CONAN: A Tool to Decode Dynamical Information from Molecular Interaction Maps

The analysis of contacts is a powerful tool to understand biomolecular function in a series of contexts, from the investigation of dynamical behavior at equilibrium to the study of nonequilibrium dynamics in which the system moves between multiple states. We thus propose a tool called CONtact ANalysis (CONAN) that, from molecular dynamics (MD) trajectories, analyzes interresidue contacts, creates videos of time-resolved contact maps, and performs correlation, principal component, and cluster analysis, revealing how specific contacts relate to functionally relevant states sampled by MD. We present how CONAN can identify features describing the dynamics of ubiquitin both at equilibrium and during mechanical unfolding. Additionally, we show the analysis of MD trajectories of an α-synuclein mutant peptide that undergoes an α-β conformational transition that can be easily monitored using CONAN, which identifies the multiple states that the peptide explores along its conformational dynamics. The high versatility and ease of use of the software make CONAN a tool that can significantly facilitate the understanding of the complex dynamical behavior of proteins or other biomolecules. CONAN and its documentation are freely available for download on GitHub.     

The time course of cellular responses to iontophoretically applied drugs.

Simple models of two-species ecosystems are usually analyzed in terms of the existence and stability of a static equilibrium state. We examine the way in which perturbations, in the form of periodic reductions in both species, lead to stable coexistence in a state of dynamic equilibrium. We establish general criteria for the occurrence of such dynamic equilibrium states. We show that coexistence in a dynamic equilibrium occurs for a fairly wide range of model parameters, and that dynamic equilibrium states are a rather robust feature of simple models.     

Growth-induced buckling of an epithelial layer

We use a proof-of-concept experiment and two mathematical models to explore growth-induced tissue buckling, as may occur in colorectal crypt formation. Our experiment reveals how growth of a cultured epithelial monolayer on a thin flexible substrate can cause out-of-plane substrate deflections. We describe this system theoretically using a ‘bilayer’ model in which a growing cell layer adheres to a thin compressible elastic beam. We compare this with the ‘supported-monolayer’ model due to Edwards and Chapman (Bull Math Biol 69:1927–1942, 2007) for an incompressible expanding beam (representing crypt epithelium), which incorporates viscoelastic tethering to underlying stroma. We show that the bilayer model can exhibit buckling via parametric growth (in which the system passes through a sequence of equilibrium states, parameterised by the total beam length); in this case, non-uniformities in cell growth and variations in cell–substrate adhesion are predicted to have minimal effect on the shape of resulting buckled states. The supported-monolayer model reveals how competition between lateral supports and stromal adhesion influences the wavelength of buckled states (in parametric growth), and how non-equilibrium relaxation of tethering forces influences post-buckled shapes. This model also predicts that non-uniformities in growth patterns have a much weaker influence on buckled shapes than non-uniformities in material properties. Together, the experiment and models support the concept of patterning by growth-induced buckling and suggest that targeted softening of a growing cell layer provides greater control in shaping tissues than non-uniform growth.     

Combining Perturbations and Parameter Variation to Influence Mean First Passage Times

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.     

Some factors affecting oxygen uptake by red blood cells in the pulmonary capillaries

In this study we investigate the equations governing the transport of oxygen in pulmonary capillaries. We use a mathematical model consisting of a red blood cell completely surrounded by plasma within a cylindrical pulmonary capillary. This model takes account of convection and diffusion of oxygen through plasma, diffusion of oxygen through the red blood cell, and the reaction between oxygen and haemoglobin molecules. The velocity field within the plasma is calculated by solving the slow flow equations. We investigate the effect on the solution of the governing equations of: (i) mixed-venous blood oxygen partial pressure (the initial conditions); (ii) alveolar gas oxygen partial pressure (the boundary conditions); (iii) neglecting the convection term; and (iv) assuming an instantaneous reaction between the oxygen and haemoglobin molecules. It is found that: (a) equilibrium is reached much more rapidly for high values of mixed-venous blood and alveolar gas oxygen partial pressure; (b) the convection term has a negligible effect on the time taken to reach a prescribed degree of equilibrium; and (c) an instantaneous reaction may be assumed. Explanations are given for each of these results.     

