Asymptotic behavior of some stochastic difference equation population models

We consider a general class of Markov population models formulated as stochastic difference equations. The population density is shown to converge either to 0, to +, or to a unique stationary distribution concentrated on (0, +), depending on the signs of the mean log growth rates near 0 and +. These results are applied to the Watkinson-MacDonald bottleneck model of annual plants with a seedbank, extended to allow for random environmental fluctuations and competition among co-occurring species. We obtain criteria for long-term persistence of single-species populations, and for coexistence of two competing species, and the biological significance of the criteria is discussed. The lamentably few applications to the problem at hand of classical limit-theory for Markov chains are surveyed.     

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.     

Molecular dynamics simulations of helix denaturation.

An understanding of the structural transitions that an alpha-helix undergoes will help to elucidate such motions in proteins and their role in protein folding. We present the results of molecular dynamics simulations to investigate these transitions in a short polyalanine peptide (13 residues) both in vacuo and in the presence of solvent. The denaturation of this peptide was monitored as a function of temperature (ranging from 5 to 200 degrees C). In vacuo, the helical state predominated at all temperatures, whereas in solution the helix melted with increasing temperature. The peptide was predominantly helical at low temperature in solution, while at intermediate temperatures the peptide spent the bulk of the time fluctuating between different conformations with intermediate amounts of helix, e.g. not completely helical nor entirely non-helical. Many of these conformations consisted of short helical segments with intervening non-helical residues. At high temperature the peptide unfolded and adopted various collapsed unstructured states. The intrahelical hydrogen bonds that break at high temperature were not fully compensated by hydrogen bonds with water molecules in the partially unfolded forms of the peptide. Increases in temperature disrupted both the helical structure and the peptide-water interactions. Water played a major but indirect role in facilitating unfolding, as opposed to specifically competing for the intrapeptide hydrogen bonds. The implications of our results to protein folding are discussed.     

Atomically detailed lipid bilayer models for the interpretation of small angle neutron and X-ray scattering data

We present a new atom density profile (ADP) model and a statistical approach for extracting structural characteristics of lipid bilayers from X-ray and neutron scattering data. Models for five lipids with varying head and tail chemical composition in the fluid phase, 1,2-dioleoyl-sn-glycero-3-phosphatidylcholine (DOPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC), 1,2-dipalmitoyl-sn-glycero-3-phosphatidylcholine (DPPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylserine (POPS), and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylglycerol (POPG), are optimized using a simplex based method to simultaneously reproduce both neutron and X-ray scattering data. Structural properties are determined using statistical analysis of multiple optimal model structures. The method and models presented make minimal assumptions regarding the atomic configuration, while taking into account the underlying physical properties of the system. The more general model and statistical approach yield data with well defined uncertainties, indicating the precision in determining density profiles, atomic locations, and bilayer structural characteristics. Resulting bilayer structures include regions exhibiting large conformational variation. Due to the increased detail in the model, the results demonstrate the possibility of a distinct hydration layer within the interfacial (backbone) region.     

Thermodynamics of left-handed helix formation

The thermodynamics of right- and left-handed helix formation by poly[d(G-C)] X poly[d(G-C)] and by poly-(dG-m5dC) X poly(dG-m5dC) were measured spectrophotometrically and calorimetrically. From the spectrophotometric measurements the thermal stabilities of the alternative helical conformations were evaluated as a function of counterion concentration. From the calorimetric measurements the enthalpies of either right-handed or left-handed helix formation were determined. The corresponding experimental delta H values are -8.6 and -11.2 kcal/mol base pairs for the two conformations in poly[dG-C)] X poly[d(G-C)], and -9.0 and -12.7 kcal/mol base pairs, respectively, for poly(dG-m5dC) X poly(dG-m5dC).     

