arXiv subjects

D. K. Klimov

Publications and source records attributed to D. K. Klimov.

16 recordsLinked to original sources

arXiv2008

Probing the Mechanisms of Fibril Formation Using Lattice Models

Using exhaustive Monte Carlo simulations we study the kinetics and mechanism of fibril formation using lattice models as a function of temperature and the number of chains. While these models are, at best, caricatures of peptides, we show that a number of generic features thought to govern fibril assembly are present in the toy model. The monomer, which contains eight beads made from three letters (hydrophobic, polar, and charged), adopts a compact conformation in the native state. The kinetics of fibril assembly occurs in three distinct stages. In each stage there is a cascade of events that transforms the monomers and oligomers to ordered structures. In the first "burst" stage highly mobile oligomers of varying sizes form. The conversion to the aggregation-prone conformation occurs within the oligomers during the second stage. As time progresses, a dominant cluster emerges that contains a majority of the chains. In the final stage, the aggregation-prone conformation particles serve as a template onto which smaller oligomers or monomers can dock and undergo conversion to fibril structures. The overall time for growth in the latter stages is well described by the Lifshitz-Slyazov growth kinetics for crystallization from super-saturated solutions.

q-bio.BM↗

arXiv2004

Finite size effects on calorimetric cooperativity of two-state proteins

Finite size effects on the calorimetric cooperatity of the folding-unfolding transition in two-state proteins are considered using the Go lattice models with and without side chains. We show that for models without side chains a dimensionless measure of calorimetric cooperativity kappa2 defined as the ratio of the van't Hoff to calorimetric enthalpy does not depend on the number of amino acids N. The average value of kappa2 is about 3/4 which is lower than the experimental value kappa2=1. For models with side chains kappa2 approaches unity as kappa2 \sim N^mu, where exponent mu=0.17. Above the critical chain length Nc =135 these models can mimic the truly all-or-non folding-unfolding transition.

q-bio.BM↗

arXiv2004

Finite size effects on thermal denaturation of globular proteins

Finite size effects on the cooperative thermal denaturation of proteins are considered. A dimensionless measure of cooperativity, Omega, scales as N^zeta, where N is the number of amino acids. Surprisingly, we find that zeta is universal with zeta = 1 + gamma, where the exponent gamma characterizes the divergence of the susceptibility for a self-avoiding walk. Our lattice model simulations and experimental data are consistent with the theory. Our finding rationalizes the marginal stability of proteins and substantiates the earlier predictions that the efficient folding of two-state proteins requires the folding transition temperature to be close to the collapse temperature.

q-bio.BM↗

arXiv2003

Thermal denaturation and folding rates of single domain proteins: size matters

We analyze the dependence of thermal denaturation transition and folding rates of globular proteins on the number of amino acid residues, N. Using lattice Go models we show that DeltaT/T_F ~ N^-1, where T_F is the folding transition temperature and DeltaT is the folding transition width. This finding is consistent with finite size effects expected for the systems undergoing a phase transition from a disordered to an ordered phase. The dependence of the folding rates k_F on N for lattice models and the dataset of 57 proteins and peptides shows that k_F = k_F^0 exp(-CN^beta) provides a good fit, if 0 < beta <= 2/3 and C is a constant. We find that k_F = k_F^0 exp(-1.1N^0.5) with k_F^0 =(0.4x10^-6 s)^-1 can estimate optimal protein folding rates to within an order of magnitude in most cases. By using this fit for a set of proteins with beta-sheet topology we find that k_F^0 is approximately equal to k_U^0, the prefactor for unfolding rates. The maximum ratio of k_U^0/k_F^0 is 10 for this class of proteins.

q-bio.BM↗

arXiv2002

Dependence of folding rates on protein length

Using three-dimensional Go lattice models with side chains for proteins, we investigate the dependence of folding times on protein length. In agreement with previous theoretical predictions, we find that the folding time grows as a power law with the chain length N with exponent $λ\approx 3.6$ for the Go model, in which all native interactions (i.e., between all side chains and backbone atoms) are uniform. If the interactions between side chains are given by pairwise statistical potentials, which introduce heterogeneity in the contact energies, then the power law fits yield large $λ$ values that typically signifies a crossover to an underlying activated process. Accordingly, the dependence of folding time is best described by the stretched exponential \exp(\sqrt{N}). The study also shows that the incorporation of side chains considerably slows down folding by introducing energetic and topological frustration.

cond-mat.soft↗

arXiv2002

Folding in lattice models with side chains

The folding kinetics of three-dimensional lattice Go models with side chains is studied using two different Monte Carlo move sets. A flexible move set based on single, double and triple backbone moves is found to be far superior compared to the standard Monte Carlo dynamics. It was found that the folding time grows as a power law with the chain length and the corresponding exponent $λ\approx 3.6$ for Go models. The study shows that the incorporation of side chains dramatically slows down folding rates.

