General protocol for predicting outbreaks of infectious diseases in social networks.

Epidemic spreading on social networks with quenched connections is strongly influenced by dynamic correlations between connected nodes, posing theoretical challenges in predicting outbreaks of infectious diseases. The quenched connections introduce dynamic correlations, indicating that the infection of one node increases the likelihood of infection among its neighboring nodes. These dynamic correlations pose significant difficulties in developing comprehensive theories for threshold determination. Determining the precise epidemic threshold is pivotal for diseases control. In this study, we propose a general protocol for accurately determining epidemic thresholds by introducing a new set of fundamental conditions, where the number of connections between individuals of each type remains constant in the stationary state, and by devising a rescaling method for infection rates. Our general protocol is applicable to diverse epidemic models, regardless of the number of stages and transmission modes. To validate our protocol's effectiveness, we apply it to two widely recognized standard models, the susceptible-infected-recovered-susceptible model and the contact process model, both of which have eluded precise threshold determination using existing sophisticated theories. Our results offer essential tools to enhance disease control strategies and preparedness in an ever-evolving landscape of infectious diseases.

Translocation of Hydrophobic Polyelectrolytes under Electrical Field: Molecular Dynamics Study.

We studied the translocation of polyelectrolyte (PE) chains driven by an electric field through a pore by means of molecular dynamics simulations of a coarse-grained HP model mimicking high salt conditions. Charged monomers were considered as polar (P) and neutral monomers as hydrophobic (H). We considered PE sequences that had equally spaced charges along the hydrophobic backbone. Hydrophobic PEs were in the globular form in which H-type and P-type monomers were partially segregated and they unfolded in order to translocate through the narrow channel under the electric field. We provided a quantitative comprehensive study of the interplay between translocation through a realistic pore and globule unraveling. By means of molecular dynamics simulations, incorporating realistic force fields inside the channel, we investigated the translocation dynamics of PEs at various solvent conditions. Starting from the captured conformations, we obtained distributions of waiting times and drift times at various solvent conditions. The shortest translocation time was observed for the slightly poor solvent. The minimum was rather shallow, and the translocation time was almost constant for medium hydrophobicity. The dynamics were controlled not only by the friction of the channel, but also by the internal friction related to the uncoiling of the heterogeneous globule. The latter can be rationalized by slow monomer relaxation in the dense phase. The results were compared with those from a simplified Fokker-Planck equation for the position of the head monomer.

Translocation, Rejection and Trapping of Polyampholytes.

Polyampholytes (PA) are a special class of polymers comprising both positive and negative monomers along their sequence. Most proteins have positive and negative residues and are PAs. Proteins have a well-defined sequence while synthetic PAs have a random charge sequence. We investigated the translocation behavior of random polyampholyte chains through a pore under the action of an electric field by means of Monte Carlo simulations. The simulations incorporated a realistic translocation potential profile along an extended asymmetric pore and translocation was studied for both directions of engagement. The study was conducted from the perspective of statistics for disordered systems. The translocation behavior (translocation vs. rejection) was recorded for all 220 sequences comprised of N = 20 charged monomers. The results were compared with those for 107 random sequences of N = 40 to better demonstrate asymptotic laws. At early times, rejection was mainly controlled by the charge sequence of the head part, but late translocation/rejection was governed by the escape from a trapped state over an antagonistic barrier built up along the sequence. The probability distribution of translocation times from all successful attempts revealed a power-law tail. At finite times, there was a population of trapped sequences that relaxed very slowly (logarithmically) with time. If a subensemble of sequences with prescribed net charge was considered the power-law decay was steeper for a more favorable net charge. Our findings were rationalized by theoretical arguments developed for long chains. We also provided operational criteria for the translocation behavior of a sequence, explaining the selection by the translocation process. From the perspective of protein translocation, our findings can help rationalize the behavior of intrinsically disordered proteins (IDPs), which can be modeled as polyampholytes. Most IDP sequences have a strong net charge favoring translocation. Even for sequences with those large net charges, the translocation times remained very dispersed and the translocation was highly sequence-selective.

Derivation of evolutionary entropy in the steady-state thermodynamics of evolutionary dynamics.