Controls exerted by the Aib residue: helix formation and helix reversal

The helix forming properties of the achiral alpha-amino isobutyric residue (Aib) have been demonstrated by numerous crystal structure analyses of designed and naturally occurring peptides containing one or more Aib residues in the sequence. Experimental and computational results concerning the type of helix obtained, whether the 3(10)-helix with 4 --> 1 type hydrogen bonds or the alpha-helix with 5 --> 1 hydrogen bonds or mixtures of the two, have been published. This paper deals with residues that, if inserted into a sequence, could perturb the helix-forming propensity afforded by the presence of Aib residues. Examples of structures will be presented in which Pro, Hyp, Gly-Gly, d-Ala-Gly, and Lac have been centrally placed in the sequence. In addition to the formation of helices, detailed experimentally obtained conformation information is presented for the role of the Aib residue in reversing the sense of the helix (the Schellman motif) with the consequent formation of the 6 --> 1 type hydrogen bond or a solvated 6 --> 1 hydrogen bond. Data are presented for 13 molecules with helix reversals at the C-terminus or near the center of the sequence.     

Kinetics of cytochrome C folding: atomically detailed simulations

The vast range of time scales (from nanoseconds to seconds) during protein folding is a challenge for experiments and computations. To make concrete predictions on folding mechanisms, atomically detailed simulations of protein folding, using potentials derived from chemical physics principles, are desired. However, due to their computational complexity, straightforward molecular dynamics simulations of protein folding are impossible today. An alternative algorithm is used that makes it possible to compute approximate atomically detailed long time trajectories (the Stochastic Difference Equation in Length). This algorithm is used to compute 26 atomically detailed folding trajectories of cytochrome c (a millisecond process). The early collapse of the protein chain (with marginal formation of secondary structure), and the earlier formation of the N and C helices (compare to the 60's helix) are consistent with the experiment. The existence of an energy barrier upon entry to the molten globule is examined as well. In addition to (favorable) comparison to experiments, we show that non-native contacts drive the formation of the molten globule. In contrast to popular folding models, the non-native contacts do not form off-pathway kinetic traps in cytochrome c.     

Numerical simulations of stochastic circulatory models

Properties of two of the stochastic circulatory models theoretically introduced by Smith et al., 1997, Bull. Math. Biol. 59, 1–22 were investigated. The models assumed the gamma distribution of the cycle time under either the geometric or Poisson elimination scheme. The reason for selecting these models was the fact that the probability density functions of the residence time of these models are formally similar to those of the Bateman and gamma-like function models, i.e., the two common deterministic models. Using published data, the analytical forms of the probability density functions of the residence time and the distributions of the simulated values of the residence time were determined on the basis of the deterministic models and the stochastic circulatory models, respectively. The Kolmogorov-Smirnov test revealed that even for 1000 xenobiotic particles, i.e., a relatively small number if the particles imply drug molecules, the probability density functions of the residence time based on the deterministic models closely matched the distributions of the simulated values of the residence time obtained on the basis of the stochastic circulatory models, provided that parameters of the latter models fulfilled selected conditions.     

Applying the stochastic difference equation to DNA conformational transitions: a study of B-Z and B-A DNA transitions

Despite the existence of numerous models to account for the B-Z DNA transition, experimenters have not yet arrived at a conclusive answer to the structural and dynamical features of the B-Z transition. By applying the stochastic difference equation to simulate the B-Z DNA transition, we have shown that the stretched intermediate model of the B-Z transition is more probable than other B-Z transition models such as the Harvey model. This is accomplished by comparing potential energy profiles of various B-Z DNA transition models and calculating relative probabilities based on the stochastic difference equation with respect to length (SDEL) formalism. The results garnered in this article allow for new approaches in determining the structural transition of B-DNA to Z-DNA experimentally. We have also simulated the B-A DNA transition using the stochastic difference equation. Unlike the B-Z DNA transition, the mechanism for the B-A DNA transition is well established. The variation in the pseudorotation angle during the transition is in good agreement with experimental results. Qualitative features of the simulated B-A transition also agree well with experimental data. The SDEL approach is thus a suitable numerical technique to compute long-time molecular dynamics trajectory for DNA molecules.     

