The complexity of the gapped consecutive-ones property problem for matrices of bounded maximum degree

The Gapped Consecutive-Ones Property (C1P) Problem, or the (k, δ)-C1P Problem is: given a binary matrix M and integers k and δ, decide if the columns of M can be ordered such that each row contains at most k blocks of 1's, and no two neighboring blocks of 1's are separated by a gap of more than δ 0's. This problem was introduced by Chauve et al. ( 2009b ). The classical polynomial-time solvable C1P Problem is equivalent to the (1, 0)-C1P problem. It has been shown that, for every unbounded or bounded k ≥ 2 and unbounded or bounded δ ≥ 1, except when (k, δ) = (2, 1), the (k, δ)-C1P Problem is NP-complete (Maňuch et al., 2011 ; Goldberg et al., 1995 ). In this article, we study the Gapped C1P Problem with a third parameter d, namely the bound on the maximum number of 1's in any row of M, or the bound on the maximum degree of M. This is motivated by the reconstruction of ancestral genomes (Ma et al., 2006 ; Chauve and Tannier, 2008 ), where, in binary matrices obtained from the experiments of Chauve and Tannier ( 2008 ), we have observed that the majority of the rows have low degree, while each high degree row contains many rows of low degree. The (d, k, δ)-C1P Problem has been shown to be polynomial-time solvable when all three parameters are fixed (Chauve et al., 2009b ). Since fixing d also fixes k (k ≤ d), the only case left to consider is the case when δ is unbounded, or the (d, k, ∞)-C1P Problem. Here we show that for every d > k ≥ 2, the (d, k, ∞)-C1P Problem is NP-complete.     

Comparative models for human apolipoprotein A-I bound to lipid in discoidal high-density lipoprotein particles

We have constructed a series of models for apolipoprotein A-I (apo A-I) bound to discoidal high-density lipoprotein (HDL) particles, based upon the molecular belt model [Segrest, J. P., et al. (1999) J. Biol. Chem. 274, 31755-31758] and helical hairpin models [Rogers, D. P., et al. (1998) Biochemistry 37, 11714-11725], and compared these with picket fence models [Phillips, J. C., et al. (1997) Biophys. J. 73, 2337-2346]. Molecular belt models for discoidal HDL particles with differing diameters are presented, illustrating that the belt model can explain the discrete changes in HDL particle size observed experimentally. Hairpin models are discussed for the binding of apo A-I to discoidal HDL particles with diameters identical to those for the molecular belt model. Two models are presented for the binding of three monomers of apo A-I to a 150 A diameter discoidal HDL particle. In one model, two monomers of apo A-I bind to the exterior of the HDL particle in an antiparallel belt, with a third monomer of apo A-I bound to the disk in a hairpin conformation. In the second model, all three monomers of apo A-I are bound to the discoidal HDL particle in a hairpin conformation. Previously published experimental data for each model are reviewed, with FRET favoring either the belt or hairpin models over the picket fence models for HDL particles with diameters of 105 A. Naturally occurring mutations appear to favor the belt model for the 105 A particles, while the 150 A HDL particles favor the presence of at least one hairpin.     

Operability region equivalence: simultaneous confidence bands for the equivalence of two regression models over restricted regions

In many scientific problems the purpose of comparing two linear regression models is to demonstrate that they have only negligible differences and so can be regarded as being practically equivalent. The frequently used statistical approach of testing the homogeneity null hypothesis of the two models by using a partial F test is not appropriate for this purpose. In this paper, a simultaneous confidence band is proposed which provides an upper bound on the largest possible difference between the two models, in units of the standard error of the observations, over a given region of the covariates. This is demonstrated to be a more practical method for assessing the equivalence of the two regression models.     

Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences

In this paper methods from differential algebra are used to study the structural identifiability of biological and pharmacokinetics models expressed in state-space form and with a structure given by rational functions. The focus is on the examples presented and on the application of efficient, automatic methods to test for structural identifiability for various input-output experiments. Differential algebra methods are coupled with Gr?bner bases, Lie derivatives and the Taylor series expansion in order to obtain efficient algorithms. In particular, an upper bound on the number of derivatives needed for the Taylor series approach for a structural identifiability analysis of rational function models is given.     

