Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites

The general Markov plus invariable sites (GM+I) model of biological sequence evolution is a two-class model in which an unknown proportion of sites are not allowed to change, while the remainder undergo substitutions according to a Markov process on a tree. For statistical use it is important to know if the model is identifiable; can both the tree topology and the numerical parameters be determined from a joint distribution describing sequences only at the leaves of the tree? We establish that for generic parameters both the tree and all numerical parameter values can be recovered, up to clearly understood issues of 'label swapping'. The method of analysis is algebraic, using phylogenetic invariants to study the variety defined by the model. Simple rational formulas, expressed in terms of determinantal ratios, are found for recovering numerical parameters describing the invariable sites.     

Accelerated Search Kinetics Mediated by Redox Reactions of DNA Repair Enzymes

A charge transport (CT) mechanism has been proposed in several articles to explain the localization of base excision repair (BER) enzymes to lesions on DNA. The CT mechanism relies on redox reactions of iron-sulfur cofactors that modify the enzyme's binding affinity. These redox reactions are mediated by the DNA strand and involve the exchange of electrons between BER enzymes along DNA. We propose a mathematical model that incorporates enzyme binding/unbinding, electron transport, and enzyme diffusion along DNA. Analysis of our model within a range of parameter values suggests that the redox reactions can increase desorption of BER enzymes not already bound to lesions, allowing the enzymes to be recycled—thus accelerating the overall search process. This acceleration mechanism is most effective when enzyme copy numbers and enzyme diffusivity along the DNA are small. Under such conditions, we find that CT BER enzymes find their targets more quickly than simple passive enzymes that simply attach to the DNA without desorbing.     

The rational parameterisation theorem for multisite post-translational modification systems

Post-translational modification of proteins plays a central role in cellular regulation but its study has been hampered by the exponential increase in substrate modification forms (“modforms”) with increasing numbers of sites. We consider here biochemical networks arising from post-translational modification under mass-action kinetics, allowing for multiple substrates, having different types of modification (phosphorylation, methylation, acetylation, etc.) on multiple sites, acted upon by multiple forward and reverse enzymes (in total number L), using general enzymatic mechanisms. These assumptions are substantially more general than in previous studies. We show that the steady-state modform concentrations constitute an algebraic variety that can be parameterised by rational functions of the L free enzyme concentrations, with coefficients which are rational functions of the rate constants. The parameterisation allows steady states to be calculated by solving L algebraic equations, a dramatic reduction compared to simulating an exponentially large number of differential equations. This complexity collapse enables analysis in contexts that were previously intractable and leads to biological predictions that we review. Our results lay a foundation for the systems biology of post-translational modification and suggest deeper connections between biochemical networks and algebraic geometry.     

Prediction of PK-specific phosphorylation site based on information entropy

Phosphorylation is a crucial way to control the activity of proteins in many eukaryotic organisms in vivo. Experimental methods to determine phosphorylation sites in substrates are usually restricted by the in vitro condition of enzymes and very intensive in time and labor. Although some in silico methods and web servers have been introduced for automatic detection of phosphorylation sites, sophisticated methods are still in urgent demand to further improve prediction performances. Protein primary se-quences can help predict phosphorylation sites catalyzed by different protein kinase and most com-putational approaches use a short local peptide to make prediction. However, the useful information may be lost if only the conservative residues that are not close to the phosphorylation site are consid-ered in prediction, which would hamper the prediction results. A novel prediction method named IEPP (Information-Entropy based Phosphorylation Prediction) is presented in this paper for automatic de-tection of potential phosphorylation sites. In prediction, the sites around the phosphorylation sites are selected or excluded by their entropy values. The algorithm was compared with other methods such as GSP and PPSP on the ABL, MAPK and PKA PK families. The superior prediction accuracies were ob-tained in various measurements such as sensitivity (Sn) and specificity (Sp). Furthermore, compared with some online prediction web servers on the new discovered phosphorylation sites, IEPP also yielded the best performance. IEPP is another useful computational resource for identification of PK-specific phosphorylation sites and it also has the advantages of simpleness, efficiency and con-venience.     