Stability analysis of the elbow with a load

We present a model of the human elbow and study the problem of existence and stability of equilibrium states. Our main goal is to demonstrate that stable equilibrium states exist just on grounds of the mechanical properties of the muscles and the skeleton. In particular, additional control mechanisms such as reflexes are not necessary to obtain stability. We assume that the activation of flexor and extensor muscles is constant and such that the right angle is an equilibrium state. We give a complete bifurcation diagram of all equilibrium states in terms of the elbow angle, the activation of the muscles and the mass of a load. Moreover, we define a dimensionless model parameter which allows to determine whether or not there are stable equilibria at an angle of ninety degrees. It turns out that the dependency of the muscle forces on the length of the muscles is the crucial factor for the stability of such an equilibrium.     

Infinite hidden Markov models for multiple multivariate time series with missing data

Exposure to air pollution is associated with increased morbidity and mortality. Recent technological advancements permit the collection of time-resolved personal exposure data. Such data are often incomplete with missing observations and exposures below the limit of detection, which limit their use in health effects studies. In this paper, we develop an infinite hidden Markov model for multiple asynchronous multivariate time series with missing data. Our model is designed to include covariates that can inform transitions among hidden states. We implement beam sampling, a combination of slice sampling and dynamic programming, to sample the hidden states, and a Bayesian multiple imputation algorithm to impute missing data. In simulation studies, our model excels in estimating hidden states and state-specific means and imputing observations that are missing at random or below the limit of detection. We validate our imputation approach on data from the Fort Collins Commuter Study. We show that the estimated hidden states improve imputations for data that are missing at random compared to existing approaches. In a case study of the Fort Collins Commuter Study, we describe the inferential gains obtained from our model including improved imputation of missing data and the ability to identify shared patterns in activity and exposure among repeated sampling days for individuals and among distinct individuals.     

The asymptotic behavior of solutions of the buffered bistable system

In this paper, we study a model for calcium buffering with bistable nonlinearity. We present some results on the stability of equilibrium states and show that there exists a threshold phenomenon in our model. In comparing with the model without buffers, we see that stationary buffers cannot destroy the asymptotic stability of the associated equilibrium states and the threshold phenomenon. Moreover, we also investigate the propagation property of solutions with initial data being a disturbance of one of the stable states which is confined to a half-line. We show that the more stable state will eventually dominate the whole dynamics and that the speed of this propagation (or invading process) is positive.     

Analysis of gene network robustness based on saturated fixed point attractors

The analysis of gene network robustness to noise and mutation is important for fundamental and practical reasons. Robustness refers to the stability of the equilibrium expression state of a gene network to variations of the initial expression state and network topology. Numerical simulation of these variations is commonly used for the assessment of robustness. Since there exists a great number of possible gene network topologies and initial states, even millions of simulations may be still too small to give reliable results. When the initial and equilibrium expression states are restricted to being saturated (i.e., their elements can only take values 1 or −1 corresponding to maximum activation and maximum repression of genes), an analytical gene network robustness assessment is possible. We present this analytical treatment based on determination of the saturated fixed point attractors for sigmoidal function models. The analysis can determine (a) for a given network, which and how many saturated equilibrium states exist and which and how many saturated initial states converge to each of these saturated equilibrium states and (b) for a given saturated equilibrium state or a given pair of saturated equilibrium and initial states, which and how many gene networks, referred to as viable, share this saturated equilibrium state or the pair of saturated equilibrium and initial states. We also show that the viable networks sharing a given saturated equilibrium state must follow certain patterns. These capabilities of the analytical treatment make it possible to properly define and accurately determine robustness to noise and mutation for gene networks. Previous network research conclusions drawn from performing millions of simulations follow directly from the results of our analytical treatment. Furthermore, the analytical results provide criteria for the identification of model validity and suggest modified models of gene network dynamics. The yeast cell-cycle network is used as an illustration of the practical application of this analytical treatment.     