Constructing and validating initial Cα models from subnanometer resolution density maps with pathwalking

A significant number of macromolecular structures solved by electron cryo-microscopy and X-ray crystallography obtain resolutions of 3.5-6?, at which direct atomistic interpretation is difficult. To address this, we developed pathwalking, a semi-automated protocol to enumerate reasonable Cα models from near-atomic resolution density maps without a structural template or sequence-structure correspondence. Pathwalking uses an approach derived from the Traveling Salesman Problem to rapidly generate an ensemble of initial models for individual proteins, which can later be optimized to produce full atomic models. Pathwalking can also be used to validate and identify potential structural ambiguities in models generated from near-atomic resolution density maps. In this work, examples from the EMDB and PDB are used to assess the broad applicability and accuracy of our method. With the growing number of near-atomic resolution density maps from cryo-EM and X-ray crystallography, pathwalking can become an important tool in modeling protein structures.     

Brownian adhesive dynamics (BRAD) for simulating the receptor-mediated binding of viruses

Current viral docking models have relied upon the assumption that bond formation and breakage are independent of viral and docking surface geometry, as well as the forces exerted on the bonds. This assumption, known as the equivalent site hypothesis (ESH), is examined in detail using a newly developed simulation technique-Brownian adhesive dynamics (BRAD). The simulation couples the thermal motion of viral particles with adhesive dynamics models to characterize the effect of bonding on viral motion. We use the binding of HIV-like particles to CD4 expressing cells as a model system to illustrate the utility of BRAD. Comparison of the transition rates between bound states predicted by ESH and the rates resulting from BRAD simulations show dramatic differences; at values of the equilibrium crosslinking constant, K(x)R(T), where ESH suggests all virus adhesion proteins will be bound (K(x)R(T) = 10(6)), BRAD predicts not all virus adhesion proteins will be bound. At values of the equilibrium crosslinking constant used in typical ESH calculations of virus docking (K(x)R(T) = 1) we find BRAD simulations predict no binding. The mean bond density from BRAD models is often much lower than that predicted by ESH for equivalent parameter values. BRAD suggests that the viruses are much less well bound than ESH predicts. The differences suggest that binding models for viruses need to be reexamined closely. BRAD is a simulation technique that will be useful for quantifying the receptor-mediated binding of a wide variety of viruses to cells.     

Tumor escape from immune elimination: simplified precursor bound cytotoxicity models

In this paper we present a series of models on cytotoxic T-cell activation derived, by successive simplifications, from the model for Tumor Escape from Immune Elimination of Grossman & Berke (1980). In their model Grossman & Berke (1980) investigate the "sneaking through" phenomenon, by which they mean that small tumors grow progressively, medium-sized tumors are rejected and large ones break through again. We define precursor bound cytotoxicity models as systems incapable of infinite proliferation. We show that sneaking through can occur in a broad class of very simple precursor bound cytotoxicity models due to the depletion of the precursor cells. The simplest process by which precursors can be depleted is long-lasting antigenic stimulation. We conclude that in precursor bound cytotoxicity models sneaking through does not need the rather intricate combination of counteracting feedback loops, memory and blocking described by Grossman & Berke (1980).     

Orientation, accessibility, and mobility of equilenin bound to the active site of steroid isomerase

The fluorescent aromatic steroid equilenin, which contains a beta-naphthol moiety, is bound by 3-oxo-delta 5-steroid isomerase. The excitation and emission fluorescence spectra of equilenin when bound to the enzyme, as well as the fluorescence decay time, are indicative of ground-state ionization. In view of the high efficiency of tyrosine quenching, which approaches 100%, the beta-naphthol moiety of equilenin must be in proximity to all three tyrosines of steroid isomerase to account for the observed efficiency of radiationless energy transfer. From the observed response to an external quencher, it appears that enzyme-bound equilenin is largely shielded from solvent. Fluorescence anisotropy measurements indicate a high degree of immobilization of the bound ligand. These models are consistent with proposed models of the enzyme-substrate complex.     

A 2-approximation for the minimum duplication speciation problem

We consider the following problem: given a set of gene family trees, spanning a given set of species, find a first speciation which splits these species into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, we propose a polynomial time 2-approximation algorithm for the Minimum Duplication Bipartition Problem. We apply this algorithm to the inference of species trees on synthetic datasets and on two datasets of eukaryotic species.     

Efficient approximations for learning phylogenetic HMM models from data

MOTIVATION: We consider models useful for learning an evolutionary or phylogenetic tree from data consisting of DNA sequences corresponding to the leaves of the tree. In particular, we consider a general probabilistic model described in Siepel and Haussler that we call the phylogenetic-HMM model which generalizes the classical probabilistic models of Neyman and Felsenstein. Unfortunately, computing the likelihood of phylogenetic-HMM models is intractable. We consider several approximations for computing the likelihood of such models including an approximation introduced in Siepel and Haussler, loopy belief propagation and several variational methods. RESULTS: We demonstrate that, unlike the other approximations, variational methods are accurate and are guaranteed to lower bound the likelihood. In addition, we identify a particular variational approximation to be best-one in which the posterior distribution is variationally approximated using the classic Neyman-Felsenstein model. The application of our best approximation to data from the cystic fibrosis transmembrane conductance regulator gene region across nine eutherian mammals reveals a CpG effect.     

