networks. By applying generalized notions of orthogonality in Hessian geometry to chemical reaction networks, we obtain two generalized decompositions of the entropy production rate, each of which captures gradient-flow and minimum-dissipation aspects in nonequilibrium dynamics.
Chemical Thermodynamics for Growing Systems
Research article with Y. Sughiyama, D. Loutchko, and T. J. Kobayashi
Physical Review Research 4, 033191 (2022): Arxiv:2201.09417 (2022)
We consider growing open chemical reaction systems (CRSs), in which autocatalytic chemical reactions are encapsulated in a finite volume and its size can change in conjunction with the reactions. The thermodynamics of growing CRSs is indispensable for understanding biological cells and designing protocells by clarifying the physical conditions and costs for their growing states. In this work, we establish a thermodynamic theory of growing CRSs by extending the Hessian geometric structure of non-growing CRSs. The theory provides the environmental conditions to determine the fate of the growing CRSs; growth, shrinking or equilibration. We also identify thermodynamic constraints; one to restrict the possible states of the growing CRSs and the other to further limit the region where a nonequilibrium steady growing state can exist. Moreover, we evaluate the entropy production rate in the steady growing state. The growing nonequilibrium state has its origin in the extensivity of thermodynamics, which is different from the conventional nonequilibrium states with constant volume. These results are derived from general thermodynamic considerations without assuming any specific thermodynamic potentials or reaction kinetics; i.e., they are obtained based solely on the second law of thermodynamics.
A Hessian Geometric Structure of Chemical Thermodynamic Systems with Stoichiometric Constraints
Research article with Y. Sughiyama, D. Loutchko, and T. J. Kobayashi
Physical Review Research 4, 033065 (2022): Arxiv:2112.12403 (2021)
We establish a Hessian geometric structure in chemical thermodynamics, which describes chemical reaction networks (CRNs) with equilibrium states. In our setup, the ideal gas assumption and mass action kinetics are not required. The existence and uniqueness condition of the equilibrium state is derived by using the Legendre duality inherent to the Hessian structure. The entropy production during a relaxation to the equilibrium state can be evaluated by the Bregman divergence. Furthermore, the equilibrium state is characterized by four distinct minimization problems of the divergence, which are obtained from the generalized Pythagorean theorem originating in the dual flatness. For the ideal gas case, we confirm that our existence and uniqueness condition implies Birch's theorem, and that the entropy production represented by the divergence coincides with the generalized Kullback-Leibler divergence. In addition, under mass action kinetics, our general framework reproduces the local detailed balance condition.
Kinetic derivation of the Hessian geometric structure in chemical reaction networks
Research article with T. J. Kobayashi, D. Loutchko, and Y. Sughiyama
Physical Review Research 4, 033066 (2022): Arxiv:2112.14910 (2021)
The theory of chemical kinetics forms the basis to describe the dynamics of chemical reaction networks. Owing to physical and thermodynamic constraints, the networks possess various structures, which can be utilized to characterize important properties of the networks. In this work, we reveal the Hessian geometry which underlies chemical reaction networks and demonstrate how it originates from the interplay of stoichiometric and thermodynamic constraints. Our derivation is based on kinetics, we assume the law of mass action and characterize the equilibrium states by the detailed balance condition. The obtained geometric structure is then related to thermodynamics via the Hessian geometry appearing in a pure thermodynamic derivation. We demonstrate, based on the fact that both equilibrium and complex balanced states form toric varieties, how the Hessian geometric framework can be extended to nonequilibrium complex balanced steady states. We conclude that Hessian geometry provides a natural framework to capture the thermodynamic aspects of chemical reaction networks.
