arXiv subjects

Eugene I. Shakhnovich

Publications and source records attributed to Eugene I. Shakhnovich.

71 records · Page 4Linked to original sources

arXiv2003

Semiconservative Replication in the Quasispecies Model

This paper extends Eigen's quasispecies equations to account for the semiconservative nature of DNA replication. We solve the equations in the limit of infinite sequence length for the simplest case of a static, sharply peaked fitness landscape. We show that the error catastrophe occurs when $ μ$, the product of sequence length and per base pair mismatch probability, exceeds $ 2 \ln \frac{2}{1 + 1/k} $, where $ k > 1 $ is the first order growth rate constant of the viable ``master'' sequence (with all other sequences having a first-order growth rate constant of $ 1 $). This is in contrast to the result of $ \ln k $ for conservative replication. In particular, as $ k \to \infty $, the error catastrophe is never reached for conservative replication, while for semiconservative replication the critical $ μ$ approaches $ 2 \ln 2 $. Semiconservative replication is therefore considerably less robust than conservative replication to the effect of replication errors. We also show that the mean equilibrium fitness of a semiconservatively replicating system is given by $ k (2 e^{-μ/2} - 1) $ below the error catastrophe, in contrast to the standard result of $ k e^{-μ} $ for conservative replication (derived by Kimura and Maruyama in 1966).

cond-mat↗

arXiv2003

The Error and Repair Catastrophes: A Two-Dimensional Phase Diagram in the Quasispecies Model

This paper develops a two gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by $ σ_{via} $, yields a viable organism with first order growth rate constant $ k > 1 $ if it is equal to some target ``master'' sequence $ σ_{via, 0} $. The second gene, denoted by $ σ_{rep} $, yields an organism capable of genetic repair if it is equal to some target ``master'' sequence $ σ_{rep, 0} $. This model is analytically solvable in the limit of infinite sequence length, and gives an equilibrium distribution which depends on $ μ\equiv L\eps $, the product of sequence length and per base pair replication error probability, and $ \eps_r $, the probability of repair failure per base pair. The equilibrium distribution is shown to exist in one of three possible ``phases.'' In the first phase, the population is localized about the viability and repairing master sequences. As $ \eps_r $ exceeds the fraction of deleterious mutations, the population undergoes a ``repair'' catastrophe, in which the equilibrium distribution is still localized about the viability master sequence, but is spread ergodically over the sequence subspace defined by the repair gene. Below the repair catastrophe, the distribution undergoes the error catastrophe when $ μ$ exceeds $ \ln k/\eps_r $, while above the repair catastrophe, the distribution undergoes the error catastrophe when $ μ$ exceeds $ \ln k/f_{del} $, where $ f_{del} $ denotes the fraction of deleterious mutations.

cond-mat.stat-mech↗

arXiv2003

Protein Evolution within a Structural Space

Understanding of the evolutionary origins of protein structures represents a key component of the understanding of molecular evolution as a whole. Here we seek to elucidate how the features of an underlying protein structural "space" might impact protein structural evolution. We approach this question using lattice polymers as a completely characterized model of this space. We develop a measure of structural comparison of lattice structures that is analgous to the one used to understand structural similarities between real proteins. We use this measure of structural relatedness to create a graph of lattice structures and compare this graph (in which nodes are lattice structures and edges are defined using structural similarity) to the graph obtained for real protein structures. We find that the graph obtained from all compact lattice structures exhibits a distribution of structural neighbors per node consistent with a random graph. We also find that subgraphs of 3500 nodes chosen either at random or according to physical constraints also represent random graphs. We develop a divergent evolution model based on the lattice space which produces graphs that, within certain parameter regimes, recapitulate the scale-free behavior observed in similar graphs of real protein structures.

cond-mat.soft↗

arXiv2003

Equilibrium Distribution of Mutators in the Single Fitness Peak Model

This paper develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak (SFP) model, which has been used in Eigen's quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This "repair catastrophe" occurs at a critical repair error probability of $ ε_r = L_{via}/L $, where $ L_{via} $ denotes the length of the genome controlling viability, while $ L $ denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in {\it Escherichia coli}.