Hypothesis generation in signaling networks.

Biological signaling networks comprise the chemical processes by which cells detect and respond to changes in their environment. Such networks have been implicated in the regulation of important cellular activities, including cellular reproduction, mobility, and death. Though technological and scientific advances have facilitated the rapid accumulation of information about signaling networks, utilizing these massive information resources has become infeasible except through computational methods and computer-based tools. To date, visualization and simulation tools have received significant emphasis. In this paper, we present a graph-theoretic formalization of biological signaling network models that are in wide but informal use, and formulate two problems on the graph: the Constrained Downstream and Minimum Knockout Problems. Solutions to these problems yield qualitative tools for generating hypotheses about the networks, which can then be experimentally tested in a laboratory setting. Using established graph algorithms, we provide a solution to the Constrained Downstream Problem. We also show that the Minimum Knockout Problem is NP-Hard, propose a heuristic, and assess its performance. In tests on the Epidermal Growth Factor Receptor (EGFR) network, we find that our heuristic reports the correct solution to the problem in seconds. Source code for the implementations of both solutions is available from the authors upon request.     

The effect of habitat fragmentation on cyclic population dynamics: a reduction to ordinary differential equations

Habitat fragmentation is known to be a key factor affecting population dynamics. In a previous study by Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009), the effect of habitat fragmentation on cyclic population dynamics was studied using spatially explicit predator?Cprey models with four different sets of reaction terms. The difficulty with spatially explicit models is that often analytical tractability is lost and the mechanisms behind the behaviour of the models are difficult to analyse. In this study, we employ a simplification procedure based on a Fourier series first-term truncation of the spatially explicit models Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009) to obtain spatially implicit models. These simpler models capture the main features of the spatially explicit models and can be used to explain the dynamics observed by Strohm and Tyson. We find that the spatially implicit models and the spatially explicit models produce similar responses to habitat fragmentation for larger high-quality patch sizes. Additionally, we find that the critical patch size of the spatially implicit models provides an upper bound on the critical patch size of the spatially explicit models. Finally, we derive an approximation of the multi-patch habitat by a single-patch habitat with partial flux boundary conditions which allows for a lower bound on the critical patch size to be calculated.     

The Crisis of Reproducibility,the Denominator Problem and the Scientific Role of Multi-scale Modeling

The “Crisis of Reproducibility” has received considerable attention both within the scientific community and without. While factors associated with scientific culture and practical practice are most often invoked, I propose that the Crisis of Reproducibility is ultimately a failure of generalization with a fundamental scientific basis in the methods used for biomedical research. The Denominator Problem describes how limitations intrinsic to the two primary approaches of biomedical research, clinical studies and preclinical experimental biology, lead to an inability to effectively characterize the full extent of biological heterogeneity, which compromises the task of generalizing acquired knowledge. Drawing on the example of the unifying role of theory in the physical sciences, I propose that multi-scale mathematical and dynamic computational models, when mapped to the modular structure of biological systems, can serve a unifying role as formal representations of what is conserved and similar from one biological context to another. This ability to explicitly describe the generation of heterogeneity from similarity addresses the Denominator Problem and provides a scientific response to the Crisis of Reproducibility.     

Donor fluorescence decay analysis for energy transfer in double-helical DNA with various acceptor concentrations

We studied fluorescence resonance energy transfer between donors and acceptors bound to double-helical DNA. The donor Hoechst 33258 binds to the minor groove of DNA and the acceptor propidium iodide (PI) is an intercalator. The time-resolved donor decays were measured in the frequency domain. The donor decays were consistent with a random 1-dimensional distribution of acceptors. The decays were analyzed in terms of three 1-dimensional models: a random continuous acceptor distribution; acceptors placed on discrete lattice sites; and a cylindrical model with the acceptor in the center, and the donors on a cylinder surface. The data were well described by all three models. Interpretation in terms of continuous distribution of acceptors revealed a minimum donor to acceptor distance of 13 A, which is 3 bp from the center of Hoechst 33252. These results suggest that PI is excluded from the 4 bp covered by Hoechst 33252 when it is bound to the minor groove of DNA.     