On homeostatic replication/growth and diversity
Representation and inference of size control laws by neural network aided point processes
Research article with T. J. Kobayashi
Physical Review Research 3, 033032: bioRxiv 2021.01.24.428011 (2021)
The regulation and coordination of cell growth and division are long-standing problems in cell physiology. Recent single-cell measurements that use microfluidic devices have provided quantitative time-series data on various physiological parameters of cells. To clarify the regulatory laws and associated relevant parameters, such as cell size, simple mathematical models have been constructed and tested based on their capabilities to reproduce the measured data. However, the models may fail to capture some aspects of data due to presumed assumptions or simplification, especially when the data are multidimensional. Furthermore, comparing a model and data for validation is not trivial when we handle noisy multidimensional data. Thus, to extract hidden laws from data, a novel method, which can handle and integrate noisy multidimensional data more flexibly and exhaustively than the conventional ones, is necessary and helpful. By using cell size control as an example, we demonstrate that this problem can be addressed by using a neural network (NN) method, originally developed for history-dependent temporal point processes. The NN can effectively segregate history-dependent deterministic factors and unexplainable noise from given data by flexibly representing the functional forms of the deterministic relation and noise distribution. By using this method, we represent and infer the birth and division cell size distributions of bacteria and fission yeast. Known size control mechanisms, such as the adder model, are revealed as the conditional dependence of the size distributions on history. Further, we show that the inferred NN model provides a better data representation for model searching than conventional descriptive statistics. Thus, the NN method can work as a powerful tool for processing noisy data to uncover hidden dynamic laws.
Horizontal transfer between loose compartments stabilizes replication of fragmented ribozymes
Research article with Y. J. Matsubara, K. Kaneko, and N. Takeuchi
PLoS Computational Biology 15(6): e1007094: Arxiv:1901.06772 (2019)
The emergence of replicases that can replicate themselves is a central issue in the origin of life. Recent experiments suggest that such replicases can be realized if an RNA polymerase ribozyme is divided into fragments short enough to be replicable by the ribozyme and if these fragments self-assemble into a functional ribozyme. However, the continued self-replication of such replicases requires that the production of every essential fragment be balanced and sustained. Here, we use mathematical modeling to investigate whether and under what conditions fragmented replicases achieve continued self-replication. We first show that under a simple batch condition, the replicases fail to display continued self-replication owing to positive feedback inherent in these replicases. This positive feedback inevitably biases replication toward a subset of fragments, so that the replicases eventually fail to sustain the production of all essential fragments. We then show that this inherent instability can be resolved by small rates of random content exchange between loose compartments (i.e., horizontal transfer). In this case, the balanced production of all fragments is achieved through negative frequency-dependent selection operating in the population dynamics of compartments. The horizontal transfer also ensures the presence of all essential fragments in each compartment, sustaining self-replication. Taken together, our results underline compartmentalization and horizontal transfer in the origin of the first self-replicating replicases.
On diversity and resources limitation/competition
Molecular Diversity and Network Complexity in Growing Protocells
Research article with K. Kaneko
Life, 2019, 9(2), 53: Arxiv:1904.08094 (2019), bioRxiv 611996 (2019)
A great variety of molecular components is encapsulated in cells. Each of these components is replicated for cell reproduction. To address the essential role of the huge diversity of cellular components, we studied a model of protocells that convert resources into catalysts with the aid of a catalytic reaction network. As the resources were limited, the diversity in the intracellular components was found to be increased to allow the use of diverse resources for cellular growth. A scaling relation was demonstrated between resource abundances and molecular diversity. In the present study, we examined how the molecular species diversify and how complex catalytic reaction networks develop through an evolutionary course. At some generations, molecular species first appear as parasites that do not contribute to the replication of other molecules. Later, the species turn into host species that contribute to the replication of other species, with further diversification of molecular species. Thus, a complex joint network evolves with this successive increase in species. The present study sheds new light on the origin of molecular diversity and complex reaction networks at the primitive stage of a cell.
Negative scaling relationship between molecular diversity and resource abundances
Research article with K. Kaneko
Physical Review E, 93, 062419 (2016)
Cell reproduction involves replication of diverse molecule species, in contrast to a simple replication system with fewer components. To address this question of diversity, we study theoretically a cell system with catalytic reaction dynamics that grows by uptake of environmental resources. It is shown that limited resources lead to increased diversity of components within the system, and the number of coexisting species increases with a negative power of the resource uptake. The relationship is explained from the optimum growth speed of the cell, determined by a tradeoff between the utility of diverse resources and the concentration onto fewer components to increase the reaction rate.
Transition to diversification by competition for multiple resources in catalytic reaction networks
Research article with K. Kaneko
Journal of Systems Chemistry, 6:5 (2015)
All life, including cells and artificial protocells, must integrate diverse molecules into a single unit in order to reproduce. Despite expected pressure to evolve a simple system with the fastest replication speed, the mechanism by which the use of a great variety of components, and the coexistence of diverse cell-types with different compositions are achieved is as yet unknown.
