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 $\mu$, 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 $\mu$ 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 $k (2 e^{-\mu/2} - 1)$ below the error catastrophe, in contrast to the standard result of $k e^{-\mu}$ for conservative replication (derived by Kimura and Maruyama in 1966).Comment: 15 pages, 7 figures, to be submitted to Phys. Rev.

The structural properties of non-traditional drug targets present new challenges for virtual screening

Traditional drug targets have historically included signaling proteins that respond to small-molecules and enzymes that use small-molecules as substrates. Increasing attention is now being directed towards other types of protein targets, in particular those that exert their function by interacting with nucleic acids or other proteins rather than small-molecule ligands. Here, we systematically compare existing examples of inhibitors of protein–protein interactions to inhibitors of traditional drug targets. While both sets of inhibitors bind with similar potency, we find that the inhibitors of protein–protein interactions typically bury a smaller fraction of their surface area upon binding to their protein targets. The fact that an average atom is less buried suggests that more atoms are needed to achieve a given potency, explaining the observation that ligand efficiency is typically poor for inhibitors of protein– protein interactions. We then carried out a series of docking experiments, and found a further consequence of these relatively exposed binding modes is that structure-based virtual screening may be more difficult: such binding modes do not provide sufficient clues to pick out active compounds from decoy compounds. Collectively, these results suggest that the challenges associated with such non-traditional drug targets may not lie with identifying compounds that potently bind to the target protein surface, but rather with identifying compounds that bind in a sufficiently buried manner to achieve good ligand efficiency, and thus good oral bioavailability. While the number of available crystal structures of distinct protein interaction sites bound to small-molecule inhibitors is relatively small at present (only 21 such complexes were included in this study), these are sufficient to draw conclusions based on the current state of the field; as additional data accumulate it will be exciting to refine the viewpoint presented here. Even with this limited perspective however, we anticipate that these insights, together with new methods for exploring protein conformational fluctuations, may prove useful for identifying the “low-hanging fruit” amongst non-traditional targets for therapeutic intervention

Equilibrium Distribution of Mutators in the Single Fitness Peak Model

This is the publisher's version, also available electronically from http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.138105.This Letter 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 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 ϵr=Lvia/L, where Lvia 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 Escherichia coli

Phosphatase Specificity and Pathway Insulation in Signaling Networks

AbstractPhosphatases play an important role in cellular signaling networks by regulating the phosphorylation state of proteins. Phosphatases are classically considered to be promiscuous, acting on tens to hundreds of different substrates. We recently demonstrated that a shared phosphatase can couple the responses of two proteins to incoming signals, even if those two substrates are from otherwise isolated areas of the network. This finding raises a potential paradox: if phosphatases are indeed highly promiscuous, how do cells insulate themselves against unwanted crosstalk? Here, we use mathematical models to explore three possible insulation mechanisms. One approach involves evolving phosphatase KM values that are large enough to prevent saturation by the phosphatase’s substrates. Although this is an effective method for generating isolation, the phosphatase becomes a highly inefficient enzyme, which prevents the system from achieving switch-like responses and can result in slow response kinetics. We also explore the idea that substrate degradation can serve as an effective phosphatase. Assuming that degradation is unsaturatable, this mechanism could insulate substrates from crosstalk, but it would also preclude ultrasensitive responses and would require very high substrate turnover to achieve rapid dephosphorylation kinetics. Finally, we show that adaptor subunits, such as those found on phosphatases like PP2A, can provide effective insulation against phosphatase crosstalk, but only if their binding to substrates is uncoupled from their binding to the catalytic core. Analysis of the interaction network of PP2A’s adaptor domains reveals that although its adaptors may isolate subsets of targets from one another, there is still a strong potential for phosphatase crosstalk within those subsets. Understanding how phosphatase crosstalk and the insulation mechanisms described here impact the function and evolution of signaling networks represents a major challenge for experimental and computational systems biology

Semiconservative replication in the quasispecies model

This is the publisher's version, also available electronically from http://journals.aps.org/pre/abstract/10.1103/PhysRevE.69.061916.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 2ln[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 lnk for conservative replication. In particular, as k→∞, the error catastrophe is never reached for conservative replication, while for semiconservative replication the critical μ approaches 2ln2. 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(2e−μ∕2−1) below the error catastrophe, in contrast to the standard result of ke−μ for conservative replication (derived by Kimura and Maruyama in 1966). From this result it is readily shown that semiconservative replication is necessary to account for the observation that, at sufficiently high mutagen concentrations, faster replicating cells will die more quickly than more slowly replicating cells. Thus, in contrast to Eigen’s original model, the semiconservative quasispecies equations are able to provide a mathematical basis for explaining the efficacy of mutagens as chemotherapeutic agents

Prokaryotic phylogenies inferred from protein structural domains

This is the publisher's version, also available electronically from http://genome.cshlp.org/content/15/3/393.The determination of the phylogenetic relationships among microorganisms has long relied primarily on gene sequence information. Given that prokaryotic organisms often lack morphological characteristics amenable to phylogenetic analysis, prokaryotic phylogenies, in particular, are often based on sequence data. In this work, we explore a new source of phylogenetic information, the distribution of protein structural domains within fully sequenced prokaryotic genomes. The evolution of the structural domains we use has been studied extensively, allowing us to base our phylogenetic methods on testable theoretical models of structural evolution. We find that the methods that produce reasonable phylogenetic relationships are indeed the methods that are most consistent with theoretical evolutionary models. This work represents, to our knowledge, the first such theoretically motivated phylogeny, as well as the first application of structural information to phylogeny on this scale. Our results have strong implications for the phylogenetic relationships among prokaryotic organisms and for the understanding of protein evolution as a whole

Sizing Up Allometric Scaling Theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.EJD acknowledges financial support from a National Institutes of Health/National Research Service Award (1F32 GM080123-01)

The Emergence of Scaling in Sequence-based Physical Models of Protein Evolution

It has recently been discovered that many biological systems, when represented as graphs, exhibit a scale-free topology. One such system is the set of structural relationships among protein domains. The scale-free nature of this and other systems has previously been explained using network growth models that, while motivated by biological processes, do not explicitly consider the underlying physics or biology. In the present work we explore a sequence-based model for the evolution protein structures and demonstrate that this model is able to recapitulate the scale-free nature observed in graphs of real protein structures. We find that this model also reproduces other statistical feature of the protein domain graph. This represents, to our knowledge, the first such microscopic, physics-based evolutionary model for a scale-free network of biological importance and as such has strong implications for our understanding of the evolution of protein structures and of other biological networks.Comment: 20 pages (including figures), 4 figures, to be submitted to PNA

Crosstalk and the evolvability of intracellular communication

Metazoan signalling networks are complex, with extensive crosstalk between pathways. It is unclear what pressures drove the evolution of this architecture. We explore the hypothesis that crosstalk allows different cell types, each expressing a specific subset of signalling proteins, to activate different outputs when faced with the same inputs, responding differently to the same environment. We find that the pressure to generate diversity leads to the evolution of networks with extensive crosstalk. Using available data, we find that human tissues exhibit higher levels of diversity between cell types than networks with random expression patterns or networks with no crosstalk. We also find that crosstalk and differential expression can influence drug activity: no protein has the same impact on two tissues when inhibited. In addition to providing a possible explanation for the evolution of crosstalk, our work indicates that consideration of cellular context will likely be crucial for targeting signalling networks