Time-dependent solutions of the spatially implicit neutral model of biodiversity

Previous research into the neutral theory of biodiversity has focused mainly on equilibrium solutions rather than time-dependent solutions. Understanding the time-dependent solutions is essential for applying neutral theory to ecosystems in which time-dependent processes, such as succession and invasion, are driving the dynamics. Time-dependent solutions also facilitate tests against data that are stronger than those based on static equilibrium patterns. Here I investigate the time-dependent solutions of the classic spatially implicit neutral model, in which a small local community is coupled to a much larger metacommunity through immigration. I present explicit general formulas for the eigenvalues, left eigenvectors and right eigenvectors of the models’s transition matrix. The time-dependent solutions can then be expressed in terms of these eigenvalues and eigenvectors. Some of these results are translated directly from existing results for the classic Moran model of population genetics (the Moran model is equivalent to the spatially implicit neutral model after a reparameterization); others of the results are new. I demonstrate that the asymptotic time-dependent solution corresponding to just these first two eigenvectors can be a good approximation to the full time-dependent solution. I also demonstrate the feasibility of a partial eigendecomposition of the transition matrix, which facilitates direct application of the results to a biologically relevant example in which a newly invading species is initially present in the metacommunity but absent from the local community.     

One- and two-soliton solutions for DNA

One- and two-soliton solutions of the nonlinear differential equation imitating rotational oscillations of DNA bases were found. It was shown that the equation is analogous to the simple sine-Gordon equation, which is solved by the inverse scattering method using the L-A pair. One- and two-soliton solutions found by the method were presented in terms of DNA parameters. The profiles of nonlinear one-soliton waves were calculated for two cases: for a homogeneous double polynucleotide chain consisting of A-T base pairs and for a homogeneous double polynucleotide chain consisting of G-C base pairs.     

Sequence dependent phase separation of protein-polynucleotide mixtures elucidated using molecular simulations

Ribonucleoprotein (RNP) granules are membraneless organelles (MLOs), which majorly consist of RNA and RNA-binding proteins and are formed via liquid–liquid phase separation (LLPS). Experimental studies investigating the drivers of LLPS have shown that intrinsically disordered proteins (IDPs) and nucleic acids like RNA and other polynucleotides play a key role in modulating protein phase separation. There is currently a dearth of modelling techniques which allow one to delve deeper into how polynucleotides play the role of a modulator/promoter of LLPS in cells using computational methods. Here, we present a coarse-grained polynucleotide model developed to fill this gap, which together with our recently developed HPS model for protein LLPS, allows us to capture the factors driving protein-polynucleotide phase separation. We explore the capabilities of the modelling framework with the LAF-1 RGG system which has been well studied in experiments and also with the HPS model previously. Further taking advantage of the fact that the HPS model maintains sequence specificity we explore the role of charge patterning on controlling polynucleotide incorporation into condensates. With increased charge patterning we observe formation of structured or patterned condensates which suggests the possible roles of polynucleotides in not only shifting the phase boundaries but also introducing microscopic organization in MLOs.     

Transitions of poly(dI-dC), poly(dI-methyl5dC) and poly(dI-bromo5dC) among and within the B-, Z-, A- and X-DNA families of conformations.

It is shown, using circular dichroism spectroscopy, that poly(dI-dC) is capable to isomerize into both Z-DNA and A-DNA in concentrated NaCl + NiCl2 and trifluoroethanol solutions, respectively. This polynucleotide also undergoes a cooperative, two-state transition in ethanol into a structure which most probably is a canonical B-DNA. This implies that the conformation of poly(dI-dC) is unusual in low-salt aqueous solution. The canonical B-DNA is also adopted by poly(dI-methyl5dC) in trifluoroethanol while this polynucleotide adopts Z-DNA not only in NaCl + NiCl2 but also in the presence of MgCl2. Poly(dI-methyl5dC) partially adopts X-DNA in concentrated CsF and mainly ethanolic solutions. Poly(dI-bromo5dC) isomerizes into Z-DNA not only in concentrated NaCl even in the absence of NiCl2 but also in concentrated MgCl2. This polynucleotide transforms between two distinct variants of Z-DNA in ethanol or trifluoroethanol solutions.     