Here we show that coexistence of such diverse compositions and cell-types is the result of competitions for a variety of limited resources. We find that a transition to diversity occurs both in chemical compositions and in protocell types, as the resource supply is decreased, when the maximum inflow and consumption of resources are balanced.
Our results indicate that a simple physical principle of competition for a variety of limiting resources can be a strong driving force to diversify intracellular dynamics of a catalytic reaction network and to develop diverse protocell types in a primitive stage of life.
Nonequilibrium dynamics of a reacting network system
Research article with T. Shimada and N. Ito
Artificial Life and Robotics, 13, 474 (2009)
The nonlinear non-equilibrium properties of reacting network systems are studied by computer simulations. It is shown that the fluctuation in the population of each chemical species obeys a log-normal distribution, not the normal Gaussian distribution. The reaction rate shows power-law decay with activation cost (energy), not the Arrhenius-type exponential decay observed in a linear non-equilibrium regime. These two characteristic features will explain the diversity, plasticity, and adaptivity observed in complex biological reaction networks.
Journal of the Physical Society of Japan, 75, 024005 (2006)
Rate Constant of Kaneko–Yomo Model
Journal of the Physical Society of Japan, 74, 1071 (2005)
Research articles with S. Yukawa and N. Ito
As a first step to study reaction dynamics in far-from-equilibrium open systems, we propose a stochastic protocell model in which two mutually catalyzing chemicals are replicating depending on the external flow of energy resources J . This model exhibits an Arrhenius type reaction; furthermore, it produces a non-Arrhenius reaction that exhibits a power-law reaction rate with regard to the activation energy. These dependences are explained using the dynamics of J ; the asymmetric random walk of J results in the Arrhenius equation and conservation of J results in a power-law dependence. Further, we find that the discreteness of molecules results in the power change. Effects of cell divisions are also discussed in our model.
On minority control and protocell growth/division
Reproduction of a Protocell by Replication of a Minority Molecule in a Catalytic Reaction Network
Research article with K. Kaneko
Physical Review Letters, 105, 268103 (2010)
Covered in Science and New Scientist
For understanding the origin of life, it is essential to explain the development of a compartmentalized structure, which undergoes growth and division, from a set of chemical reactions. In this study, a hypercycle with two chemicals that mutually catalyze each other is considered in order to show that the reproduction of a protocell with a growth-division process naturally occurs when the replication speed of one chemical is considerably slower than that of the other chemical, and molecules are crowded as a result of replication. It is observed that the protocell divides after a minority molecule is replicated at a slow synthesis rate, and thus, a synchrony between the reproduction of a cell and molecule replication is achieved. The robustness of such protocells against the invasion of parasitic molecules is also demonstrated.
Research article with K. Kaneko
Life, 4, 586 (2014)
Explanation of the emergence of primitive cellular structures from a set of chemical reactions is necessary to unveil the origin of life and to experimentally synthesize protocells. By simulating a cellular automaton model with a two-species hypercycle, we demonstrate the reproduction of a localized cluster; that is, a protocell with a growth-division process emerges when the replication and degradation speeds of one species are respectively slower than those of the other species, because of overcrowding of molecules as a natural outcome of the replication. The protocell exhibits synchrony between its division process and replication of the minority molecule. We discuss the effects of the crowding molecule on the formation of primitive structures. The generality of this result is demonstrated through the extension of our model to a hypercycle with three molecular species, where a localized layered structure of molecules continues to divide, triggered by the replication of a minority molecule at the center.
Research article with K. Kaneko
New Journal of Physics, 20, 035001(2018); arXiv:1711.08988 (2017)
Explanation of exponential growth in self-reproduction is an important step toward elucidation of the origins of life because optimization of the growth potential across rounds of selection is necessary for Darwinian evolution. To produce another copy with approximately the same composition, the exponential growth rates for all components have to be equal. How such balanced growth is achieved, however, is not a trivial question, because this kind of growth requires orchestrated replication of the components in stochastic and nonlinear catalytic reactions. By considering a mutually catalyzing reaction in two- and three-dimensional lattices, as represented by a cellular automaton model, we show that self-reproduction with exponential growth is possible only when the replication and degradation of one molecular species is much slower than those of the others, i.e., when there is a minority molecule. Here, the synergetic effect of molecular discreteness and crowding is necessary to produce the exponential growth. Otherwise, the growth curves show superexponential growth because of nonlinearity of the catalytic reactions or subexponential growth due to replication inhibition by overcrowding of molecules. Our study emphasizes that the minority molecular species in a catalytic reaction network is necessary for exponential growth at the primitive stage of life.