We investigate the parallel mutation-selection model with varying population size, which is formulated in terms of individuals undergoing the evolution processes of reproduction and mutation, to derive evolutionary entropy. Under the framework of the steady-state thermodynamics for evolutionary dynamics, the excess growth (the difference between the maximum growth rate and the total growth rate) can be interpreted as the evolutionary entropy defined in terms of the probability distributions characteristic of evolutionary dynamics. The Clausius inequality states that the excess growth is always less than or equal to the entropy difference in evolutionary dynamics. Analytically, by using the genome sequence length L=3, we derive the growth after evolutionary dynamics with the finite number of environmental changes and calculate the entropy difference during this evolutionary dynamics, and we verify the Clausius inequality. Furthermore, by taking the infinite limit of the number of environmental changes, we verify that the equality holds for the quasistatic environmental change. By using the derived evolutionary entropy, we propose the thermodynamic relation between the free fitness and evolutionary entropy, where the free fitness is the maximum growth rate possible. Numerically, we use the Gillespie-type simulations, which provides direct realizations of the master equation governing evolutionary dynamics, to verify the Clausius inequality and we find that the simulation results are in good agreement with the analytic results.

The Measurement of Information and Free Energy in Mechanical-Force-Driven Coil-Globule Transitions.

We study the role of information (the relative entropy) for polymers undergoing coil-globule transitions driven by a time-dependent force. Pulling experiments at various speeds are performed by Brownian dynamics simulations. We obtain the work distributions for the forward and time-reversed backward processes and information stored at the end of the nonequilibrium pulling processes. We present the systematic method to measure the information from the pulling experiments and extract the information by analyzing slowly relaxing modes. When the information is incorporated, the work distributions modified by the information allow access to the proper free energy via the formulation of the generalized fluctuation theorems even if the initial states of the forward and time-reversed backward processes are out of equilibrium. This demonstrates that the work-information conversion works well for a single-molecule system with many degrees of freedom.

Free energy measurements by the generalized fluctuation theorems: Theory and numerical study of a model filament.

We measure the free energy of a model filament, which undergoes deformations and structural transitions, as a function of its extension, in silico. We perform Brownian Dynamics (BD) simulations of pulling experiments at various speeds, following a protocol close to experimental ones. The results from the fluctuation theorems are compared with the estimates from Monte Carlo (MC) simulation, where the rugged free energy landscape is produced by the density of states method. The fluctuation theorems (FT) give accurate estimates of the free energy up to moderate pulling speeds. At higher pulling speeds, the work distributions do not efficiently sample the domain of small work and FT slightly overestimates free energy. In order to comprehend the differences, we analyze the work distributions from the BD simulations in the framework of trajectory thermodynamics and propose the generalized fluctuation theorems that take into account the information (relative entropy) evaluated in the expanded phase space. The measured work - free energy relation is consistent with the results obtained from the generalized fluctuation theorems. We discuss operational methods to improve the estimates at high pulling speed.

Modularity enhances the rate of evolution in a rugged fitness landscape.

Biological systems are modular, and this modularity affects the evolution of biological systems over time and in different environments. We here develop a theory for the dynamics of evolution in a rugged, modular fitness landscape. We show analytically how horizontal gene transfer couples to the modularity in the system and leads to more rapid rates of evolution at short times. The model, in general, analytically demonstrates a selective pressure for the prevalence of modularity in biology. We use this model to show how the evolution of the influenza virus is affected by the modularity of the proteins that are recognized by the human immune system. Approximately 25% of the observed rate of fitness increase of the virus could be ascribed to a modular viral landscape.

Quasispecies theory for evolution of modularity.

Biological systems are modular, and this modularity evolves over time and in different environments. A number of observations have been made of increased modularity in biological systems under increased environmental pressure. We here develop a quasispecies theory for the dynamics of modularity in populations of these systems. We show how the steady-state fitness in a randomly changing environment can be computed. We derive a fluctuation dissipation relation for the rate of change of modularity and use it to derive a relationship between rate of environmental changes and rate of growth of modularity. We also find a principle of least action for the evolved modularity at steady state. Finally, we compare our predictions to simulations of protein evolution and find them to be consistent.

Evolution dynamics of a model for gene duplication under adaptive conflict.