Predicting RNA pseudoknot folding thermodynamics

Based on the experimentally determined atomic coordinates for RNA helices and the self-avoiding walks of the P (phosphate) and C₄ (carbon) atoms in the diamond lattice for the polynucleotide loop conformations, we derive a set of conformational entropy parameters for RNA pseudoknots. Based on the entropy parameters, we develop a folding thermodynamics model that enables us to compute the sequence-specific RNA pseudoknot folding free energy landscape and thermodynamics. The model is validated through extensive experimental tests both for the native structures and for the folding thermodynamics. The model predicts strong sequence-dependent helix-loop competitions in the pseudoknot stability and the resultant conformational switches between different hairpin and pseudoknot structures. For instance, for the pseudoknot domain of human telomerase RNA, a native-like and a misfolded hairpin intermediates are found to coexist on the (equilibrium) folding pathways, and the interplay between the stabilities of these intermediates causes the conformational switch that may underlie a human telomerase disease.     

On the behavior of solutions in viral dynamical models

We consider simple mathematical models for the early population dynamics of the human immunodefficiency type 1 virus (HIV-1). Although these systems of differential equations may be solved by numerical methods, few general theoretical results are available due to nonlinearities. We analyze a model whose components are plasma densities of uninfected CD4+ T-cells and infected cells (assumed in this model to be proportional to virion density). In addition to analyzing the nature of the equilibrium points, we show that there are no periodic or limit-cycle solutions. Depending on the values of the parameters, solutions either tend without oscillation to an equilibrium point with zero virion density or to an equilibrium point in which there are a nonzero number of virions. In the latter case the approach to equilibrium may be through damped oscillations or without oscillation.     

New Reaction of <Emphasis Type="Italic">O</Emphasis>-Benzoquinone at the Thioether Group of Methionine

THERE are two biochemical systems which probably evolved before the development of accurate polynucleotide-specified protein synthesis: these are the system for polynucleotide replication and the machinery of protein synthesis itself^{1, 2}. Before accurately specified proteins became available, these processes were perhaps catalysed by polynucleotide enzymes. Both tRNA and rRNA, which can be viewed as polynucleotide enzymes, have persisted as indispensable components of the contemporary apparatus. This has led me to wonder whether polynucleotide enzymes might still be operative in DNA replication. Moreover, in view of the complexity which would have been required for even a rudimentary form of protein synthesis, it seems unlikely that tRNA and rRNA arose by chance in a single evolutionary step¹. More probably they have evolved from the replicative machinery for polynucleotides and thus it seems likely that the machinery of DNA replication may have many features in common with the polynucleotide components of protein synthesis.     

The possible role of assignment catalysts in the origin of the genetic code

A model is presented for the emergence of a primitive genetic code through the selection of a family of proteins capable of executing the code and catalyzing their own formation from polynucleotide templates. These proteins are assignment catalysts capable of modulating the rate of incorporation of different amino acids at the position of different codons. The starting point of the model is a polynucleotide based polypeptide construction process which maintains colinearity between template and product, but may not maintain a coded relationship between amino acids and codons. Among the primitive proteins made are assumed to be assignment catalysts characterized by structural and functional parameters which are used to formulate the production kinetics of these catalysts from available templates. Application of the model to the simple case of two letter codon and amino acid alphabets has been analyzed in detail. As the structural, functional, and kinetic parameters are varied, the dynamics undergoes many bifurcations, allowing an initially ambiguous system of catalysts to evolve to a coded, self-reproductive system. The proposed selective pressure of this evolution is the efficiency of utilization of monomers and energy. The model also simulates the qualitative features of suppression, in which a deleterious mutation is partly corrected by the introduction of translational error.     