On cellular information processing and decision making
Information processing and integration with intracellular dynamics near critical point
Research article with T. J. Kobayashi
Frontiers in fractal physiology, 3, 203 (2012)
Recent experimental observations suggest that cells can show relatively precise and reliable responses to external signals even though substantial noise is inevitably involved in the signals. An intriguing question is the way how cells can manage to do it.
One possible way to realize such response for a cell is to evolutionary develop and optimize its intracellular signaling pathways so as to extract relevant information from the noisy signal. We recently demonstrated that certain intracellular signaling reactions could actually conduct statistically optimal information processing. In this paper, we clarify that such optimal reaction operates near bifurcation point. This result suggests that critical-like phenomena in the single-cell level may be linked to efficient information processing inside a cell. In addition, improving the performance of response in the single-cell level is not the only way for cells to realize reliable response.
Another possible strategy is to integrate information of individual cells by cell-to-cell interaction such as quorum sensing. Since cell-to-cell interaction is a common phenomenon, it is equally important to investigate how cells can integrate their information by cell-to-cell interaction to realize efficient information processing in the population level. In this paper, we consider roles and benefits of cell-to-cell interaction by considering integrations of obtained information of individuals with the other cells from the viewpoint of information processing. We also demonstrate that, by introducing cell movement, spatial organizations can spontaneously emerge as a result of efficient responses of the population to external signals.
Theoretical Aspects of Cellular Decision-Making and Information Processing
Review with T. J. Kobayashi
Advances in Systems Biology, 736, 275 (2011)
Microscopic biological processes have extraordinary complexity and variety at the sub-cellular, intra-cellular, and multi-cellular levels. In dealing with such complex phenomena, conceptual and theoretical frameworks are crucial, which enable us to understand seemingly different intra- and inter-cellular phenomena from unified viewpoints.
Decision-making is one such concept that has attracted much attention recently. Since a number of cellular behavior can be regarded as processes to make specific actions in response to external stimuli, decision-making can cover and has been used to explain a broad range of different cellular phenomena [Balázsi et al. (Cell 144(6):910, 2011), Zeng et al. (Cell 141(4):682, 2010)]. Decision-making is also closely related to cellular information-processing because appropriate decisions cannot be made without exploiting the information that the external stimuli contain. Efficiency of information transduction and processing by intra-cellular networks determines the amount of information obtained, which in turn limits the efficiency of subsequent decision-making. Furthermore, information-processing itself can serve as another concept that is crucial for understanding of other biological processes than decision-making. In this work, we review recent theoretical developments on cellular decision-making and information-processing by focusing on the relation between these two concepts.
Dynamics of intracellular information decoding
Research article with T. J. Kobayashi
Physical Biology, 8, 055007 (2011)
A variety of cellular functions are robust even to substantial intrinsic and extrinsic noise in intracellular reactions and the environment that could be strong enough to impair or limit them. In particular, of substantial importance is cellular decision-making in which a cell chooses a fate or behavior on the basis of information conveyed in noisy external signals. For robust decoding, the crucial step is filtering out the noise inevitably added during information transmission. As a minimal and optimal implementation of such an information decoding process, the autocatalytic phosphorylation and autocatalytic dephosphorylation (aPadP) cycle was recently proposed.
Here, we analyze the dynamical properties of the aPadP cycle in detail. We describe the dynamical roles of the stationary and short-term responses in determining the efficiency of information decoding and clarify the optimality of the threshold value of the stationary response and its information-theoretical meaning. Furthermore, we investigate the robustness of the aPadP cycle against the receptor inactivation time and intrinsic noise. Finally, we discuss the relationship among information decoding with information-dependent actions, bet-hedging and network modularity.