Markov invariants, plethysms, and phylogenetics

We explore model-based techniques of phylogenetic tree inference exercising Markov invariants. Markov invariants are group invariant polynomials and are distinct from what is known in the literature as phylogenetic invariants, although we establish a commonality in some special cases. We show that the simplest Markov invariant forms the foundation of the Log–Det distance measure. We take as our primary tool group representation theory, and show that it provides a general framework for analyzing Markov processes on trees. From this algebraic perspective, the inherent symmetries of these processes become apparent, and focusing on plethysms, we are able to define Markov invariants and give existence proofs. We give an explicit technique for constructing the invariants, valid for any number of character states and taxa. For phylogenetic trees with three and four leaves, we demonstrate that the corresponding Markov invariants can be fruitfully exploited in applied phylogenetic studies.     

Prediction of PK-specific phosphorylation site based on information entropy

Phosphorylation is a crucial way to control the activity of proteins in many eukaryotic organisms in vivo. Experimental methods to determine phosphorylation sites in substrates are usually restricted by the in vitro condition of enzymes and very intensive in time and labor. Although some in silico methods and web servers have been introduced for automatic detection of phosphorylation sites, sophisticated methods are still in urgent demand to further improve prediction performances. Protein primary sequences can help predict phosphorylation sites catalyzed by different protein kinase and most computational approaches use a short local peptide to make prediction. However, the useful information may be lost if only the conservative residues that are not close to the phosphorylation site are considered in prediction, which would hamper the prediction results. A novel prediction method named IEPP (Information-Entropy based Phosphorylation Prediction) is presented in this paper for automatic detection of potential phosphorylation sites. In prediction, the sites around the phosphorylation sites are selected or excluded by their entropy values. The algorithm was compared with other methods such as GSP and PPSP on the ABL, MAPK and PKA PK families. The superior prediction accuracies were obtained in various measurements such as sensitivity (Sn) and specificity (Sp). Furthermore, compared with some online prediction web servers on the new discovered phosphorylation sites, IEPP also yielded the best performance. IEPP is another useful computational resource for identification of PK-specific phosphorylation sites and it also has the advantages of simpleness, efficiency and convenience.     

Intracellular detection assays for high-throughput screening

New optical assay methods promise to accelerate the use of living cells in screens for drug discovery. Most of these methods employ either fluorescent or luminescent read-outs and allow cell-based assays for most targets, including receptors, ion channels and intracellular enzymes. Furthermore, genetically encoded probes offer the possibility of custom-engineered biosensors for intracellular biochemistry, specifically localized targets, and protein—protein interactions.     

Multistationarity in Sequential Distributed Multisite Phosphorylation Networks

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.     

Alternative algebraic rate‐integration approach for progress‐curve analysis of enzyme kinetics

A recent article of Zavrel et al. in this journal (Eng. Life Sci. 2010, 10, 191–200) described a comparison of several computer programs for progress‐curve analysis with respect to different computational approaches for parameter estimation. The authors applied both algebraic and dynamic parameter estimations, although they omitted time‐course analysis through the integrated rate equation. Recently, it was demonstrated that progress‐curve analysis through the integrated rate equation can be considered a simple and useful alternative for enzymes that obey the generalized Michaelis–Menten reaction mechanism. To complete this gap, the time‐dependent solution of the generalized Michaelis–Menten equation is here fitted to the progress curves from the Zavrel et al. reference article. This alternative rate‐integration approach for determining the kinetics parameters of Michaelis–Menten‐type enzymes yields the values with the greatest accuracy, as compared with the results obtained by other (algebraic or dynamic) parameter estimations.     

A novel domain arrangement in a monomeric cyclodextrin-hydrolyzing enzyme from the hyperthermophile Pyrococcus furiosus

PFTA (Pyrococcus furiosus thermostable amylase) is a hyperthermophilic amylase isolated from the archaeon Pyrococcus furiosus. This enzyme possesses characteristics of both α-amylase- and cyclodextrin (CD)-hydrolyzing enzymes, allowing it to degrade pullulan, CD and acarbose—activities that are absent in most α-amylases—without the transferring activity that is common in CD-hydrolyzing enzymes. The crystal structure of PFTA revealed a unique monomeric subunit with an extended N-terminal region and an N′-domain folded into its own active site—a significantly altered domain configuration relative to that of the conventional dimeric CD-hydrolyzing amylases in glycoside hydrolase family 13. The active site is formed by the interface of the N′-domain and the catalytic domain and exhibits a broad and wide-open geometry without the concave pocket that is commonly found in the active sites of maltogenic amylases. The mutation of a residue (Gly415 to Glu) located at the domain interface between the N′- and catalytic domains yielded an enzyme that produced a significantly higher purity maltoheptaose (G7) from β-CD, supporting the involvement of this interface in substrate recognition and indicating that this mutant enzyme is a suitable candidate for the production of pure G7. The unique configuration of the active site distinguishes this archaic monomeric enzyme from classical bacterial CD-hydrolyzing amylases and provides a molecular basis for its enzymatic characteristics and for its potential use in industrial applications.     

