Mathematical models for continuous culture growth dynamics of mixed populations subsisting on a heterogeneous resource base. II. Predation and trophic structure

Models are presented for the joint dynamics of predators and prey, maintained in continuous flow chemostat culture. The predators are visualized as subsisting on one or more prey organisms, which in turn are visualized as subsisting on one or more substrate resources supplied by the investigator. The dynamic equations are translated into an analogous Lotka-Volterra predation model, and the criteria for the existence and stability of various equilibria are indicated. Denoting the number of different predator organisms as N_H, the number of different prey organisms by N_I and the number of different substrates as N_J, it is shown that the joint coexistence of all components requires 0 ? N_I ? N_H ? N_J. The model is extended to more complex situations by including additional trophic layers and by allowing trophic layer “leap-frogging.” The model may always be translated into an approximately quadratic differential equation of the Lotka-Volterra type. The α- and β-coefficients of these latter are really variables, and become quite complex for some of the multi-layered models.     

Forced-unfolding and force-quench refolding of RNA hairpins

Nanomanipulation of individual RNA molecules, using laser optical tweezers, has made it possible to infer the major features of their energy landscape. Time-dependent mechanical unfolding trajectories, measured at a constant stretching force (f_S) of simple RNA structures (hairpins and three-helix junctions) sandwiched between RNA/DNA hybrid handles show that they unfold in a reversible all-or-none manner. To provide a molecular interpretation of the experiments we use a general coarse-grained off-lattice Gō-like model, in which each nucleotide is represented using three interaction sites. Using the coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of f_S and the loading rate (r_f). The simulations and theoretical analysis have been done both with and without the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and timescales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by f_S results in a sequential unfolding of the hairpin starting from their ends, whereas thermal denaturation occurs stochastically. From the dependence of the unfolding rates on r_f and f_S we show that the position of the unfolding transition state is not a constant but moves dramatically as either r_f or f_S is changed. The transition-state movements are interpreted by adopting the Hammond postulate for forced-unfolding. Forced-unfolding simulations of RNA, with handles attached to the two ends, show that the value of the unfolding force increases (especially at high pulling speeds) as the length of the handles increases. The pathways for refolding of RNA from stretched initial conformation, upon quenching f_S to the quench force f_Q, are highly heterogeneous. The refolding times, upon force-quench, are at least an order-of-magnitude greater than those obtained by temperature-quench. The long f_Q-dependent refolding times starting from fully stretched states are analyzed using a model that accounts for the microscopic steps in the rate-limiting step, which involves the trans to gauche transitions of the dihedral angles in the GAAA tetraloop. The simulations with explicit molecular model for the handles show that the dynamics of force-quench refolding is strongly dependent on the interplay of their contour length and persistence length and the RNA persistence length. Using the generality of our results, we also make a number of precise experimentally testable predictions.     

Competitive blocking of apical sodium channels in epithelia

Apical sodium-selective channels in frog skin, when blocked by amiloride or triamterene, exhibit fluctuations in current, the spectra of which are Lorentzian. These effects have been modeled previously with two-state and three-state models by Lindemann and Van Driessche. A recent observation by Hoshiko and Van Driessche that corner frequencies are lowered by increasing the apical sodium concentration cannot be accounted for by these models. We explore the possibility that sodium (S) and amiloride (A) compete for a site at the mouth of the channel. A new three-state channel model (sodium-occupied, open/unoccupied, open/amiloride-blocked) is analyzed. Its corner frequency is of the form f_c = f_co[1 + (A/K_A)/(1 + S/K_S)], consistent with the observed sodium dependence of the corner frequency. The minimum frequency, f_co, and the inhibition constants, K_A and K_S, are expressed in terms of the rate constants of the model. To account for sodium self-inhibition, we postulate that two sodium ions in the channel may result in clogging — a fourth state. The two corner frequencies are calculated; so are the plateau values of the noise power. The noise power shows a maximum as a function of blocker concentration, as observed previously using triamterene. The four-state model predicts the observed suppression by small amounts of blocker of the low-frequency sodium (clogging) noise.     