Group chase and escape
-Fusion of Pursuits-Escapes and Collective Motions-
with T. Ohira
Available from Springer (2019)
This book presents a unique fusion of two different research topics. One is related to the traditional mathematical problem of chases and escapes. The problem mainly deals with a situation where a chaser pursues an evader to analyze their trajectories and capture time. It dates back more than 300 years and has developed in various directions such as differential games. The other topic is the recently developing field of collective behavior, which investigates origins and properties of emergent behavior in groups of self-driving units. Applications include schools of fish, flocks of birds, and traffic jams. This book first reviews representative topics, both old and new, from these two areas. Then it presents the combined research topic of "group chase and escape", recently proposed by the authors. Although the combination is simple and straightforward, the book describes the emergence of rather intricate behavior, provoking the interest of readers for further developments and applications of related topics.
Research article with T. Ohira
New Journal of Physics, 12(5) 053013 (2010)
Covered by Dr. T. Vicsek in Nature
We describe here a new concept of one group chasing another, called 'group chase and escape', by presenting a simple model. We will show that even a simple model can demonstrate rather rich and complex behavior. In particular, there are cases where an optimal number of chasers exists for a given number of escapees (or targets) to minimize the cost of catching all targets. We have also found an indication of self-organized spatial structures formed by both groups.
Group chase and escape with conversion from targets to chasers
Research article with R. Nishi, K. Nishinari and T. Ohira
Physica A, 391, 337 (2012)
We study the effect of converting caught targets into new chasers in the context of the recently proposed ‘group chase and escape’ problem. Numerical simulations have shown that this conversion can substantially reduce the lifetimes of the targets when a large number of them are initially present. At the same time, it also leads to a non-monotonic dependence on the initial number of targets, resulting in the existence of a maximum lifetime. As a counter-effect for this conversion, we further introduce self-multiplying abilities to the targets. We found that the longest lifetime exists when a suitable combination of these two effects is created.
Delayed pursuit-escape as a model for virtual stick balancing
Research article with J. G. Milton, A. Fuerte, C. Belair, J. Lippai and T. Ohira
Nonlinear Theory and Its Applications(NOLTA) IEICE, 4(2), 129 (2013)
The process of pursuit and escape underlies many biological phenomena ranging from predator-prey interactions, combat and sporting activities. Time delays, τ, arise as a consequence of the time taken to identify the opponent, formulate a strategy, and then act upon it.
Here we consider virtual stick balancing (VSB) as a delayed pursuit-escape task. The movements of the target in VSB are programmed to resemble those of balancing a stick at the fingertip. A model of delayed pursuit-escape is developed by assuming that the target movements are governed by a simple random walk and the movements of the pursuer by a delayed random walk biased towards the target when τ=0. When τ > 0 the movements can become transiently biased away from the target. The model reproduces the oscillatory dynamics and statistical properties of VSB. For both model and VSB, transients occur in which the pursuer moves inappropriately causing increases in tracking error. The presence of a signature, or trigger, for impending escape suggests the possibility that escapes can be predicted before they occur.
Chasers and Escapes: From Singles to Groups
Review with S. Matsumoto and T. Ohira
Mathematical Approaches to Biological Systems: Networks, Oscillations and Collective Motions, pp. 139 (2015, Springer, Tokyo)
On transport phenomena by molecular dynamics simulations
Distribution in flowing reaction-diffusion systems
Research article with H. J. Herrmann and N. Ito
Physical Review E, 83, 061132 (2009)
A power-law distribution is found in the density profile of reacting systems A+B→C+D and 2A→2C under a flow in two and three dimensions. Different densities of reactants A and B are fixed at both ends. For the reaction A+B, the concentration of reactants asymptotically decay in space as x−1/2 and x−3/4 in two dimensions and three dimensions, respectively. For 2A, it decays as log(x)/x in two dimensions. The decay of A+B is explained considering the effect of segregation of reactants in the isotropic case. The decay for 2A is explained by the marginal behavior of two-dimensional diffusion. A logarithmic divergence of the diffusion constant with system size is found in two dimensions.
Research article with N. Ito
Journal of the Physical Society of Japan, 77, 125001 (2008)
In this paper, we have performed non-equilibrium molecular dynamics simulations on thermal diffusions of binary mixtures of hard spheres. It is confirmed that a heavier component accumulates in the cooler region and the Soret coefficient is positive. The Soret coefficient increases with mass ratios. The values of thermal diffusion factor obtained from our results are almost consistent with experimental observations of dilute gas systems. We also directly observe that the Soret coefficient shows 1/T-dependences and the profiles of inverse of the heavier component obey linear gradients under linear temperature gradients. The results are in agreement with a phenomenological expression.
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