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Topological constraints can affect both equilibrium and dynamic properties of polymer systems, and can play a role in the organization of chromosomes. Despite many theoretical studies, the effects of topological constraints on the equilibrium state of a single compact polymer have not been systematically studied. Here we use simulations to address this longstanding problem. We find that sufficiently long unknotted polymers differ from knotted ones in the spatial and topological states of their subchains. The unknotted globule has subchains that are mostly unknotted and form asymptotically compact $R_G(s) \sim s^{1/3}$ crumples. However, crumples display high fractal dimension of the surface $d_b = 2.8$, forming excessive contacts and interpenetrating each other. We conclude that this topologically constrained equilibrium state resembles a conjectured crumpled globule [Grosberg et al., Journal de Physique, 1988, 49, 2095], but differs from its idealized hierarchy of self-similar, isolated and compact crumples.

The classic model of eukaryotic gene expression requires direct spatial contact between a distal enhancer and a proximal promoter. Recent Chromosome Conformation Capture (3C) studies show that enhancers and promoters are embedded in a complex network of looping interactions. Here we use a polymer model of chromatin fiber to investigate whether, and to what extent, looping interactions between elements in the vicinity of an enhancer-promoter pair can influence their contact frequency. Our equilibrium polymer simulations show that a chromatin loop, formed by elements flanking either an enhancer or a promoter, suppresses enhancer-promoter interactions, working as an insulator. A loop formed by elements located in the region between an enhancer and a promoter, on the contrary, facilitates their interactions. We find that different mechanisms underlie insulation and facilitation; insulation occurs due to steric exclusion by the loop, and is a global effect, while facilitation occurs due to an effective shortening of the enhancer-promoter genomic distance, and is a local effect. Consistently, we find that these effects manifest quite differently for in silico 3C and microscopy. Our results show that looping interactions that do not directly involve an enhancer-promoter pair can nevertheless significantly modulate their interactions. This phenomenon is analogous to allosteric regulation in proteins, where a conformational change triggered by binding of a regulatory molecule to one site affects the state of another site., Comment: Main text and all figures combined

Cancer progression is an example of a rapid adaptive process where evolving new traits is essential for survival and requires a high mutation rate. Precancerous cells acquire a few key mutations that drive rapid population growth and carcinogenesis. Cancer genomics demonstrates that these few 'driver' mutations occur alongside thousands of random 'passenger' mutations-a natural consequence of cancer's elevated mutation rate. Some passengers can be deleterious to cancer cells, yet have been largely ignored in cancer research. In population genetics, however, the accumulation of mildly deleterious mutations has been shown to cause population meltdown. Here we develop a stochastic population model where beneficial drivers engage in a tug-of-war with frequent mildly deleterious passengers. These passengers present a barrier to cancer progression that is described by a critical population size, below which most lesions fail to progress, and a critical mutation rate, above which cancers meltdown. We find support for the model in cancer age-incidence and cancer genomics data that also allow us to estimate the fitness advantage of drivers and fitness costs of passengers. We identify two regimes of adaptive evolutionary dynamics and use these regimes to rationalize successes and failures of different treatment strategies. We find that a tumor's load of deleterious passengers can explain previously paradoxical treatment outcomes and suggest that it could potentially serve as a biomarker of response to mutagenic therapies. Collective deleterious effect of passengers is currently an unexploited therapeutic target. We discuss how their effects might be exacerbated by both current and future therapies.

Cancer progression is driven by a small number of genetic alterations accumulating in a neoplasm. These few driver alterations reside in a cancer genome alongside tens of thousands of other mutations that are widely believed to have no role in cancer and termed passengers. Many passengers, however, fall within protein coding genes and other functional elements and can possibly have deleterious effects on cancer cells. Here we investigate a potential of mildly deleterious passengers to accumulate and alter the course of neoplastic progression. Our approach combines evolutionary simulations of cancer progression with the analysis of cancer sequencing data. In our simulations, individual cells stochastically divide, acquire advantageous driver and deleterious passenger mutations, or die. Surprisingly, despite selection against them, passengers accumulate and largely evade selection during progression. Although individually weak, the collective burden of passengers alters the course of progression leading to several phenomena observed in oncology that cannot be explained by a traditional driver-centric view. We tested predictions of the model using cancer genomic data. We find that many passenger mutations are likely to be damaging and that, in agreement with the model, they have largely evaded purifying selection. Finally, we used our model to explore cancer treatments that exploit the load of passengers by either 1) increasing the mutation rate; or 2) exacerbating their deleterious effects. While both approaches lead to cancer regression, the later leads to less frequent relapse. Our results suggest a new framework for understanding cancer progression as a balance of driver and passenger mutations., Comment: 10 pages, 4 figures and Supplemental Information

The rapid growth of cancer genome structural information provides an opportunity for a better understanding of the mutational mechanisms of genomic alterations in cancer and the forces of selection that act upon them. Here we test the evidence for two major forces, spatial chromosome structure and purifying (or negative) selection, that shape the landscape of somatic copy-number alterations (SCNAs) in cancer1. Using a maximum likelihood framework we compare SCNA maps and three-dimensional genome architecture as determined by genome-wide chromosome conformation capture (HiC) and described by the proposed fractal-globule (FG) model2. This analysis provides evidence that the distribution of chromosomal alterations in cancer is spatially related to three-dimensional genomic architecture and additionally suggests that purifying selection as well as positive selection shapes the landscape of SCNAs during somatic evolution of cancer cells.

