Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, is presented. Korzuhin's Theorem. which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem ‘for beginners’ is also given.     

A model of physiological and allometric factors in the self-thinning curve

A generalized self-thinning curve for plants is derived from the modified Von Bertallanfy equation. When an asymptotic relation between photosynthesis per unit of leaf area and stocking density is assumed, the self-thinning curve thus derived is also asymptotic on a log-log scale but is fitted quite well by a log-linear approximation. The model predicts that the slope of the log-linear approximation is a function of (a) photosynthetic response to density and (b) the relation between leaf area and total aboveground biomass. Intercept of the log-linear approximation is a function of these plus maximum attainable biomass, site productivity, density at which maximum photosynthesis is attained, and the nature of carbon loss within the plant community. Linkages between various parameters within the model act to reduce differences in slope and intercept for species with different life history's and physiological requirements.     

The Relationship between Corvis ST Tonometry Measured Corneal Parameters and Intraocular Pressure,Corneal Thickness and Corneal Curvature

The purpose of the study was to investigate the correlation between Corneal Visualization Scheimpflug Technology (Corvis ST tonometry: CST) parameters and various other ocular parameters, including intraocular pressure (IOP) with Goldmann applanation tonometry. IOP with Goldmann applanation tonometry (IOP-G), central corneal thickness (CCT), axial length (AL), corneal curvature, and CST parameters were measured in 94 eyes of 94 normal subjects. The relationship between ten CST parameters against age, gender, IOP-G, AL, CST-determined CCT and average corneal curvature was investigated using linear modeling. In addition, the relationship between IOP-G versus CST-determined CCT, AL, and other CST parameters was also investigated using linear modeling. Linear modeling showed that the CST measurement ‘A time-1’ is dependent on IOP-G, age, AL, and average corneal curvature; ‘A length-1’ depends on age and average corneal curvature; ‘A velocity-1’ depends on IOP-G and AL; ‘A time-2’ depends on IOP-G, age, and AL; ‘A length-2’ depends on CCT; ‘A velocity-2’ depends on IOP-G, age, AL, CCT, and average corneal curvature; ‘peak distance’ depends on gender; ‘maximum deformation amplitude’ depends on IOP-G, age, and AL. In the optimal model for IOP-G, A time-1, A velocity-1, and highest concavity curvature, but not CCT, were selected as the most important explanatory variables. In conclusion, many CST parameters were not significantly related to CCT, but IOP usually was a significant predictor, suggesting that an adjustment should be made to improve their usefulness for clinical investigations. It was also suggested CST parameters were more influential for IOP-G than CCT and average corneal curvature.     

A macroscopic approach to population genetics

Conventional population genetics uses as primitive variables the frequencies and fitnesses of individual genes. This paper develops a formalism whose primitive variables are the frequencies and fitnesses of genotypes and environmental histories in a population. From the mathematical relation that describes genetic variation and selection of genotypes and environmental histories we derive a sequence of more specialized equations, including those of the conventional theory. Some familiar formulas of the conventional theory (including Fisher's fundamental theorem, the formula relating the rate of change of a metric character to selection pressure, and the definitions of broad and narrow heritability) are shown to be special cases of simpler and more general formulas. It is shown that the “genotypic value” of a trait, together with its heritability, may depend strongly on genotype-environment correlations.A generalization of Fisher's fundamental theorem shows that the rate of evolution of a trait depends on the skewness of its fitness distribution. An equation relating the second derivative of the mean fitness to the skewness is derived.Finally, the formalism is applied in a preliminary way to a recent theory of genetic variation (Layzer,1978a), according to which the genetic variability of a trait is selected along with the trait itself. It is shown that there is positive feedback between the two kinds of selection.     

Effect of phenylhydrazine on the multiple hemoglobins of Rana catesbeiana tadpoles

Tadpoles of Rana catesbeiana immersed in or injected with phenylhydrazine show a preferential loss from the circulation of the electrophoretically fastest hemoglobin (Hb) and retention of the electrophoretically slower Hb's on polyacrylamide disk-gels.The electrophoretically slowest Hb has previously been found to be synthesized in the kidney while the other Hb's are synthesized in the liver. We postulate that the differential response of the tadpole hemoglobins to the action of phenylhydrazine results from the different Hb's being found in different red blood cell lines that originate in different erythropoietic tissues.     