Probabilistic Model of Microbial Cell Growth,Division, and Mortality

After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability P_d(t)δt, dead with probability P_m(t)δt, or alive but undivided with the probability 1 − [P_d(t) + P_m(t)]δt, where t is time, P_d(t) expresses the probability of division for an individual cell per unit of time, and P_m(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat''s properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population''s size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With P_d(t) and P_m(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.     

The origin of fibonacci phyllotaxis—an analysis of Adler's contact pressure model and Mitchison's expanding apex model

Adler's contact pressure model for Fibonacci phyllotaxis is examined theoretically. It is shown that the model, as it stands, does not account for Fibonacci phyllotaxis, since it requires, but does not provide, a mechanism for initiating new primordia with increasingly greater precision as phyllotaxis rises. Modifications are suggested which remedy this deficiency in the model; one of these modifications involves a combination of Adler's model with Mitchison's model.From a comparison of the ranges of divergence angles permitted by Adler's model against Fujita's measurements of divergence angles in plants with low phyllotaxis, it is shown that the modified contact pressure model, if based on the concept of mechanical pressures between primordia in contact, cannot account for the divergence angles found in low phyllotaxis systems. However it is shown that this deficiency can be overcome if the contact pressure effect is regarded as a chemical phenomenon, mediated by a growth inhibitor produced by the prirnordia and moving more readily in vertical directions than in other directions.Mitchison's model, which is based on the concepts of an expanding apex and primordium initiation by existing primordia, is shown to account for Fibonacci phyllotaxis only if phyllotaxis rises sufficiently slowly; to guarantee that an F_n + F_n+1 system can develop there must already be at least F_n+1 primordia present in an F_n?1 + F_n system, at least F_n primordia in an F_n?2 + F_n?1 system, and so on down to at least three primordia in a 1 + 2 system, making a total of at least F_n+3?5 primordia (where F_n = nth term of the Fibonacci series with F₁ = F₂ = 1). Adler's model, modified, requires only that F_{n + 1} primordia be present with divergence angles in the range 120–180° to guarantee that an F_n + F_{n + 1} system can develop.     

Electron spin resonance and electron nuclear double resonance studies of cation radicals derived from tocopherol model compounds

Electron spin resonance (ESR) and electron nuclear double resonance (ENDOR) measurements were performed for the cation radicals obtained from the model compounds of α-, β-, γ- and δ-tocopherol (vitamin E) by oxidizing the tocopherol precursors in an AlCl₃-CH₂Cl₂ solution. The proton hyperfine coupling constants g-values were precisely determined. The ENDOR spectra of the cation radicals of α-, β-, γ- and δ-tocopherol models in CH₂Cl₂ at ?100°C clearly show 10, 6, 6 and 12 different proton hyperfine couplings, respectively. By varying the temperature, the ESR spectra of the α- and δ-tocopherol model cations exhibit line-width alternation phenomena characteristic of the hindered rotation of the OH group. However, neither the β- nor the γ-tocopherol model cation radical ESR spectra show any sign of an alternative line-width effect. These results are interpreted by assuming that the β- and γ-tocopherol model cations are stabilized in the trans and cis conformations, respectively. On tocopherol model cations are stabilized in the trans and cis conformations, respectively. On the other hand, both the α- and δ-tocopherol model cations exist as cis and trans isomers.     