Modeling Superhydrophobic Contact Angles and Wetting Transition

It is well known that surface roughness has a very important effect on superhydrophobicity.The Wenzel and Cassie-Baxtermodels,which correspond to the homogeneous and heterogeneous wetting respectively,are currently primary instructions fordesigning superhydrophobic surfaces.However,the particular drop shape that a drop exhibits might depend on how it is formed.A water drop can occupy multiple equilibrium states,which relate to different local minimal energy.In some cases,both equilibriumstates can even co-exist on a same substrate.Thus the apparent contact angles may vary and have different values.Wediscuss how the Wenzel and Cassie-Baxter equations determine the homogeneous and heterogeneous wetting theoretically.Contact angle analysis on hierarchical surface structure and contact angle hysteresis has been put specific attention.In particular,we study the energy barrier of transition from Cassie-Baxter state to Wenzel state,based on existing achievement by previousresearchers,to determine the possibility of the transition and how it can be interpreted.It has been demonstrated that surfaceroughness and geometry will influence the energy required for a drop to get into equilibrium,no matter it is homogeneous orheterogeneous wetting.     

Quantum-like model of brain's functioning: decision making from decoherence

We present a quantum-like model of decision making in games of the Prisoner's Dilemma type. By this model the brain processes information by using representation of mental states in a complex Hilbert space. Driven by the master equation the mental state of a player, say Alice, approaches an equilibrium point in the space of density matrices (representing mental states). This equilibrium state determines Alice's mixed (i.e., probabilistic) strategy. We use a master equation in which quantum physics describes the process of decoherence as the result of interaction with environment. Thus our model is a model of thinking through decoherence of the initially pure mental state. Decoherence is induced by the interaction with memory and the external mental environment. We study (numerically) the dynamics of quantum entropy of Alice's mental state in the process of decision making. We also consider classical entropy corresponding to Alice's choices. We introduce a measure of Alice's diffidence as the difference between classical and quantum entropies of Alice's mental state. We see that (at least in our model example) diffidence decreases (approaching zero) in the process of decision making. Finally, we discuss the problem of neuronal realization of quantum-like dynamics in the brain; especially roles played by lateral prefrontal cortex or/and orbitofrontal cortex.     

Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells

Cancer cells within individual tumors often exist in distinct phenotypic states that differ in functional attributes. While cancer cell populations typically display distinctive equilibria in the proportion of cells in various states, the mechanisms by which this occurs are poorly understood. Here, we study the dynamics of phenotypic proportions in human breast cancer cell lines. We show that subpopulations of cells purified for a given phenotypic state return towards equilibrium proportions over time. These observations can be explained by a Markov model in which cells transition stochastically between states. A prediction of this model is that, given certain conditions, any subpopulation of cells will return to equilibrium phenotypic proportions over time. A second prediction is that breast cancer stem-like cells arise de novo from non-stem-like cells. These findings contribute to our understanding of cancer heterogeneity and reveal how stochasticity in single-cell behaviors promotes phenotypic equilibrium in populations of cancer cells.     

Networks of forest ecosystems: Mathematical modeling of their biotic pump mechanism and resilience to certain patch deforestation

We propose an innovative mathematical model for studying the dynamics of a complex network of forest ecosystems, in which two forest patches interact which each other through water exchanges. Our model reproduces a recently analyzed principle of constant precipitation quantity over densely forested areas. We perform a stability and bifurcation analysis and show that the distance separating two forest ecosystems can attract a part of the network to an extinction state. We incorporate a randomly generated perturbation modeling deforestation and investigate the effect of the level of deforestation on the equilibrium states of the network. We also exhibit an original type of synchronization in the case of densely distributed forest ecosystems.     

