Ramzi R. Khuri
I use a mean-field approximation to show the rapid collapse of heteropolymers from random walk size to a much smaller, molten globule state due to hydrophobic interactions, to be followed by a slower annealing process in which there is little change in overall size.
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Various subsets of self-avoiding walks naturally appear when investigating existing methods designed to predict the 3D conformation of a protein of interest. Two such subsets, namely the folded and the unfoldable self-avoiding walks, are studied computationally in this article. We show that these two sets are equal and correspond to the whole $n$-step self-avoiding walks for $n\leqslant 14$, but that they are different for numerous $n \geqslant 108$, which are common protein lengths. Concrete counterexamples are provided and the computational methods used to discover them are completely detail…
Superparamagnetism of tryptophan implying the presence of magnetic domain is reported. The observation helps us to conceive assembly of proteins as a physical lattice gas with multidimensional Ising character, each lattice points assuming discrete spin states. When magnetic field is applied the equilibrium is lost and the population density of one spin state increases (unidirectional alignment), resulting in net magnetization. Spatial coherence between the identical spin states further imparts a ferromagnetic memory. This effect is observed using direct nanoscale video imaging. Out of the thre…
We present a numerical Monte Carlo analysis of a continuos spin Ising chain that can describe the statistical proterties of folded proteins. We find that depending on the value of the Metropolis temperature, the model displays the three known nontrivial phases of polymers: At low temperatures the model is in a collapsed phase, at medium temperatures it is in a random walk phase, and at high temperatures it enters the self-avoiding random walk phase. By investigating the temperature dependence of the specific energy we confirm that the transition between the collapsed phase and the random walk …
CSAW (conditioned self-avoiding walk) is a model of protein folding that combines SAW (self-avoiding walk) with Monte-Carlo. It simulates the Brownian motion of a chain molecule in the presence of interactions, both among chain residues, and with the environment. In a first model that includes the hydrophobic effect and hydrogen bonding, a chain of 30 residues folds into a native state with stable secondary and tertiary structures. The process starts with a rapid collapse into an intermediate "molten globule", which slowly decays into the native state afer a relatively long quiescent period. T…
In the traditional random-conformational-search model, various hypotheses with a series of meta-stable intermediate states were often proposed to resolve the Levinthal paradox. Here we introduce a quantum strategy to formulate protein folding as a quantum walk on a definite graph, which provides us a general framework without making hypotheses. Evaluating it by the mean of first passage time, we find that the folding time via our quantum approach is much shorter than the one obtained via classical random walks. This idea is expected to evoke more insights for future studies.
We have studied a random walk model based on majority rule. At a given instant, the moving direction of a cargo is determined by motor coordination mediated by a tug-of-war mechanism between two kinds of competing motor proteins. We have demonstrated that the probability distribution $P(t)$ for unidirectional run time $t$ of a cargo can be remarkably described by Levy walk for $t<γ_u^{-1}$ as $P(t)\propto t^{-3/2} e^{-γ_u t}$ with $γ_u$ being the unbinding rate of a motor protein from microtubule. The mean squared displacement of a cargo changes from super-diffusive behavior $\langle X^2\rangl…
Single-particle tracking offers detailed information about the motion of molecules in complex environments such as those encountered in live cells, but the interpretation of experimental data is challenging. One of the most powerful tools in the characterization of random processes is the power spectral density. However, because anomalous diffusion processes in complex systems are usually not stationary, the traditional Wiener-Khinchin theorem for the analysis of power spectral densities is invalid. Here, we employ a recently developed tool named aging Wiener-Khinchin theorem to derive the pow…
All known life forms are based upon a hierarchy of interwoven feedback loops, operating over a cascade of space, time and energy scales. Among the most basic loops are those connecting DNA and proteins. For example, in genetic networks, DNA genes are expressed as proteins, which may bind near the same genes and thereby control their own expression. In this molecular type of self-reference, information is mapped from the DNA sequence to the protein and back to DNA. There is a variety of dynamic DNA-protein self-reference loops, and the purpose of this remark is to discuss certain geometrical an…
The proposal of this paper is to provide a simple angular random walk model to build up polypeptide structures, which encompass properties of dihedral angles of folded proteins. From this model, structures will be built with lengths ranging from 125 up to 400 amino acids for the different fractions of secondary structure motifs, which dihedral angles were randomly chosen according to narrow Gaussian probability distributions. In order to measure the fractal dimension of proteins three different cases were analyzed. The first contained alpha-helix structures only, the second beta-strands struct…
We present an exact enumeration algorithm for identifying the {\it native} configuration - a maximally compact self avoiding walk configuration that is also the minimum energy configuration for a given set of contact-energy schemes; the process is implicitly sequence-dependent. In particular, we show that the 25-step native configuration on a diamond lattice consists of two sheet-like structures and is the same for all the contact-energy schemes, ${(-1,0,0);(-7,-3,0); (-7,-3,-1); (-7,-3,1)}$; on a square lattice also, the 24-step native configuration is independent of the energy schemes consid…
We report on atomistic simulation of the folding of a natively-knotted protein, MJ0366, based on a realistic force field. To the best of our knowledge this is the first reported effort where a realistic force field is used to investigate the folding pathways of a protein with complex native topology. By using the dominant-reaction pathway scheme we collected about 30 successful folding trajectories for the 82-amino acid long trefoil-knotted protein. Despite the dissimilarity of their initial unfolded configuration, these trajectories reach the natively-knotted state through a remarkably simila…
The symport of lactose and H+ is an important physiological process in E. coli, for it is closely related to cellular energy supply. In this paper, the symport of H+ and lactose by E. coli LacY protein is computationally simulated using a newly proposed cotransport model that takes the "leakage" phenomenon (uncoupled sugar translocation) into account and also satisfies the static head equilibrium condition. Then, we study the equilibrium properties of this cotransport process including equilibrium solution and the time required to reach equilibrium state, when varying the parameters of the ini…
Natural protein sequences that self-assemble to form globular structures are compact with high packing densities in the folded states. It is known that proteins unfold upon addition of denaturants, adopting random coil structures. The dependence of the radii of gyration on protein size in the folded and unfolded states obeys the same scaling laws as synthetic polymers. Thus, one might surmise that the mechanism of collapse in proteins and polymers ought to be similar. However, because the number of amino acids in single domain proteins is not significantly greater than about two hundred, it ha…
Geometric and structural constraints greatly restrict the selection of folds adapted by protein backbones, and yet, folded proteins show an astounding diversity in functionality. For structure to have any bearing on function, it is thus imperative that, apart from the protein backbone, other tunable degrees of freedom be accountable. Here, we focus on side-chain interactions, which non-covalently link amino acids in folded proteins to form a network structure. At a coarse-grained level, we show that the network conforms remarkably well to realizations of random graphs and displays associated p…
A theoretical calculation is presented to describe the confined motion of transmembrane molecules in cell membranes. The study is analytic, based on Master equations for the probability of the molecules moving as random walkers, and leads to explicit usable solutions including expressions for the molecular mean square displacement and effective diffusion constants. One outcome is a detailed understanding of the dependence of the time variation of the mean square displacement on the initial placement of the molecule within the confined region. How to use the calculations is illustrated by extra…
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