Analyse des courbes de survie de nématodes phytoparasites selon le modèle de Teissier

Four survival curves of plant parasitic nematodes are analysed with Teissier's model. The model is based upon the assumption of an exponential decrease with time of the life expectancy, E(t) = E₀ exp(−αt), with E₀ the life expectancy at time 0 and α a decay coefficient. By integrating this formula we obtain an expression for the number of survivors at time t, S(t), as a function of the time, t, and S₀, the number of individuals at time 0, and the parameters E₀ and α: S(t) = S₀exp[αt −(exp(αt) −1)/αE₀. S₀ is a scaling parameter, related to the initial number of individuals in the population, thus the form of the curve depends only on the parameters E₀ and α, which are readily understandable in a biological model. These parameters E₀ and α are estimated by fitting the model to experimental data using a non-linear regression based on a least squares procedure. The results show a highly significant fit of the model to the data indicating the ability of the model to describe the plant parasitic nematodes survival curve. Results are discussed and the hypothesis of Teissier is compared with other related hypotheses.     

Effects on hypothalamus when CPG is fed back to basal ganglia based on KIV model

The KIV model approximates the operation of the basic vertebrate forebrain together with the basal ganglia and motor systems. In KIV model, the hypothalamus and the basal ganglia which are two important parts in the midline forebrain are closely associated with the locomotion. The CPG model with time delay is established in this paper and the stability of this CPG model is discussed. The CPG output is treated as the proprioception and fed back to the basal ganglia. We focus on the effects on the hypothalamus and the basal ganglia when the time delay parameter a_d, the CPG amplitude parameter e and the CPG frequency parameter T_r are changed. Through analysis, we find that there exists optimum value of the parameters a_d or T_r which can make the synchronization of the hypothalamus optimum when the CPG is added into the basal ganglia. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.     

The dynamics of a discrete population model with threshold

The dynamics of the discrete, scalar population model x_{k + 1} = ax²_k(1 – x_k) are investigated. In addition to density dependence, which has been studied previously by many, this equation models the threshold phenomenon. Some similarities to and differences from previous models are observed. In particular, for large a values this model exhibits chaos which is restricted to a nowhere dense Cantor set of measure 0. In order to explain this, a piecewise linear simplification of the model is considered. Other models exhibiting similar dynamics are also mentioned.     

A new attempt to study the oxygen evolving system of photosynthesis: Determination of transition probabilities of a state i

In order to interpret the behaviour of the oxygen evolving system, a complete analysis of the model of Forbush et al. (1971) led Lavorel (1976) to characterize the period four oscillations by four coefficients obeying the relation δ₁ ? δ₂ + δ₃ ? δ₄ = 1. A critical examination of the analysis method showed that the model was extremely sensitive to experimental inaccuracy. From this fact, the δ₁'s have to be considered as free variables: in this case, the above relation which is a consequence of the model, was verified by better than 3 %. Furthermore the matrix on which the δ₁'s calculation is based is an ill-conditioned matrix, a property of which is to amplify the errors. Consequently, Lavorel's conclusions must be accepted with care.A new δ₁'s calculation method is presented. It shows that the first flash after following a period of darkness, acts in a specific way which is not consistent with the model. On the contrary the effects of the following flashes are in good agreement with those forecasted by the model and with the homogeneous hypothesis of Kok where the transition probabilities are equal for all the states S_i. The quadratic deviation between theoretical and experimental sequences is better than 1 %. It has been verified that the misses are largely smaller with chloroplasts than with Chlorella and that the double hits are strongly dependent on the flash duration.     

关帝山天然次生针叶林林隙径高比

林隙径高比(D_EG/H)是指林隙直径与林隙高度的比值。它是林隙的一个主要特征因子,是研究森林动态及评价森林采伐强度的一个重要指标。以关帝山三种天然次生针叶林(华北落叶松、云杉、油松林)林隙作为研究对象,分析了3种林分林隙径高比结构,结果为:云杉林林隙径高比D_EG/H以0.6-1.6之间分布最多,占81.82%,油松林林隙径高比主要分布在0.8-1.6之间,占70.72%。华北落叶松林林隙径高比主要分布在0.4-1之间,占97.06%;通过林隙大小与林隙下幼树数量及林隙敏感度与幼树密度之间的散点分布趋势对林隙大小和林隙敏感度两个特征因子进行比较分析,结果为:林隙大小与林隙下幼树数量之间的散点分布无规律,很难反映各自林隙大小与幼树数量之间的具体关系。而D_EG/H与幼树密度之间的散点分布很有规律,能较好的反映幼树密度与林隙径高比之间的关系;利用线性模型、对数模型以及二阶多项分布模型分别对幼树密度和D_EG/H进行回归分析,并利用各自模型的相关系数、参数P值对它们进行比较,结果为:3种模型都可以用来拟合3种林分的幼树密度和林隙敏感度,其中对数模型拟合效果最好。     