Cooperative binding of transcription factors (TFs) to cis-regulatory regions (CRRs) is essential for precision of gene expression in development and other processes. The classical model of cooperativity requires direct interactions between TFs, thus constraining the arrangement of TFs sites in a CRR. On the contrary, genomic and functional studies demonstrate a great deal of flexibility in such arrangements with variable distances, numbers of sites, and identities of the involved TFs. Such flexibility is inconsistent with the cooperativity by direct interactions between TFs. Here we demonstrate that strong cooperativity among non-interacting TFs can be achieved by their competition with nucleosomes. We find that the mechanism of nucleosome-mediated cooperativity is mathematically identical to the Monod-Wyman-Changeux (MWC) model of cooperativity in hemoglobin. This surprising parallel provides deep insights, with parallels between heterotropic regulation of hemoglobin (e.g. Bohr effect) and roles of nucleosome-positioning sequences and chromatin modifications in gene regulation. Characterized mechanism is consistent with numerous experimental results, allows substantial flexibility in and modularity of CRRs, and provides a rationale for a broad range of genomic and evolutionary observations. Striking parallels between cooperativity in hemoglobin and in transcription regulation point at a new design principle that may be used in range of biological systems., Comment: 7 pages 3 figures and the Supplement

To regulate a particular gene, a transcription factor (TF) needs to bind a specific genome location. How is this genome address specified amid the presence of ~10^6-10^9 decoy sites? Our analysis of 319 known TF binding motifs clearly demonstrates that prokaryotes and eukaryotes use strikingly different strategies to target TFs to specific genome locations; eukaryotic TFs exhibit widespread nonfunctional binding and require clustering of sites in regulatory regions for specificity., Comment: 4 pages, 2 figures, expanded Supplementary Methods, Figures and Tables

The binding of a transcription factor (TF) to a DNA operator site can initiate or repress the expression of a gene. Computational prediction of sites recognized by a TF has traditionally relied upon knowledge of several cognate sites, rather than an ab initio approach. Here, we examine the possibility of using structure-based energy calculations that require no knowledge of bound sites but rather start with the structure of a protein-DNA complex. We study the PurR E. coli TF, and explore to which extent atomistic models of protein-DNA complexes can be used to distinguish between cognate and non-cognate DNA sites. Particular emphasis is placed on systematic evaluation of this approach by comparing its performance with bioinformatic methods, by testing it against random decoys and sites of homologous TFs. We also examine a set of experimental mutations in both DNA and the protein. Using our explicit estimates of energy, we show that the specificity for PurR is dominated by direct protein-DNA interactions, and weakly influenced by bending of DNA., Comment: 26 pages, 3 figures

A ubiquitous building block of signaling pathways is a cycle of covalent modification (e.g., phosphorylation and dephosphorylation in MAPK cascades). Our paper explores the kind of information processing and filtering that can be accomplished by this simple biochemical circuit. Signaling cycles are particularly known for exhibiting a highly sigmoidal (ultrasensitive) input-output characteristic in a certain steady-state regime. Here we systematically study the cycle's steady-state behavior and its response to time-varying stimuli. We demonstrate that the cycle can actually operate in four different regimes, each with its specific input-output characteristics. These results are obtained using the total quasi-steady-state approximation, which is more generally valid than the typically used Michaelis-Menten approximation for enzymatic reactions. We invoke experimental data that suggests the possibility of signaling cycles operating in one of the new regimes. We then consider the cycle's dynamic behavior, which has so far been relatively neglected. We demonstrate that the intrinsic architecture of the cycles makes them act - in all four regimes - as tunable low-pass filters, filtering out high-frequency fluctuations or noise in signals and environmental cues. Moreover, the cutoff frequency can be adjusted by the cell. Numerical simulations show that our analytical results hold well even for noise of large amplitude. We suggest that noise filtering and tunability make signaling cycles versatile components of more elaborate cell signaling pathways., Comment: to appear in PLoS Computational Biology

Strong experimental and theoretical evidence shows that transcription factors and other specific DNA-binding proteins find their sites using a two-mode search: alternating between 3D diffusion through the cell and 1D sliding along the DNA. We consider the role spatial effects in the mechanism on two different scales. First, we reconcile recent experimental findings by showing that the 3D diffusion of the transcription factor is often local, i.e. the transcription factor lands quite near its dissociation site. Second, we discriminate between two types of searches: global searches and local searches. We show that these searches differ significantly in average search time and the variability of search time. Using experimentally measured parameter values, we also show that 1D and 3D search is not optimally balanced, leading to much larger estimates of search time. Together, these results lead to a number of biological implications including suggestions of how prokaryotes and eukaryotes achieve rapid gene regulation and the relationship between the search mechanism and noise in gene expression., Comment: 16 pages, 4 figures