We present and solve the dynamics of a model for gene duplication showing escape from adaptive conflict. We use a Crow-Kimura quasispecies model of evolution where the fitness landscape is a function of Hamming distances from two reference sequences, which are assumed to optimize two different gene functions, to describe the dynamics of a mixed population of individuals with single and double copies of a pleiotropic gene. The evolution equations are solved through a spin coherent state path integral, and we find two phases: one is an escape from an adaptive conflict phase, where each copy of a duplicated gene evolves toward subfunctionalization, and the other is a duplication loss of function phase, where one copy maintains its pleiotropic form and the other copy undergoes neutral mutation. The phase is determined by a competition between the fitness benefits of subfunctionalization and the greater mutational load associated with maintaining two gene copies. In the escape phase, we find a dynamics of an initial population of single gene sequences only which escape adaptive conflict through gene duplication and find that there are two time regimes: until a time t single gene sequences dominate, and after t double gene sequences outgrow single gene sequences. The time t is identified as the time necessary for subfunctionalization to evolve and spread throughout the double gene sequences, and we show that there is an optimum mutation rate which minimizes this time scale.

Evolutionary processes in finite populations.

We consider the evolution of large but finite populations on arbitrary fitness landscapes. We describe the evolutionary process by a Markov-Moran process. We show that to O(1/N), the time-averaged fitness is lower for the finite population than it is for the infinite population. We also show that fluctuations in the number of individuals for a given genotype can be proportional to a power of the inverse of the mutation rate. Finally, we show that the probability for the system to take a given path through the fitness landscape can be nonmonotonic in system size.

Effect of robustness on selection of a mutation-rate regulating gene.

We study the evolution of a population of sequences, where each sequence is divided into a reproduction-rate (fitness) encoding part and a mutation-rate regulating part. Evolutionary selection acts on the sequence both by a direct fitness landscape and by indirect selection on a mutation landscape through which the sequence's mutation rate is determined, thereby providing a model of a mutation-rate-regulating gene. Coupling of the fitness landscape and mutation landscape leads to adaptive evolution of the sequence. We investigate the effects of robustness in the mutation landscape and fitness landscapes on selection of the sequence. We find that the effects of robustness in both the mutation and the fitness landscape can be described by an effective sequence length, defined as the mutational load divided by the per-base mutation rate, and we give expressions for the effective sequence length for various fitness and mutation landscapes. The probability that the sequence with a reduced mutation rate evolves is increased by increasing the robustness of the mutation landscape, and decreased by increasing the robustness of the fitness landscape. However, in the case of the mutation-rate-regulating part making up only a very small part of the total sequence length, we show that selection for a more robust sequence with less-reduced mutation rate is very weak, and therefore we conjecture that robust sequences play little role in selection of error-reducing mechanisms in real populations.

Optimal mutation rates in dynamic environments: The Eigen model.

We consider the Eigen quasispecies model with a dynamic environment. For an environment with sharp-peak fitness in which the most-fit sequence moves by k spin-flips each period T we find an asymptotic stationary state in which the quasispecies population changes regularly according to the regular environmental change. From this stationary state we estimate the maximum and the minimum mutation rates for a quasispecies to survive under the changing environment and calculate the optimum mutation rate that maximizes the population growth. Interestingly we find that the optimum mutation rate in the Eigen model is lower than that in the Crow-Kimura model, and at their optimum mutation rates the corresponding mean fitness in the eigenmodel is lower than that in the Crow-Kimura model, suggesting that the mutation process which occurs in parallel to the replication process as in the Crow-Kimura model gives an adaptive advantage under changing environment.

Quasispecies theory for finite populations.

We present stochastic, finite-population formulations of the Crow-Kimura and Eigen models of quasispecies theory, for fitness functions that depend in an arbitrary way on the number of mutations from the wild type. We include back mutations in our description. We show that the fluctuation of the population numbers about the average values is exceedingly large in these physical models of evolution. We further show that horizontal gene transfer reduces by orders of magnitude the fluctuations in the population numbers and reduces the accumulation of deleterious mutations in the finite population due to Muller's ratchet. Indeed, the population sizes needed to converge to the infinite population limit are often larger than those found in nature for smooth fitness functions in the absence of horizontal gene transfer. These analytical results are derived for the steady state by means of a field-theoretic representation. Numerical results are presented that indicate horizontal gene transfer speeds up the dynamics of evolution as well.

Maximum, minimum, and optimal mutation rates in dynamic environments.