On the problems of the evolutionary optimization of life history. I. Markov model of Leslie life cycle and optimization of fertility

The paper considers the properties of individual life history corresponding to the Leslie model of age-structured population. The life history is modelled as a finite Markov chain with absorption at a death state of individual. In this model, individual longevity, average number of offspring R _L (produced by an individual over the entire life), and some other known characteristics of the life history have been derived using simple probability methods that do not involve matrix calculus and their individual components have been interpreted. In the linear Leslie population model (derived by simple modification of a Markov chain), R _L determines the growth or decline of a population. Individuals with higher R _L values have evolutionary advantages in the population due to accelerated growth in their number. The selection of fertility as a factor of the increase in R _L is considered. In the Leslie model, fertility is a set of correlated quantitative traits, where the age-specific fertility components are determined both by multiple loci and the environment. According to the genetic theory of quantitative trait selection, they evolve towards an increase in R _L . Taking into account the limited resources for reproduction, selection optimizes the fertility distribution according to age. Optimal distribution corresponds to the attainment of the maximum R _L . This complies with the maximization of the rate of population growth (r-selection), which is characteristic of linear population models. The search for the R _L maximum and optimal distribution of fertility belongs to the field of linear programming.     

Chloride and Salicylate Influence Prestin-dependent Specific Membrane Capacitance: SUPPORT FOR THE AREA MOTOR MODEL*

The outer hair cell is electromotile, its membrane motor identified as the protein SLC26a5 (prestin). An area motor model, based on two-state Boltzmann statistics, was developed about two decades ago and derives from the observation that outer hair cell surface area is voltage-dependent. Indeed, aside from the nonlinear capacitance imparted by the voltage sensor charge movement of prestin, linear capacitance (C_lin) also displays voltage dependence as motors move between expanded and compact states. Naturally, motor surface area changes alter membrane capacitance. Unit linear motor capacitance fluctuation (δC_sa) is on the order of 140 zeptofarads. A recent three-state model of prestin provides an alternative view, suggesting that voltage-dependent linear capacitance changes are not real but only apparent because the two component Boltzmann functions shift their midpoint voltages (V_h) in opposite directions during treatment with salicylate, a known competitor of required chloride binding. We show here using manipulations of nonlinear capacitance with both salicylate and chloride that an enhanced area motor model, including augmented δC_sa by salicylate, can accurately account for our novel findings. We also show that although the three-state model implicitly avoids measuring voltage-dependent motor capacitance, it registers δC_sa effects as a byproduct of its assessment of C_lin, which increases during salicylate treatment as motors are locked in the expanded state. The area motor model, in contrast, captures the characteristics of the voltage dependence of δC_sa, leading to a better understanding of prestin.     

Calculation of optical signal using three-dimensional bidomain/diffusion model reveals distortion of the transmembrane potential

Optical mapping experiments allow investigators to view the effects of electrical currents on the transmembrane potential, V_m, as a shock is applied to the heart. One important consideration is whether the optical signal accurately represents V_m. We have combined the bidomain equations along with the photon diffusion equation to study the excitation and emission of photons during optical mapping of cardiac tissue. Our results show that this bidomain/diffusion model predicts an optical signal that is much smaller than V_m near a stimulating electrode, a result consistent with experimental observations. Yet, this model, which incorporates the effect of lateral averaging, also reveals an optical signal that overestimates V_m at distances >1 mm away from the electrode. Although V_m falls off with distance r from the electrode as exp(−r/λ)/r, the optical signal decays as a simple exponential, exp(−r/λ). Moreover, regions of hyperpolarization adjacent to a cathode are emphasized in the optical signal compared to the region of depolarization under the cathode. Imaging methods utilizing optical mapping techniques will need to account for these distortions to accurately reconstruct V_m.     