A number of recently discovered protein structures incorporate a rather unexpected structural feature: a knot in the polypeptide backbone. These knots are extremely rare, but their occurrence is likely connected to protein function in as yet unexplored fashion. Our analysis of the complete Protein Data Bank reveals several new knots which, along with previously discovered ones, can shed light on such connections. In particular, we identify the most complex knot discovered to date in human ubiquitin hydrolase, and suggest that its entangled topology protects it against unfolding and degradation by the proteasome. Knots in proteins are typically preserved across species and sometimes even across kingdoms. However, we also identify a knot which only appears in some transcarbamylases while being absent in homologous proteins of similar structure. The emergence of the knot is accompanied by a shift in the enzymatic function of the protein. We suggest that the simple insertion of a short DNA fragment into the gene may suffice to turn an unknotted into a knotted structure in this protein., Comment: 6 pages, 2 figures, 1 table

Recognition and binding of specific sites on DNA by proteins is central for many cellular functions such as transcription, replication, and recombination. In the process of recognition, a protein rapidly searches for its specific site on a long DNA molecule and then strongly binds this site. Here we aim to find a mechanism that can provide both a fast search (1-10 sec) and high stability of the specific protein-DNA complex ($K_d=10^{-15}-10^{-8}$ M). Earlier studies have suggested that rapid search involves the sliding of a protein along the DNA. Here we consider sliding as a one-dimensional (1D) diffusion in a sequence-dependent rough energy landscape. We demonstrate that, in spite of the landscape's roughness, rapid search can be achieved if 1D sliding is accompanied by 3D diffusion. We estimate the range of the specific and non-specific DNA-binding energy required for rapid search and suggest experiments that can test our mechanism. We show that optimal search requires a protein to spend half of time sliding along the DNA and half diffusing in 3D. We also establish that, paradoxically, realistic energy functions cannot provide both rapid search and strong binding of a rigid protein. To reconcile these two fundamental requirements we propose a search-and-fold mechanism that involves the coupling of protein binding and partial protein folding. Proposed mechanism has several important biological implications for search in the presence of other proteins and nucleosomes, simultaneous search by several proteins etc. Proposed mechanism also provides a new framework for interpretation of experimental and structural data on protein-DNA interactions.

Many biological processes involve one dimensional diffusion over a correlated inhomogeneous energy landscape with a correlation length $\xi_c$. Typical examples are specific protein target location on DNA, nucleosome repositioning, or DNA translocation through a nanopore, in all cases with $\xi_c\approx$ 10 nm. We investigate such transport processes by the mean first passage time (MFPT) formalism, and find diffusion times which exhibit strong sample to sample fluctuations. For a a displacement $N$, the average MFPT is diffusive, while its standard deviation over the ensemble of energy profiles scales as $N^{3/2}$ with a large prefactor. Fluctuations are thus dominant for displacements smaller than a characteristic $N_c \gg \xi_c$: typical values are much less than the mean, and governed by an anomalous diffusion rule. Potential biological consequences of such random walks, composed of rapid scans in the vicinity of favorable energy valleys and occasional jumps to further valleys, is discussed.

We show that the icosahedral packings of protein capsomeres proposed by Caspar and Klug for spherical viruses become unstable to faceting for sufficiently large virus size, in analogy with the buckling instability of disclinations in two-dimensional crystals. Our model, based on the nonlinear physics of thin elastic shells, produces excellent one parameter fits in real space to the full three-dimensional shape of large spherical viruses. The faceted shape depends only on the dimensionless Foppl-von Karman number \gamma=YR^2/\kappa, where Y is the two-dimensional Young's modulus of the protein shell, \kappa is its bending rigidity and R is the mean virus radius. The shape can be parameterized more quantitatively in terms of a spherical harmonic expansion. We also investigate elastic shell theory for extremely large \gamma, 10^3 < \gamma < 10^8, and find results applicable to icosahedral shapes of large vesicles studied with freeze fracture and electron microscopy., Comment: 11 pages, 12 figures

In this Communication we present statistical analysis of conservation profiles in families of homologous sequences for nine proteins whose folding nucleus was determined by protein engineering methods. We show that in all but one protein (AcP) folding nucleus residues are significantly more conserved than the rest of the protein. Two aspects of our study are especially important: 1) grouping of amino acids into classes according to their physical-chemical properties and 2) proper normalization of amino acid probabilities that reflects the fact that evolutionary pressure to conserve some amino acid types may itself affect concentration of various amino acid types in protein families. Neglect of any of those two factors may make physical and biological ``signals'' from conservation profiles disappear., Comment: submitted to Journal of Molecular Biology

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., Comment: 5 pages, 3 figures