cond-mat.stat-mech↗

arXiv2003

Structural Determinant of Protein Designability

Here we present an approximate analytical theory for the relationship between a protein structure's contact matrix and the shape of its energy spectrum in amino acid sequence space. We demonstrate a dependence of the number of sequences of low energy in a structure on the eigenvalues of the structure's contact matrix, and then use a Monte Carlo simulation to test the applicability of this analytical result to cubic lattice proteins. We find that the lattice structures with the most low-energy sequences are the same as those predicted by the theory. We argue that, given sufficiently strict requirements for foldability, these structures are the most designable, and we propose a simple means to test whether the results in this paper hold true for real proteins.

cond-mat.soft↗

arXiv2002

Phase diagram for unzipping DNA with long-range interactions

We present a critique and extension of the mean-field approach to the mechanical pulling transition in bound polymer systems. Our model is motivated by the theoretically and experimentally important examples of adsorbed polymers and double-stranded DNA, and we focus on the case in which quenched disorder in the sequence of monomers is unimportant for the statistical mechanics. We show how including excluded volume interactions in the model affects the phase diagram for the critical pulling force, and we predict a re-entrancy phase at low temperatures which has not been previously discussed. We also consider the case of non-equilibrium pulling, in which the external force probes the local, rather than the global structure of the dsDNA or adsorbed polymer. The dynamics of the pulling transition in such experiments could illuminate the polymer's loop structure, which depends on the nature of excluded volume interactions.

cond-mat.soft↗

arXiv2001

Mechanical response of random heteropolymers

We present an analytical theory for heteropolymer deformation, as exemplified experimentally by stretching of single protein molecules. Using a mean-field replica theory, we determine phase diagrams for stress-induced unfolding of typical random sequences. This transition is sharp in the limit of infinitely long chain molecules. But for chain lengths relevant to biological macromolecules, partially unfolded conformations prevail over an intermediate range of stress. These necklace-like structures, comprised of alternating compact and extended subunits, are stabilized by quenched variations in the composition of finite chain segments. The most stable arrangements of these subunits are largely determined by preferential extension of segments rich in solvophilic monomers. This predicted significance of necklace structures explains recent observations in protein stretching experiments. We examine the statistical features of select sequences that give rise to mechanical strength and may thus have guided the evolution of proteins that carry out mechanical functions in living cells.

cond-mat.stat-mech↗

arXiv2001

Reversible stretching of homopolymers and random heteropolymers

We have analyzed the equilibrium response of chain molecules to stretching. For a homogeneous sequence of monomers, the induced transition from compact globule to extended coil below the $θ$-temperature is predicted to be sharp. For random sequences, however, the transition may be smoothed by a prevalence of necklace-like structures, in which globular regions and coil regions coexist in a single chain. As we show in the context of a random copolymer, preferential solvation of one monomer type lends stability to such structures. The range of stretching forces over which necklaces are stable is sensitive to chain length as well as sequence statistics.

cond-mat.soft↗

arXiv2001

Understanding hierarchical protein evolution from first principles

We propose a model that explains the hierarchical organization of proteins in fold families. The model, which is based on the evolutionary selection of proteins by their native state stability, reproduces patterns of amino acids conserved across protein families. Due to its dynamic nature, the model sheds light on the evolutionary time scales. By studying the relaxation of the correlation function between consecutive mutations at a given position in proteins, we observe separation of the evolutionary time scales: at the short time intervals families of proteins with similar sequences and structures are formed, while at long time intervals the families of structurally similar proteins that have low sequence similarity are formed. We discuss the evolutionary implications of our model. We provide a ``profile'' solution to our model and find agreement between predicted patterns of conserved amino acids and those actually observed in nature.

cond-mat.stat-mech↗

arXiv2001

Excluded Volume in Protein Sidechain Packing

To examine the relationship between sidechain geometry and sidechain packing, we use an all-atom Monte Carlo simulation to sample the large space of sidechain conformations. We study three models of excluded volume and use umbrella sampling to effectively explore the entire space. We find that while excluded volume constraints reduce the size of conformational space by many orders of magnitude, the number of allowed conformations is still large. An average repacked conformation has 20% of its chi angles in a non-native state. Interestingly, well-packed conformations, with up to 50% non-native chi's exist. Entropy is distributed non-uniformly over positions, and we partially explain the observed distribution using rotamer probabilities derived from the pdb database. In spite of our finding that 65% of the native rotamers and 85% of chi 1 angles can be correctly predicted on the basis of excluded volume only, 95% of positions can accomodate more than 1 rotamer in simulation. We estimate that in order to quench the sidechain entropy observed in the presence of excluded volume interactions, other interactions (hydrophobic, polar, electrostatic) must provide an additional stabilization of at least 0.6 kT per residue in order to single out the native state.