The distance of manganese to phosphorus atoms of polynucleotides, and dynamics of binding

The structure of manganese - polynucleotide complexes is studied by phosphorous NMR. An average value for the phosphorus-manganese distance is derived from the longitudinal relaxation rate. It is larger than the distance for direct coordination and decreases as a funtion of temperature. An inner-sphere/outer sphere equilibrium is proposed. The outer-sphere dominates in the double-stranded polynucleotides, whereas the inner-sphere contribution is important in single-stranded species. The kinetic parameters of the model are derived from the transverse relaxation time. The similar properties of DNA fragments and tRNA argue strongly against entrapment of manganese in special sites of tRNA.     

Effects of quarantine in six endemic models for infectious diseases

Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence. For most of these models, the endemic equilibrium is asymptotically stable, but for the SIQR model with the quarantine-adjusted incidence, the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation. The Hopf bifurcation surface and stable periodic solutions are found numerically.     

A time-delay model of single-species growth with stage structure

A single-species growth model with stage structure consisting of immature and mature stages is developed using a discrete time delay. It is shown that under suitable hypotheses there exists a globally asymptotically stable positive equilibrium. Questions concerning oscillation and nonoscillation of solutions are addressed analytically and numerically. The effect of the delay on the populations at equilibrium is also considered.     

Single-salt behavior of a symmetrical 4-site channel with barriers at its middle and ends

The theory of a symmetrical 3-barrier, 4-site, single-filing ionic channel is developed. The model goes beyond earlier models by including additional sites, as well as barriers which need not be symmetrical in the applied field, and contains the earlier models as special cases. It is itself a special case of the most general 4-site model, which has 5 barriers. By considering the barriers at the mouth and middle of the channel to be sufficiently larger than the barriers separating the sites in each channel half, these barriers can be neglected; thus this case reduces to a 3-barrier model where the sites in each channel half can then be assumed to be in equilibrium with each other. The alternative 3-barrier, 4-site case, where the barrier between the sites is considered to be larger than that at the mouth of the channel, is considered elsewhere. Pure cation permeation is considered and only single-salt properties of the system are analyzed, namely occupancy, conductance, flux ratio exponent and current-voltage relation. The concentration dependences of these properties are computed and interrelated and, where possible, also given in analytical form. The mathematical relations are obtained for a channel which is symmetrical around its middle, which is the appropriate assumption for the gramicidin channel. However, the barriers themselves are allowed to be asymmetric with respect to the potential dependence, which has been found to be essential for gramicidin. Mathematically, a straight-forward matrix formulation is used; but a general theoretical method is presented for reducing a complex model (with more than 2 sites) to a simpler cases when equilibrium exists across one or several barriers, as is often the cases. This method is a prototype which makes analytical solutions of complex barrier models possible in many cases.     

Phosphorylation of tRNA by T4 polynucleotide kinase.

The phosphorylation of various intact tRNA species by T4 polynucleotide kinase has been studied. The apparent Michaelis constant was on the average found to be 100 times lower than for some single-stranded DNAs previously studied. (J.R. Lillehaug and K. Kleppe, (1975) Biochemistry, 14, 1221). Conditions which result in complete phosphorylation of different tRNA species have also been established. Studies on equilibrium constants and the reversibility of the reaction revealed that the phosphorylation reaction is not a true equilibrium reaction under the conditions used in this work.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Covalent binding of benzo[a]pyrenediol epoxides to polynucleotides

Spectroscopic studies on the trans-7,8-dihydroxy-anti-9,10-epoxy-7,8,9,10-tetrahydrobenzo[a]pyrene- (anti-BPDE-) modified synthetic polynucleotide solutions reveal interesting sequence-dependent stereoselective covalent binding of anti-BPDE to DNA. Absorption spectral results indicate that the G.C polymers are much more reactive than the A.T polymers toward this metabolite and the homopolymer suffers higher modification than its corresponding alternating polynucleotide. The covalently attached anti-BPDE exhibits only a 2-3-nm red shift in the guanine-containing polynucleotide and native DNA solutions as opposed to the 8-nm red shift in poly(G) and none in the A.T polymers. Distinct stereoselectivities are exhibited by poly(dG-dC).poly(dG-dC) vs. poly(dG).poly(dC) as suggested by the oppositely signed CD in the pyrene spectral region. Comparison with the syn-BPDE modified polynucleotides reveals some interesting differences with its anti diastereomer. Significant contributions from the intercalated syn-BPDE are apparent in the modified guanine-containing polynucleotides as indicated by the appearance of 10-nm red-shifted shoulders. In contrast to the strong dependence on polynucleotides for anti-BPDE, the rate of hydrolysis of syn-BPDE appears to be insensitive to their presence in the solution. anti-BPDE modification on the 50 microM hexaamminecobalt-induced Z-form poly(dG-dC).poly(dG-dC) is much less extensive than its corresponding B form, possibly the consequence of both structural and ionic strength factors. The spectral characteristics of anti-BPDE bonded to these two forms are distinctly different, with the Z form resembling more closely those of A.T polymers.(ABSTRACT TRUNCATED AT 250 WORDS)     