Eukaryotic Elongation Factor 2 Kinase Activity Is Controlled by Multiple Inputs from Oncogenic Signaling

Eukaryotic elongation factor 2 kinase (eEF2K), an atypical calmodulin-dependent protein kinase, phosphorylates and inhibits eEF2, slowing down translation elongation. eEF2K contains an N-terminal catalytic domain, a C-terminal α-helical region and a linker containing several regulatory phosphorylation sites. eEF2K is expressed at high levels in certain cancers, where it may act to help cell survival, e.g., during nutrient starvation. However, it is a negative regulator of protein synthesis and thus cell growth, suggesting that cancer cells may possess mechanisms to inhibit eEF2K under good growth conditions, to allow protein synthesis to proceed. We show here that the mTORC1 pathway and the oncogenic Ras/Raf/MEK/extracellular signal-regulated kinase (ERK) pathway cooperate to restrict eEF2K activity. We identify multiple sites in eEF2K whose phosphorylation is regulated by mTORC1 and/or ERK, including new ones in the linker region. We demonstrate that certain sites are phosphorylated directly by mTOR or ERK. Our data reveal that glycogen synthase kinase 3 signaling also regulates eEF2 phosphorylation. In addition, we show that phosphorylation sites remote from the N-terminal calmodulin-binding motif regulate the phosphorylation of N-terminal sites that control CaM binding. Mutations in the former sites, which occur in cancer cells, cause the activation of eEF2K. eEF2K is thus regulated by a network of oncogenic signaling pathways.     

Mechanical Fluidity of Fully Suspended Biological Cells

Mechanical characteristics of single biological cells are used to identify and possibly leverage interesting differences among cells or cell populations. Fluidity—hysteresivity normalized to the extremes of an elastic solid or a viscous liquid—can be extracted from, and compared among, multiple rheological measurements of cells: creep compliance versus time, complex modulus versus frequency, and phase lag versus frequency. With multiple strategies available for acquisition of this nondimensional property, fluidity may serve as a useful and robust parameter for distinguishing cell populations, and for understanding the physical origins of deformability in soft matter. Here, for three disparate eukaryotic cell types deformed in the suspended state via optical stretching, we examine the dependence of fluidity on chemical and environmental influences at a timescale of ∼1 s. We find that fluidity estimates are consistent in the time and frequency domains under a structural damping (power-law or fractional-derivative) model, but not under an equivalent-complexity, lumped-component (spring-dashpot) model; the latter predicts spurious time constants. Although fluidity is suppressed by chemical cross-linking, we find that ATP depletion in the cell does not measurably alter the parameter, and we thus conclude that active ATP-driven events are not a crucial enabler of fluidity during linear viscoelastic deformation of a suspended cell. Finally, by using the capacity of optical stretching to produce near-instantaneous increases in cell temperature, we establish that fluidity increases with temperature—now measured in a fully suspended, sortable cell without the complicating factor of cell-substratum adhesion.     

Modelling cellular signal communication mediated by phosphorylation dependent interaction with 14-3-3 proteins

The 14-3-3 proteins are important effectors of Ser/Thr phosphorylation in eukaryotic cells. Using mathematical modelling we investigated the roles of these proteins as effectors in signalling pathways that involve multi-phosphorylation events. We defined optimal conditions for positive and negative cross-talk. Particularly, synergistic signal interaction was evident at very different sets of binding affinities and phosphorylation kinetics. We identified three classes of 14-3-3 targets that all have two binding sites, but displayed synergistic interaction between converging signalling pathways for different ranges of parameter values. Consequently, these protein targets will respond differently to interventions that affect 14-3-3 binding affinities or phosphorylation kinetics.     