Frictional models for stochastic simulations of proteins

As a first step toward a systematic parametrization of friction constants of atoms in proteins, a model in which frictional resistance is placed explicitly on each atom accessible to solvent is used to calculate overall translational and rotational diffusion constants. It is found that these quantities are relatively insensitive to the precise value of the atomic friction constant, as long as the effective hydrodynamic radius of the surface atoms is approximately 1 Å. However, if only protein atoms are included in the calculation, no reasonable range atomic of radii can reproduce the experimental translational diffusion constant to better than 20% for lysozyme and 5% for ribonuclease. When a hydration shell of approximately 70% coverage for lysozyme and 20% for ribonuclease is included, there is quantitative agreement with experimental results. The sensitivity of peptide diffusion to levels of hydration is also investigated; it is found that for glycine, two bound waters are required to provide agreement with experiment. These findings imply that the effects of solvent damping will be underestimated in stochastic simulations of proteins and peptides unless bound waters are taken into account.     

Hybrid stochastic and deterministic simulations of calcium blips

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.     

Membrane assembly of simple helix homo-oligomers studied via molecular dynamics simulations

The assembly of simple transmembrane helix homo-oligomers is studied by combining a generalized Born implicit membrane model with replica exchange molecular dynamics simulations to sample the conformational space of various oligomerization states and the native oligomeric conformation. Our approach is applied to predict the structures of transmembrane helices of three proteins--glycophorin A, the M2 proton channel, and phospholamban--using only peptide sequence and the native oligomerization state information. In every case, the methodology reproduces native conformations that are in good agreement with available experimental structural data. Thus, our method should be useful in the prediction of native structures of transmembrane domains of other peptides. When we ignore the experimental constraint on the native oligomerization state and attempt de novo prediction of the structure and oligomerization state based only on sequence and simple energetic considerations, we identify the pentamer as the most stable oligomer for phospholamban. However, for the glycophorin A and the M2 proton channels, we tend to predict higher oligomers as more stable. Our studies demonstrate that reliable predictions of the structure of transmembrane helical oligomers can be achieved when the observed oligomerization state is imposed as a constraint, but that further efforts are needed for the de novo prediction of both structure and oligomeric state.     

Numerical simulations of one- and two-dimensional stochastic neural field equations with delay

Journal of Computational Neuroscience - Neural Field Equations (NFE) are intended to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind...     

Regulation of the RNA-dependent protein kinase by triple helix formation

The RNA-dependent protein kinase (PKR) is an interferon-induced, RNA-activated enzyme that phosphorylates the α-subunit of the translation initiation factor eIF-2, inhibiting its function. PKR is activated in vitro by binding to double-stranded RNA (dsRNA) molecules of ~30 bp or longer. Here we show that triple helix forming oligonucleotides (TFOs) inhibit dsRNA binding to the isolated RNA binding domain of PKR. The inhibition is specific to the targeted RNA and dependent on TFO length. Binding to a 30 bp duplex is inhibited by a 28 nt TFO and a 20 nt TFO with an IC₅₀ of 35 ± 2 and 210 ± 22 nM, respectively. An 18 nt TFO partially inhibits binding. The activation of the kinase domain of PKR by a 30 bp RNA duplex is also inhibited by a 28 nt TFO. Inhibition of binding is most effective when the triple helix is formed prior to addition of the protein. These results indicate that triplex formation can be used to prevent the binding of an RNA binding protein with dsRNA-binding motifs.     

Host-guest scale of left-handed polyproline II helix formation

Despite the clear importance of the left-handed polyproline II (PPII) helical conformation in many physiologically important processes as well as its potential significance in protein unfolded states, little is known about the physical determinants of this conformation. We present here a scale of relative PPII helix-forming propensities measured for all residues, except tyrosine and tryptophan, in a proline-based host peptide system. Proline has the highest measured propensity in this system, a result of strong steric interactions that occur between adjacent prolyl rings. The other measured propensities are consistent with backbone solvation being an important component in PPII helix formation. Side chain to backbone hydrogen bonding may also play a role in stabilizing this conformation. The PPII helix-forming propensity scale will prove useful in future studies of the conformational properties of proline-rich sequences as well as provide insights into the prevalence of PPII helices in protein unfolded states.     

Double helix formation of phosphatidylcholine myelin figures.

Double helix formation of phosphatidylcholine myelin figures has been studied by use of optical microscopy. The winding of the double helices was looser than a geometrically possible one and the pitch was related proportionally to the outer radii of helical myelin figures. The regularity in the winding was explained in terms of the intermembrane binding energy and the bending elastic energy.