Thermodynamics of Error Correction

Information processing at the molecular scale is limited by thermal fluctuations. This can cause undesired consequences in copying information since thermal noise can lead to errors that can compromise the functionality of the copy. For example, a high error rate during DNA duplication can lead to cell death. Given the importance of accurate copying at the molecular scale, it is fundamental to understand its thermodynamic features. In this paper, we derive a universal expression for the copy error as a function of entropy production and {\cred work dissipated by the system during wrong incorporations}. Its derivation is based on the second law of thermodynamics, hence its validity is independent of the details of the molecular machinery, be it any polymerase or artificial copying device. Using this expression, we find that information can be copied in three different regimes. In two of them, work is dissipated to either increase or decrease the error. In the third regime, the protocol extracts work while correcting errors, reminiscent of a Maxwell demon. As a case study, we apply our framework to study a copy protocol assisted by kinetic proofreading, and show that it can operate in any of these three regimes. We finally show that, for any effective proofreading scheme, error reduction is limited by the chemical driving of the proofreading reaction.Comment: 9 pages, 5 figure

Mapping of uncertainty relations between continuous and discrete time

Lower bounds on fluctuations of thermodynamic currents depend on the nature of time: discrete or continuous. To understand the physical reason, we compare current fluctuations in discrete-time Markov chains and continuous-time master equations. We prove that current fluctuations in the master equations are always more likely, due to random timings of transitions. This comparison leads to a mapping of the moments of a current between discrete and continuous time. We exploit this mapping to obtain new uncertainty bounds. Our results reduce the quests for uncertainty bounds in discrete and continuous time to a single problem.Comment: 5 pages, 3 figure

Selective advantage of diffusing faster

We study a stochastic spatial model of biological competition in which two species have the same birth and death rates, but different diffusion constants. In the absence of this difference, the model can be considered as an off-lattice version of the Voter model and presents similar coarsening properties. We show that even a relative difference in diffusivity on the order of a few percent may lead to a strong bias in the coarsening process favoring the more agile species. We theoretically quantify this selective advantage and present analytical formulas for the average growth of the fastest species and its fixation probability.Comment: 8 pages, 5 figures (Main Text + Supplementary Information). Accepted versio

Kinetic vs. energetic discrimination in biological copying

We study stochastic copying schemes in which discrimination between a right and a wrong match is achieved via different kinetic barriers or different binding energies of the two matches. We demonstrate that, in single-step reactions, the two discrimination mechanisms are strictly alternative and can not be mixed to further reduce the error fraction. Close to the lowest error limit, kinetic discrimination results in a diverging copying velocity and dissipation per copied bit. On the opposite, energetic discrimination reaches its lowest error limit in an adiabatic regime where dissipation and velocity vanish. By analyzing experimentally measured kinetic rates of two DNA polymerases, T7 and Pol{\gamma}, we argue that one of them operates in the kinetic and the other in the energetic regime. Finally, we show how the two mechanisms can be combined in copying schemes implementing error correction through a proofreading pathwayComment: 18 pages, 10 figures, main text+supplementary information. Accepted for publication in Phys. Rev. Let

A Stochastic Model for the Species Abundance Problem in an Ecological Community

We propose a model based on coupled multiplicative stochastic processes to understand the dynamics of competing species in an ecosystem. This process can be conveniently described by a Fokker-Planck equation. We provide an analytical expression for the marginalized stationary distribution. Our solution is found in excellent agreement with numerical simulations and compares rather well with observational data from tropical forests.Comment: 4 pages, 3 figures, submitted to PR

Neutral models in ecology : species abundance and extinction dynamics

In Chap. 1, we discuss some of the statistical patterns that seem to be ubiquitous in ecosystems. Evidences for these patterns are collected both in study of living ecosystem and in quantitative study of the Fossil Record. In Chap. 2, we introduce the most important models that tried to give an explanation for the observed patterns. The important concept of neutrality will be also discussed. In Chap. 3, we present a stochastic neutral model of the populations in a singletrophic level ecosystems. A continuous version of this model is analytically solvable; we compare the analytical solution with numerical simulations and with experimental data coming from studies of tropical forests. In Chap. 4, we address the problem of calculating the species lifetime distribution function for the \u201cstandard\u201d formulation of the ecological neutral theory [4, 13]. Depending on the parameters range and on the initial condition, the solution has several different asymptotic behaviors We study them and make a comparison with evidences of the fossil record. In a realistic parameter range, we obtain the correct scaling. In Chap. 5, we discuss the possible ecological implications of the result of Chap. 4. In particular, we try to \u201cscale up\u201d the predictions of the neutral theory on a long timescale. We predict in the same framework the lifetimes of species and genera and the distribution of species among genera. Finally, we outline final conclusions and perspectives

Generalized Euler-Lotka equation for correlated cell divisions

Cell division times in microbial populations display significant fluctuations, that impact the population growth rate in a non-trivial way. If fluctuations are uncorrelated among different cells, the population growth rate is predicted by the Euler-Lotka equation, which is a classic result in mathematical biology. However, cell division times can be significantly correlated, due to physical properties of cells that are passed through generations. In this paper, we derive an equation remarkably similar to the Euler-Lotka equation which is valid in the presence of correlations. Our exact result is based on large deviation theory and does not require particularly strong assumptions on the underlying dynamics. We apply our theory to a phenomenological model of bacterial cell division in E.coli and to experimental data. We find that the discrepancy between the growth rate predicted by the Euler-Lotka equation and our generalized version is relatively small, but large enough to be measurable by our approach.Comment: 6 pages, 5 figures, combined Main Text + SI. Accepted as a Letter in Physical Review

Error-speed correlations in biopolymer synthesis

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. Performance of these enzymes is usually characterized in terms of their average error rate and speed. However, because of thermal fluctuations in these single-molecule processes, both error and speed are inherently stochastic quantities. In this paper, we study fluctuations of error and speed in biopolymer synthesis and show that they are in general correlated. This means that, under equal conditions, polymers that are synthesized faster due to a fluctuation tend to have either better or worse errors than the average. The error-correction mechanism implemented by the enzyme determines which of the two cases holds. For example, discrimination in the forward reaction rates tends to grant smaller errors to polymers with faster synthesis. The opposite occurs for discrimination in monomer rejection rates. Our results provide an experimentally feasible way to identify error-correction mechanisms by measuring the error-speed correlations.Comment: PDF file consist of the main text (pages 1 to 5) and the supplementary material (pages 6 to 12). Overall, 7 figures split between main text and S

Symbolic dynamics of biological feedback networks

We formulate general rules for a coarse-graining of the dynamics, which we term `symbolic dynamics', of feedback networks with monotone interactions, such as most biological modules. Networks which are more complex than simple cyclic structures can exhibit multiple different symbolic dynamics. Nevertheless, we show several examples where the symbolic dynamics is dominated by a single pattern that is very robust to changes in parameters and is consistent with the dynamics being dictated by a single feedback loop. Our analysis provides a method for extracting these dominant loops from short time series, even if they only show transient trajectories.Comment: 4 pages, 4 figure

Species clustering in competitive Lotka-Volterra models

We study the properties of Lotka-Volterra competitive models in which the intensity of the interaction among species depends on their position along an abstract niche space through a competition kernel. We show analytically and numerically that the properties of these models change dramatically when the Fourier transform of this kernel is not positive definite, due to a pattern forming instability. We estimate properties of the species distributions, such as the steady number of species and their spacings, for different types of kernels.Comment: 4 pages, 3 figure