Developmental autonomy and somatic niche construction promotes robust cell fate decisions

During the course of development cells undergo division producing a variety of cell types. Proliferation and differentiation are dependent on both genetic programs, encoded by the cellular genome, and environmental cues produced by the local cellular environment imposing local selection pressures on cells. We explore the role that cellular signals play over a large range of potential parameter regimes, in minimizing developmental error: errors in differentiation where an inappropriate proportion of differentiated daughter cells are generated. We find that trophic factors produced by the population of dividing cells can compensate for increased error rates when signals act through a form of positive feedback—survival signals. We operationalize these signals as the somatic niche and refer to their production as somatic niche construction. We find that tissue development switches to an autonomous state, independent of cellular signals, when errors are unmanageably high or density regulation is very strong. A signal-selective regime—strong niche dependence—is favored at low to intermediate error, assuming compartmentalized density dependence.     

Hyperdigraph-theoretic analysis of the EGFR signaling network: initial steps leading to GTP:Ras complex formation.

We construct an algebraic-combinatorial model of the SOS compartment of the EGFR biochemical network. A Petri net is used to construct an initial representation of the biochemical decision making network, which in turn defines a hyperdigraph. We observe that the linear algebraic structure of each hyperdigraph admits a canonical set of algebraic-combinatorial invariants that correspond to the information flow conservation laws governing a molecular kinetic reaction network. The linear algebraic structure of the hyperdigraph and its sets of invariants can be generalized to define a discrete algebraic-geometric structure, which is referred to as an oriented matroid. Oriented matroids define a polyhedral optimization geometry that is used to determine optimal subpaths that span the nullspace of a set of kinetic chemical reaction equations. Sets of constrained submodular path optimizations on the hyperdigraph are objectively obtained as a spanning tree of minimum cycle paths. This complete set of subcircuits is used to identify the network pinch points and invariant flow subpaths. We demonstrate that this family of minimal circuits also characteristically identifies additional significant biochemical reaction pattern features. We use the SOS Compartment A of the EGFR biochemical pathway to develop and demonstrate the application of our algebraic-combinatorial mathematical modeling methodology.     

A Grammar Inference Approach for Predicting Kinase Specific Phosphorylation Sites

Kinase mediated phosphorylation site detection is the key mechanism of post translational mechanism that plays an important role in regulating various cellular processes and phenotypes. Many diseases, like cancer are related with the signaling defects which are associated with protein phosphorylation. Characterizing the protein kinases and their substrates enhances our ability to understand the mechanism of protein phosphorylation and extends our knowledge of signaling network; thereby helping us to treat such diseases. Experimental methods for predicting phosphorylation sites are labour intensive and expensive. Also, manifold increase of protein sequences in the databanks over the years necessitates the improvement of high speed and accurate computational methods for predicting phosphorylation sites in protein sequences. Till date, a number of computational methods have been proposed by various researchers in predicting phosphorylation sites, but there remains much scope of improvement. In this communication, we present a simple and novel method based on Grammatical Inference (GI) approach to automate the prediction of kinase specific phosphorylation sites. In this regard, we have used a popular GI algorithm Alergia to infer Deterministic Stochastic Finite State Automata (DSFA) which equally represents the regular grammar corresponding to the phosphorylation sites. Extensive experiments on several datasets generated by us reveal that, our inferred grammar successfully predicts phosphorylation sites in a kinase specific manner. It performs significantly better when compared with the other existing phosphorylation site prediction methods. We have also compared our inferred DSFA with two other GI inference algorithms. The DSFA generated by our method performs superior which indicates that our method is robust and has a potential for predicting the phosphorylation sites in a kinase specific manner.     

Ultrasensitivity in multisite phosphorylation of membrane-anchored proteins

Cellular signaling is initially confined to the plasma membrane, where the cytoplasmic tails of surface receptors and other membrane-anchored proteins are phosphorylated in response to ligand binding. These proteins often contain multiple phosphorylation sites that are regulated by membrane-confined enzymes. Phosphorylation of these proteins is thought to be tightly regulated, because they initiate and regulate signaling cascades leading to cellular activation, yet how their phosphorylation is regulated is poorly understood. Ultrasensitive or switchlike responses in their phosphorylation state are not expected because the modifying enzymes are in excess. Here, we describe a novel mechanism of ultrasensitivity exhibited by multisite membrane-anchored proteins, but not cytosolic proteins, even when enzymes are in excess. The mechanism underlying this concentration-independent ultrasensitivity is the local saturation of a single enzyme by multiple sites on the substrate. Local saturation is a passive process arising from slow membrane diffusion, steric hindrances, and multiple sites, and therefore may be widely applicable. Critical to this ultrasensitivity is the brief enzymatic inactivation that follows substrate modification. Computations are presented using ordinary differential equations and stochastic spatial simulations. We propose a new role, to our knowledge, for multisite membrane-anchored proteins, discuss experiments that can be used to probe the model, and relate our findings to previous theoretical work.     