Cycles in cannibalistic egg-larval interactions

A model of a cannibalistic larval-egg interaction such as occurs in Tribolium is developed which leads to a system of nonlinear Volterra integral equations. I determine the local stability properties of the unique equilibrium point of the model. A Hopf bifurcation analysis shows that the model always undergoes a subcritical bifurcation when stability is lost. Numerical solutions confirm the presence of multiple attractors over a range of parameter values. The form of the cycles observed in the numerical solutions is analogous to that observed in laboratory populations of Tribolium.     

Separation of Two Types of Adeno-Associated Virus Particles Containing Complementary Polynucleotide Chains

An adeno-associated virus containing bromodeoxyuridine-substituted deoxyribonucleic acid has been fractionated by equilibrium centrifugation in CsCl into two classes of virions which contain complementary polynucleotide chains.     

Lead ion binding and RNA chain hydrolysis in phenylalanine tRNA

Crystalline complexes of yeast phenylalanine tRNA and Lead (II) ion were prepared by soaking pregrown orthorhombic crystals of tRNA in saturated lead chloride solutions. The locations of tightly bound lead ions on the tRNA were determined by difference Fourier methods. There are three major lead binding sites; two of these replace tightly bound magnesium ions in the native tRNA structure. Site I is located in the dihydrouridine loop of the molecule adjacent to phosphate P18 which is specifically cleaved by lead. This is evident from changes observed in the Pb-native difference electron density maps. A possible mechanism for lead ion hydrolysis of the polynucleotide chain is proposed.     

Polynucleotide evolution and branching processes

The theory of multitype branching processes is applied to the kinetics of polynucleotide replication. The results obtained are compared with the solutions of the deterministic differential equations of conventional chemical kinetics.     

Use of polynucleotide/polyacrylamide-gel electrophoresis as a sensitive technique for the detection and comparison of ribonuclease activities.

A technique is described in which the incorporation of a polynucleotide substrate into the matrix of a polyacrylamide gel allows the use of electrophoresis for the detection of polycationic ribonuclease activity rather than simply the presence of protein. Because use is made of the catalytic properties of ribonucleases, polynucleotide/polyacrylamide-gel electrophoresis is apparoximately 10(5) times more sensitive for the detection of these enzymes than conventional gel electrophoresis with the use of protein-staining dyes. Initial studies showed that the poor migration, in the gels, of highly charged polycationic ribonucleases in the presence of negatively charged synthetic polynucleotides could be overcome by high concentrations of spermine. The positively charged polyamine, by neutralizing the polyanionic polynucleotide, enabled these basic enzymes to migrate considerable distances in the gel. Electrophoresis of the RNAases under conditions of low pH, and incubation of the gel at neutral pH followed by staining for polynucleotide, resulted in coloured gels containing clear bands that define regions of enzyme activity. Alterations in spermine concentration or substrate identity caused changes in the positions of these bands, suggesting a dynamic interaction among the enzyme, polyamine and polynucleotide. Because of the advantages, in terms of selectivity and sensitivity of polynucleotide/polyacrylamide-gel electrophoresis, this technique was used to demonstrate the nuclease homogenity of three purified bovine muscle enzymes, and to compare these enzymes with each other, as well as with bovine pancreatic ribonuclease A.