cond-mat.soft↗

arXiv2001

Introducing Protein Folding Using Simple Models

We discuss recent theoretical developments in the study of simple lattice models of proteins. Such models are designed to understand general features of protein structures and mechanism of folding. Among the topics covered are (i) the use of lattice models to understand the selection of the limited set of viable protein folds; (ii) the relationship between structure and sequence spaces; (iii) the application of lattice models for studying folding mechanisms (topological frustration, kinetic partitioning mechanism). Classification of folding scenarios based on the intrinsic thermodynamic properties of a sequence (namely, the collapse and folding transition temperatures) is outlined. A brief discussion of random heteropolymer model is also presented.

cond-mat.soft↗

arXiv1999

Emergence of stable and fast folding protein structures

The number of protein structures is far less than the number of sequences. By imposing simple generic features of proteins (low energy and compaction) on all possible sequences we show that the structure space is sparse compared to the sequence space. Even though the sequence space grows exponentially with N (the number of amino acids) we conjecture that the number of low energy compact structures only scales as ln N. This implies that many sequences must map onto countable number of basins in the structure space. The number of sequences for which a given fold emerges as a native structure is further reduced by the dual requirements of stability and kinetic accessibility. The factor that determines the dual requirement is related to the sequence dependent temperatures, T_θ(collapse transition temperature) and T_F (folding transition temperature). Sequences, for which σ=(T_θ-T_F)/T_θis small, typically fold fast by generically collapsing to the native-like structures and then rapidly assembling to the native state. Such sequences satisfy the dual requirements over a wide temperature range. We also suggest that the functional requirement may further reduce the number of sequences that are biologically competent. The scheme developed here for thinning of the sequence space that leads to foldable structures arises naturally using simple physical characteristics of proteins. The reduction in sequence space leading to the emergence of foldable structures is demonstrated using lattice models of proteins.

cond-mat.soft↗

arXiv1999

Stretching Single Domain Proteins: Phase Diagram and Kinetics of Force-Induced Unfolding

Single molecule force spectroscopy reveals unfolding of domains in titin upon stretching. We provide a theoretical framework for these experiments by computing the phase diagrams for force-induced unfolding of single domain proteins using lattice models. The results show that two-state folders (at zero force) unravel cooperatively whereas stretching of non-two-state folders occurs through intermediates. The stretching rates of individual molecules show great variations reflecting the heterogeneity of force-induced unfolding pathways. The approach to the stretched state occurs in a step-wise "quantized" manner. Unfolding dynamics depends sensitively on topology. The unfolding rates increase exponentially with force f till an optimum value which is determined by the barrier to unfolding when f=0. A mapping of these results to proteins shows qualitative agreement with force-induced unfolding of Ig-like domains in titin. We show that single molecule force spectroscopy can be used to map the folding free energy landscape of proteins in the absence of denaturants.

cond-mat.soft↗

arXiv1998

Linking Rates of Folding in Lattice Models of Proteins with Underlying Thermodynamic Characteristics

We investigate the sequence-dependent properties of proteins that determine the dual requirements of stability of the native state and its kinetic accessibility using simple cubic lattice models. Three interaction schemes are used to describe the potentials between residues. We show that, under the simulation conditions when the native basin of attraction (NBA) is stable, there is an excellent correlation between folding times (τ_{F}) and the dimensionless parameter (σ_{T} = (T_θ - T_{F})/T_θ), where (T_θ) is the collapse temperature and (T_{F}) is the folding transition temperature. There is also a significant correlation between (τ_{F}) and Z-score (Z=(E_{N}-E_{ms})/δ), where (E_{N}) is the energy of the native state, (E_{ms}) is the average energy of the ensemble of misfolded structures, and (δ) is the dispersion in contact energies. An approximate relationship between (σ_{T}) and the Z-score is derived, which explains the superior correlation seen between (τ_{F}) and (σ_{T}). For two state folders (τ_{F}) is linked to the free energy difference between the unfolded states and the NBA.

cond-mat.soft↗

arXiv1998

Cooperativity in Protein Folding: From Lattice Models with Side Chains to Real Proteins

We consider equilibrium folding transitions in lattice protein models with and without side chains. A dimensionless measure, $Omega_{c}$, is introduced to quantitatively assess the degree of cooperativity in lattice models and in real proteins. We show that larger values of $Ω_{c}$ resembling those seen in proteins are obtained in lattice models with side chains (LMSC). The enhanced cooperativity in LMSC is due to the possibility of denser packing of side chains in the interior of the model protein. We also establish that $Ω_{c}$ correlates extremely well with (σ= (T_θ -T_{f} )/T_θ), where (T_θ) and (T_{f}) are collapse and folding transition temperatures, respectively. These theoretical ideas are used to analyze folding transitions in various real proteins. The values of $Ω_{c}$ extracted from experiments show a correlation with $σ$. We conclude that the degree of cooperativity can be expressed in terms of the single parameter $σ$, which can be estimated from experimental data.