Multiple phosphorylations of cytochrome c oxidase and their functions

Cytochrome c oxidase (COX), the terminal enzyme of the mitochondrial electron transport chain, is regulated by isozyme expression, allosteric effectors such as the ATP/ADP ratio, and reversible phosphorylation. Of particular interest is the "allosteric ATP-inhibition," which has been hypothesized to keep the mitochondrial membrane potential at low healthy values (<140 mV), thus preventing the formation of superoxide radical anions, which have been implicated in multiple degenerative diseases. It has been proposed that the "allosteric ATP-inhibition" is switched on by the protein kinase A-dependent phosphorylation of COX. The goal of this study was to identify the phosphorylation site(s) involved in the "allosteric ATP-inhibition" of COX. We report the mass spectrometric identification of four new phosphorylation sites in bovine heart COX. The identified phosphorylation sites include Tyr-218 in subunit II, Ser-1 in subunit Va, Ser-2 in subunit Vb, and Ser-1 in subunit VIIc. With the exception of Ser-2 in subunit Vb, the identified phosphorylation sites were found in enzyme samples with and without "allosteric ATP inhibition," making Ser-2 of subunit Vb a candidate site enabling allosteric regulation. We therefore hypothesize that additional phosphorylation(s) may be required for the "allosteric ATP-inhibition," and that these sites may be easily dephosphorylated or difficult to identify by mass spectrometry.     

Prediction of functional phosphorylation sites by incorporating evolutionary information

Protein phosphorylation is a ubiquitous protein post-translational modification, which plays an important role in cellular signaling systems underlying various physiological and pathological processes. Current in silico methods mainly focused on the prediction of phosphorylation sites, but rare methods considered whether a phosphorylation site is functional or not. Since functional phosphorylation sites are more valuable for further experimental research and a proportion of phosphorylation sites have no direct functional effects, the prediction of functional phosphorylation sites is quite necessary for this research area. Previous studies have shown that functional phosphorylation sites are more conserved than non-functional phosphorylation sites in evolution. Thus, in our method, we developed a web server by integrating existing phosphorylation site prediction methods, as well as both absolute and relative evolutionary conservation scores to predict the most likely functional phosphorylation sites. Using our method, we predicted the most likely functional sites of the human, rat and mouse proteomes and built a database for the predicted sites. By the analysis of overall prediction results, we demonstrated that protein phosphorylation plays an important role in all the enriched KEGG pathways. By the analysis of protein-specific prediction results, we demonstrated the usefulness of our method for individual protein studies. Our method would help to characterize the most likely functional phosphorylation sites for further studies in this research area.     

Mutation-selection models solved exactly with methods of statistical mechanics

We reconsider deterministic models of mutation and selection acting on populations of sequences, or, equivalently, multilocus systems with complete linkage. Exact analytical results concerning such systems are few, and we present recent and new ones obtained with the help of methods from quantum statistical mechanics. We consider a continuous-time model for an infinite population of haploids (or diploids without dominance), with N sites each, two states per site, symmetric mutation and arbitrary fitness function. We show that this model is exactly equivalent to a so-called Ising quantum chain. In this picture, fitness corresponds to the interaction energy of spins, and mutation to a temperature-like parameter. The highly elaborate methods of statistical mechanics allow one to find exact solutions for non-trivial examples. These include quadratic fitness functions, as well as 'Onsager's landscape'. The latter is a fitness function which captures some essential features of molecular evolution, such as neutrality, compensatory mutations and flat ridges. We investigate the mean number of mutations, the mutation load, and the variance in fitness under mutation-selection balance. This also yields some insight into the 'error threshold' phenomenon, which occurs in some, but not all, examples.