Molecular models for murine sperm-egg binding

Murine sperm initiate fertilization by binding to the specialized extracellular matrix of mouse eggs, known as the zona pellucida. Over the past decade, powerful genetic, biophysical, and biochemical techniques have been employed to gain new insights into this interaction. Evidence from these studies does not support either of two major models for binding first proposed over two decades ago. Two more recently established models suggest that protein-protein interactions predominate during this initial stage of fertilization. Another model proposes that about 75-80% of the murine sperm bound to zona pellucida under well defined in vitro conditions is carbohydrate dependent, with the remaining sperm bound via protein-protein interactions. Mounting evidence suggests that the carbohydrate sequences coating the murine egg could be employed as specific immune recognition markers. Continued investigation of this system may resolve many of these controversial findings and reveal novel functions for murine zona pellucida glycoproteins.     

Applications and extensions of Chao's moment estimator for the size of a closed population

This article revisits Chao's (1989, Biometrics45, 427-438) lower bound estimator for the size of a closed population in a mark-recapture experiment where the capture probabilities vary between animals (model M(h)). First, an extension of the lower bound to models featuring a time effect and heterogeneity in capture probabilities (M(th)) is proposed. The biases of these lower bounds are shown to be a function of the heterogeneity parameter for several loglinear models for M(th). Small-sample bias reduction techniques for Chao's lower bound estimator are also derived. The application of the loglinear model underlying Chao's estimator when heterogeneity has been detected in the primary periods of a robust design is then investigated. A test for the null hypothesis that Chao's loglinear model provides unbiased abundance estimators is provided. The strategy of systematically using Chao's loglinear model in the primary periods of a robust design where heterogeneity has been detected is investigated in a Monte Carlo experiment. Its impact on the estimation of the population sizes and of the survival rates is evaluated in a Monte Carlo experiment.     

Thermodynamical model for insertion and aggregate binding of caffeine to the homopolymer poly(riboadenylate) and model choice by data analysis

This paper describes the model used to estimate the parameters of caffeine-poly(riboadenylate) (poly(A)) interactions from corresponding 1H-NMR measurements. The model of insertion and aggregate binding describes the non-cooperative insertion of a molecule C into an interspace between two monomers of a homopolymer in competition with aggregate binding. It contains two binding constants, K1 for insertion and K2 for the interaction of monomeric A units of the polymer with C molecules in bound aggregates, and two cooperativity parameters, Kcc for stacking of C molecules within aggregates and tau which is thought to be due to conformational adaptation of the polymer to those bound aggregates which cover more than one A unit. In contrast to other models, the size of a binding site (within the aggregates) is less than one monomeric unit, with n denoting the maximum number of C molecules per A unit in bound aggregates. The model is developed for general n by means of the method of sequence-generating functions. For n = 2 and n = 3, the correctness of the model treatment was checked by the matrix method. The model is applicable to the binding of aggregates to homopolymers, which are flexible enough to fit their structure to the aggregates.     

Accounting for Ligand Conformational Restriction in Calculations of Protein-Ligand Binding Affinities

The conformation adopted by a ligand on binding to a receptor may differ from its lowest-energy conformation in solution. In addition, the bound ligand is more conformationally restricted, which is associated with a configurational entropy loss. The free energy change due to these effects is often neglected or treated crudely in current models for predicting binding affinity. We present a method for estimating this contribution, based on perturbation theory using the quasi-harmonic model of Karplus and Kushick as a reference system. The consistency of the method is checked for small model systems. Subsequently we use the method, along with an estimate for the enthalpic contribution due to ligand-receptor interactions, to calculate relative binding affinities. The AMBER force field and generalized Born implicit solvent model is used. Binding affinities were estimated for a test set of 233 protein-ligand complexes for which crystal structures and measured binding affinities are available. In most cases, the ligand conformation in the bound state was significantly different from the most favorable conformation in solution. In general, the correlation between measured and calculated ligand binding affinities including the free energy change due to ligand conformational change is comparable to or slightly better than that obtained by using an empirically-trained docking score. Both entropic and enthalpic contributions to this free energy change are significant.     

Analysis of agonism by dopamine at the dopaminergic D 2 G-protein coupled receptor based on comparative modelling of rhodopsin

A number of theoretical dopaminergic D 2 models were constructed using comparative modelling techniques. The models constructed were based on the recently published crystal structure of rhodopsin. D 2 models were constructed without any ligands bound in the active site and also with the endogenous agonist, dopamine, bound in the active site. Comparison of the bound and unbound models revealed the importance of the chi angle of serine 197 and the effect this has on the bending of helix 5 when an agonist is present in the binding site.