cond-mat.soft↗

arXiv1997

Kinetic Partitioning Mechanism as a Unifying Theme in the Folding of Biomolecules

We present a unified framework for folding kinetics of proteins and RNA. The basis for this framework relies on the notion of topological frustration, which gives rise to several competing basins of attraction (CBA) in addition to the native basin of attraction (NBA) on the free energy surface. A rough free energy surface results in direct and indirect pathways to the NBA, i.e., a kinetic partitioning mechanism (KPM). The unified framework for folding kinetics allows us to propose a foldability principle, according to which fast folding sequences are characterized by the folding transition temperature $T_{F}$ being close to the collapse transition temperature $T_{θ}$. Biomolecules, for which foldability principle is satisfied, such as small proteins and tRNAs, are expected to fold rapidly with two-state kinetics. Estimates for the multiple time scales in KPM are also given.

cond-mat.soft↗

arXiv1997

Viscosity Dependence of the Folding Rates of Proteins

The viscosity dependence of the folding rates for four sequences (the native state of three sequences is a beta-sheet, while the fourth forms an alpha-helix) is calculated for off-lattice models of proteins. Assuming that the dynamics is given by the Langevin equation we show that the folding rates increase linearly at low viscosities η, decrease as 1/ηat large ηand have a maximum at intermediate values. The Kramers theory of barrier crossing provides a quantitative fit of the numerical results. By mapping the simulation results to real proteins we estimate that for optimized sequences the time scale for forming a four turn α-helix topology is about 500 nanoseconds, whereas the time scale for forming a beta-sheet topology is about 10 microseconds.

cond-mat.soft↗

arXiv1996

Protein Folding Kinetics: Time Scales, Pathways, and Energy Landscapes in Terms of Sequence Dependent Properties

The folding kinetics of a number of sequences for off-lattice continuum model of proteins is studied using Langevin simulations at two values of the friction coefficient. We show that there is a remarkable correlation between folding times, $τ_{F}$, and $σ= (T_{θ} - T_{F})/T_{θ} $, where $T_{θ}$ and $T_{F}$ are the equilibrium collapse and folding transition temperatures, respectively. The microscopic dynamics reveals several scenarios for the refolding kinetics depending on the values of $σ$. Proteins with small $σ$ reach the native conformation via a nucleation collapse mechanism and their energy landscape is characterized by single dominant native basin of attraction. Proteins with large $σ$ get trapped in competing basins of attraction, in which they adopt misfolded structures. In this case only a small fraction of molecules $Φ$ access the native state rapidly, the majority of them approach the native state by a three stage multipathway mechanism. The partition factor $Φ$ is determined by $σ$: smaller the value of $σ$ larger is $Φ$. The qualitative aspects of our results are found to be independent of the friction coefficient. Estimates for time scales for folding of small proteins via a nucleation collapse mechanism are presented.

cond-mat.stat-mech↗

arXiv1996

Reply to Comment on "Criterion that Determines the Foldability of Proteins"

We point out that the correlation between folding times and $σ= (T_{θ} - T_{f})/T_{θ}$ in protein-like heteropolymer models where $T_{θ}$ and $T_{f}$ are the collapse and folding transition temperatures was already established in 1993 before the other presumed equivalent criterion (folding times correlating with $T_{f}$ alone) was suggested. We argue that the folding times for these models show no useful correlation with the energy gap even if restricted to the ensemble of compact structures as suggested by Karplus and Shakhnovich (cond-mat/9606037).

cond-mat↗

arXiv1996

Factors Governing the Foldability of Proteins

We use a three dimensional cubic lattice model of proteins to study their properties that determine folding to the native state. The protein chain is modeled as a sequence of $N$ beads. The interactions between beads are taken from a Gaussian distribution of energies. We studied 56 sequences with unique ground states for $N = 15$ and $27$. Thermodynamic and kinetic properties were determined using Monte Carlo simulations and exhaustive enumeration. For all sequences we find collapse temperature, $T_θ$, at which the protein collapses into compact structure, and folding temperature, $T_{f}$, at which the protein acquires the native state. We show that parameter $σ= (T_θ - T_{f})/T_θ$ correlates extremely well with folding times. Fast folders reach the native state via a nucleation collapse mechanism without forming any intermediates, whereas for moderate and slow folders only a fraction of molecules $Φ$ reaches the native state by this process. The remaining fraction folds via three stage multipathway process. The simultaneous requirement of native state stability and kinetic accessibility can be achieved for the sequences with small values of $σ$.

cond-mat↗