We analyze the dynamics of the parallel mutation-selection quasispecies model with a changing environment. For an environment with the sharp-peak fitness function in which the most fit sequence changes by k spin flips every period T , we find analytical expressions for the minimum and maximum mutation rates for which a quasispecies can survive, valid in the limit of large sequence size. We find an asymptotic solution in which the quasispecies population changes periodically according to the periodic environmental change. In this state we compute the mutation rate that gives the optimal mean fitness over a period. We find that the optimal mutation rate per genome, k/T , is independent of genome size, a relationship which is observed across broad groups of real organisms.

Quasispecies theory for horizontal gene transfer and recombination.

We introduce a generalization of the parallel, or Crow-Kimura, and Eigen models of molecular evolution to represent the exchange of genetic information between individuals in a population. We study the effect of different schemes of genetic recombination on the steady-state mean fitness and distribution of individuals in the population, through an analytic field theoretic mapping. We investigate both horizontal gene transfer from a population and recombination between pairs of individuals. Somewhat surprisingly, these nonlinear generalizations of quasispecies theory to modern biology are analytically solvable. For two-parent recombination, we find two selected phases, one of which is spectrally rigid. We present exact analytical formulas for the equilibrium mean fitness of the population, in terms of a maximum principle, which are generally applicable to any permutation invariant replication rate function. For smooth fitness landscapes, we show that when positive epistatic interactions are present, recombination or horizontal gene transfer introduces a mild load against selection. Conversely, if the fitness landscape exhibits negative epistasis, horizontal gene transfer or recombination introduces an advantage by enhancing selection towards the fittest genotypes. These results prove that the mutational deterministic hypothesis holds for quasispecies models. For the discontinuous single sharp peak fitness landscape, we show that horizontal gene transfer has no effect on the fitness, while recombination decreases the fitness, for both the parallel and the Eigen models. We present numerical and analytical results as well as phase diagrams for the different cases.

Generating function, path integral representation, and equivalence for stochastic exclusive particle systems.

We present the path integral representation of the generating function for classical exclusive particle systems. By introducing hard-core bosonic creation and annihilation operators and appropriate commutation relations, we construct the Fock space structure. Using the state vector, the generating function is defined and the master equation of the system is transformed into the equation for the generating function. Finally, the solution of the linear equation for the generating function is derived in the form of the path integral. Applying the formalism, the equivalence of reaction-diffusion processes of single species and two species is described.

Universality classification of restricted solid-on-solid type surface growth models.

We consider the restricted solid-on-solid (RSOS) type surface growth models and classify them into dynamic universality classes according to their symmetry and conservation law. Four groups of RSOS-type microscopic models--asymmetric (A), asymmetric-conserved (AC), symmetric (S), and symmetric-conserved (SC) groups--are introduced and the corresponding stochastic differential equations (SDEs) are derived. Analyzing these SDEs using dynamic renormalization group theory, we confirm the previous results that A-RSOS, AC-RSOS, and S-RSOS groups belong to the Kardar-Parisi-Zhang class, the Villain-Lai-Das Sarma class, and the Edwards-Wilkinson class, respectively. We also find that SC-RSOS group belongs to a new universality class featuring the conserved-cubic nonlinearity.

Universality class of the restricted solid-on-solid model with hopping.

We study the restricted solid-on-solid (RSOS) model with finite hopping distance l(0), using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E 62, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the l(0)=infinity case, which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of l(0). Consequently, we find that the CRSOS model belongs to the VLD universality class and that the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.

Derivation of continuum stochastic equations for discrete growth models.

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

Physical model for the gating mechanism of ionic channels.

We propose a physical model for the gating mechanism of ionic channels. First, we investigate the fluctuation-mediated interactions between two proteins imbedded in a cellular membrane and find that the interaction depends on their orientational configuration as well as the distance between them. The orientational dependence of interactions arises from the fact that the noncircular cross-sectional shapes of individual proteins constrain fluctuations of the membrane differently according to their orientational configuration. Then, we apply these interactions to ionic channels composed of four, five, and six proteins. As the gating stimulus creates the changes in the structural shape of proteins composing ionic channels, the orientational configuration of the ionic channels changes due to the free energy minimization, and ionic channels are open or closed according to the conformation thereof.