cond-mat.soft↗

arXiv2000

Understanding conserved amino acids in proteins

It has been conjectured that evolution exerted pressure to preserve amino acids bearing thermodynamic, kinetic, and functional roles. In this letter we show that the physical requirement to maintain protein stability gives rise to a sequence conservatism pattern that is in remarkable agreement with that found in natural proteins. Based on the physical properties of amino acids, we propose a model of evolution that explains conserved amino acids across protein families sharing the same fold.

cond-mat.stat-mech↗

arXiv1999

Mode-coupling theory for heteropolymers

We study the Langevin dynamics of a heteropolymer by means of a mode-coupling approximation scheme, giving rise to a set of coupled integro-differential equations relating the response and correlation functions. The analysis shows that there is a regime at low temperature characterized by out-of-equilibrium dynamics, with violation of time-translational invariance and of the fluctuation-dissipation theorem. The onset of ageing dynamics at low temperatures gives new insight into the nature of the slow dynamics of a disordered polymer. We also introduce a renormalization-group treatment of our mode-coupling equations, which supports our analysis, and might be applicable to other systems.

cond-mat.dis-nn↗

arXiv1999

An Analytical Approach to the Protein Designability Problem

We present an analytical method for determining the designability of protein structures. We apply our method to the case of two-dimensional lattice structures, and give a systematic solution for the spectrum of any structure. Using this spectrum, the designability of a structure can be estimated. We outline a heirarchy of structures, from most to least designable, and show that this heirarchy depends on the potential that is used.

cond-mat.soft↗

arXiv1998

Identifying the protein folding nucleus using molecular dynamics

Molecular dynamics simulations of folding in an off-lattice protein model reveal a nucleation scenario, in which a few well-defined contacts are formed with high probability in the transition state ensemble of conformations. Their appearance determines folding cooperativity and drives the model protein into its folded conformation.

cond-mat.stat-mech↗

arXiv1998

Discrete molecular dynamics studies of the folding of a protein-like model

Background: Many attempts have been made to resolve in time the folding of model proteins in computer simulations. Different computational approaches have emerged. Some of these approaches suffer from the insensitivity to the geometrical properties of the proteins (lattice models), while others are computationally heavy (traditional MD). Results: We use a recently-proposed approach of Zhou and Karplus to study the folding of the protein model based on the discrete time molecular dynamics algorithm. We show that this algorithm resolves with respect to time the folding --- unfolding transition. In addition, we demonstrate the ability to study the coreof the model protein. Conclusion: The algorithm along with the model of inter-residue interactions can serve as a tool to study the thermodynamics and kinetics of protein models.

cond-mat.stat-mech↗

arXiv1996

Cooperativity and Stability in a Langevin Model of Protein Folding

We present two simplified models of protein dynamics based on Langevin's equation of motion in a viscous medium. We explore the effect of the potential energy function's symmetry on the kinetics and thermodynamics of simulated folding. We find that an isotropic potential energy function produces, at best, a modest degree of cooperativity. In contrast, a suitable anisotropic potential energy function delivers strong cooperativity.

cond-mat.stat-mech↗

arXiv1996

Design of Proteins with Specified Thermal Properties

We propose a new and effective means for designing stable and fast-folding polypeptide sequences using a cumulant expansion of the molecular partition function. This method is unique in that $T_{Z}$, the ``cumulant design temperature'' entered as a parameter in the design process, is predicted also to be the optimal folding temperature. The method was tested using monte-carlo folding simulations of the designed sequences, at various folding temperatures $T_{F}$. (Folding simulations were run on a cubic lattice for computational convenience, but the design process itself is lattice-independent.) Simulations confirmed that, over a wide range of $T_{Z}$, all designed sequences folded rapidly when $T_{F} \approx T_{Z}$. Additionally, highly thermostable model proteins were created simply by designing with high $T_{Z}$. The mechanism proposed in these studies provides a plausible pathway for the evolutionary design of biologically active proteins, which {\em must} fold and remain stable within a relatively narrow range of temperatures.

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