The thermodynamic landscape of testosterone binding to cytochrome P450 3A4: ligand binding and spin state equilibria

Human cytochrome P450 (CYP) 3A4 catalyzes the oxygen-dependent metabolism of greater than 60% of known drugs. CYP3A4 binds multiple ligands simultaneously, and this contributes to complex allosteric kinetic behavior. Substrates that bind to this enzyme change the ferric spin state equilibrium of the heme, which can be observed by optical absorbance and electron paramagnetic resonance (EPR) spectroscopy. The ligand-dependent spin state equilibrium has not been quantitatively understood for any ligands that exhibit multiple binding. The CYP3A4 substrate testosterone (TST) has been shown previously by absorbance spectroscopy to induce spin state changes that are characteristic of a low spin to high spin conversion. Here, EPR was used to examine the equilibrium binding of TST to CYP3A4 at [CYP3A4] > K(D), which allows for characterization of the singly occupied state (i.e., CYP3A4.TST). We also have used absorbance spectroscopy to examine equilibrium binding, where [CYP3A4] < K(D), which allows for determination of K(D)'s. The combination of absorbance and EPR spectroscopy at different CYP3A4 concentrations relative to K(D) and curve fitting of the resultant equilibrium binding titration curves to the Adair-Pauling equations, and modifications of it, reveals that the first equivalent of TST binds with higher affinity than the second equivalent of TST and its binding is positively cooperative with respect to ligand-dependent spin state conversion. Careful analysis of the EPR and absorbance spectral results suggests that the binding of the second TST induces a shift to the high spin state and thus that the second TST binding causes displacement of the bound water. A model involving six thermodynamic states is presented and this model is related to the turnover of the enzyme.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

On the evolution of polygyny: a theoretical examination of the polygyny threshold model

The polygyny threshold model states that if costs incurred areless than the benefits gained from mating polygynously in termsof breeding-situation quality, then polygyny is favored andcould evolve. We constructed mathematical models and computersimulations to evaluate this hypothesis. In the basic model,there is a single locus with two alleles, which regulates whetherthe female is receptive to polygyny. There are two breedingsituations of differing quality on which males randomly assort.Females then select a mate based on the associated breedingsituation and whether the male already has mates. This basicmodel is extended mathematically to include a cost for the initialfemale of a male with multiple mates and again to include geneexpression in males. The computer simulations extend the basicmodel to multiple loci and alleles and to multiple breedingsituations. The results presented here suggest that the polygynythreshold model is valid in a population genetic context: ifthe fitness of females that actually mate polygynously is greaterthan the fitness of monogamous females on poorer breeding situations,polygyny evolves. However, this approach reveals interestingdynamics not apparent from the verbal model. If the trait isexpressed in males and females, then polygyny can evolve evenif females mating polygynously have a lower fitness than femalesmating monogamously. In the multiple breeding-situations model,the polygyny allele increases to some equilibrium value abovewhich it experiences no selection. Surprisingly, as the costof polygyny increases, the equilibrium frequency of the polygynyallele also increases. The difference between this evolutionarymodel and the ideal free distribution is discussed.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Immune network behavior—I. From stationary states to limit cycle oscilations

We develop a model for the idiotypic interaction between two B cell clones. This model takes into account B cell proliferation, B cell maturation, antibody production, the formation and subsequent elimination of antibody-antibody complexes and recirculation of antibodies between the spleen and the blood. Here we investigate, by means of stability and bifurcation analysis, how each of the processes influences the model's behavior. After appropriate nondimensinalization, the model consists of eight ordinary differential equations and a number of parameters. We estimate the parameters from experimental sources. Using a coordinate system that exploits the pairwise symmetry of the interactions between two clones, we analyse two simplified forms of the model and obtain bifurcation diagrams showing how their five equilibrium states are related. We show that the so-called immune states lose stability if B cell and antibody concentrations change on different time scales. Additionally, we derive the structure of stable and unstable manifolds of saddle-tye equilibria, pinpoint their (global) bifurcations and show that these bifurcations play a crucial role in determining the parameter regimes in which the model exhibits oscillatory behavior.