Quantal basis of vesicle growth and information content, a unified approach

Secretory vesicles express a periodic multimodal size distribution. The successive modes are integral multiples of the smallest mode (G₁). The vesicle content ranges from macromolecules (proteins, mucopolysaccharides and hormones) to low molecular weight molecules (neurotransmitters). A steady-state model has been developed to emulate a mechanism for the introduction of vesicles of monomer size, which grow by a unit addition mechanism, G₁+G_n→G_n+1 which, at a later stage are eliminated from the system. We describe a model of growth and elimination transition rates which adequately illustrates the distributions of vesicle population size at steady-state and upon elimination. Consequently, prediction of normal behavior and pathological perturbations is feasible. Careful analysis of spontaneous secretion, as compared to short burst-induced secretion, suggests that the basic character-code for reliable communication should be within a range of only 8-10 vesicles’ burst which may serve as a yes/no message.     

Ising Model of Cardiac Thin Filament Activation with Nearest-Neighbor Cooperative Interactions

We have developed a model of cardiac thin filament activation using an Ising model approach from equilibrium statistical physics. This model explicitly represents nearest-neighbor interactions between 26 troponin/tropomyosin units along a one-dimensional array that represents the cardiac thin filament. With transition rates chosen to match experimental data, the results show that the resulting force-pCa (F-pCa) relations are similar to Hill functions with asymmetries, as seen in experimental data. Specifically, Hill plots showing (log(F/(1-F)) vs. log [Ca]) reveal a steeper slope below the half activation point (Ca₅₀) compared with above. Parameter variation studies show interplay of parameters that affect the apparent cooperativity and asymmetry in the F-pCa relations. The model also predicts that Ca binding is uncooperative for low [Ca], becomes steeper near Ca₅₀, and becomes uncooperative again at higher [Ca]. The steepness near Ca₅₀ mirrors the steep F-pCa as a result of thermodynamic considerations. The model also predicts that the correlation between troponin/tropomyosin units along the one-dimensional array quickly decays at high and low [Ca], but near Ca₅₀, high correlation occurs across the whole array. This work provides a simple model that can account for the steepness and shape of F-pCa relations that other models fail to reproduce.     

A computational model for functional mapping of genes that regulate intra-cellular circadian rhythms

Background

Development of the mathematical models that adequately describe biochemical reactions and molecular-genetic mechanisms is one of the most important tasks in modern bioinformatics. Because the enzyme adenylosuccinate synthetase (AdSS) has long been extensively studied, a wealth of kinetic data has been accumulated.

Results

We describe a mathematical model for the reaction catalyzed by AdSS. The model's parameters were fitted to experimental data obtained from published literature. The advantage of our model is that it includes relationships between the reaction rate, the concentrations of three substrates (GTP, IMP and ASP), the effects of five inhibitors (GMP, GDP, AMP, ASUC and SUCC), and the influence of Mg²⁺ ions.

Conclusion

Our model describes the reaction catalyzed by AdSS as a fully random process. The model structure implies that each of the inhibitors included in it is only competitive to one of the substrates. The model was tested for adequacy using experimental data published elsewhere. The values obtained for the parameters are as follows: V _max = 1.35·10^-3 mM/min, Km _GTP = 0.023 mM, Km _IMP = 0.02 mM, Km _ASP = 0.3 mM, Ki _GMP = 0.024 mM, Ki _GDP = 8·10^-3 mM, Ki _AMP = 0.01 mM, Ki _ASUC = 7.5·10^-3 mM, Ki _SUCC = 8 mM, Km _Mg = 0.08 mM.