Theoretical effects of radioligand diffusional gradients and microscopic neuroreceptor distribution inin vivo kinetic studies

A simplified one-dimensional model system was used to test the possibility that physically realistic parameters would lead to the prediction of microscopic heterogeneity of radioligand distribution in the brain and that microscopic heterogeneity of radioligand and neuroreceptor distribution could influence the macroscopically observedin vivo kinetics. The model was represented mathematically by a partial differential equation which is similar to the heat diffusion equation, but with special boundary conditions. The equation was solved analytically under the condition of negligible receptor occupancy by inversion of the Laplace transform and in the more general case of arbitrary receptor occupancy by cubic spline approximation. In simulations with physically reasonable values for rate constants and parameters, we find that significant radioligand gradients can occur. Thus, the level of radioligand in the immediate vicinity of the receptor may be substantially different from the average level in a macroscopically measured region of interest. In order to analyze the simulated data, we derived a rigorous steady-state solution, including both a statement of necessary and sufficient conditions for the validity of the steady-state approximation as well as a demonstration of the proper technique for assessing the consistency of the derived parameter with the requirements of the approximation. The radioligand heterogeneity leads to significant errors in the parameters estimated in the steady-state kinetic analysis. In particular, the pseudo first-order rate constant for radioligand-neuroreceptor association, which is often used as a measure of the total amount of neuroreceptor, is underestimated. The first-order rate constant for radioligand-neuroreceptor dissociation is also underestimated. These effects can partially account for the experimentally-observed discrepancy betweenin vivo andin vitro estimates of these kinetic parameters.     

Plant growth and respiration re-visited: maintenance respiration defined - it is an emergent property of, not a separate process within, the system - and why the respiration : photosynthesis ratio is conservative

Background and Aims

Plant growth and respiration still has unresolved issues, examined here using a model. The aims of this work are to compare the model''s predictions with McCree''s observation-based respiration equation which led to the ‘growth respiration/maintenance respiration paradigm’ (GMRP) – this is required to give the model credibility; to clarify the nature of maintenance respiration (MR) using a model which does not represent MR explicitly; and to examine algebraic and numerical predictions for the respiration:photosynthesis ratio.

Methods

A two-state variable growth model is constructed, with structure and substrate, applicable on plant to ecosystem scales. Four processes are represented: photosynthesis, growth with growth respiration (GR), senescence giving a flux towards litter, and a recycling of some of this flux. There are four significant parameters: growth efficiency, rate constants for substrate utilization and structure senescence, and fraction of structure returned to the substrate pool.

Key Results

The model can simulate McCree''s data on respiration, providing an alternative interpretation to the GMRP. The model''s parameters are related to parameters used in this paradigm. MR is defined and calculated in terms of the model''s parameters in two ways: first during exponential growth at zero growth rate; and secondly at equilibrium. The approaches concur. The equilibrium respiration:photosynthesis ratio has the value of 0·4, depending only on growth efficiency and recycling fraction.

Conclusions

McCree''s equation is an approximation that the model can describe; it is mistaken to interpret his second coefficient as a maintenance requirement. An MR rate is defined and extracted algebraically from the model. MR as a specific process is not required and may be replaced with an approach from which an MR rate emerges. The model suggests that the respiration:photosynthesis ratio is conservative because it depends on two parameters only whose values are likely to be similar across ecosystems.     

The influence of the type 3 (sigmoid) functional response upon the stability of predator-prey difference models

Sigmoid functional responses are found to exert a stabilizing influence upon a discrete-generation predator-prey model in a way analogous to that found in continuous predator-prey models. The precise effect depends upon the degree to which a predator's feeding history influences its reproductive success. The time delay intrinsic in difference equation models imposes constraints not found in differential models, however, it is shown that in an otherwise unstable model the inclusion of a sigmoid functional response can result in local stability. With the addition of prey self-regulation the stabilizing influence of the functional response acts in concert with self-regulation, as it does in continuous models. These results show that the effect of the sigmoid response upon stability is not dependent upon the assumption of continuity, and reinforces the view that sigmoid responses could be an important factor stabilizing natural communities.     

Dynamic Model of Heat Inactivation Kinetics for Bacterial Adaptation

The Weibullian-log logistic (WeLL) inactivation model was modified to account for heat adaptation by introducing a logistic adaptation factor, which rendered its “rate parameter” a function of both temperature and heating rate. The resulting model is consistent with the observation that adaptation is primarily noticeable in slow heat processes in which the cells are exposed to sublethal temperatures for a sufficiently long time. Dynamic survival patterns generated with the proposed model were in general agreement with those of Escherichia coli and Listeria monocytogenes as reported in the literature. Although the modified model''s rate equation has a cumbersome appearance, especially for thermal processes having a variable heating rate, it can be solved numerically with commercial mathematical software. The dynamic model has five survival/adaptation parameters whose determination will require a large experimental database. However, with assumed or estimated parameter values, the model can simulate survival patterns of adapting pathogens in cooked foods that can be used in risk assessment and the establishment of safe preparation conditions.Combined with heat transfer data or models, microbial survival kinetics, especially of bacteria or spores, is extensively used to determine the safety of industrial heat preservation processes like canning, extant or planned. The same is true for milder heat processes such as milk and fruit pasteurization. However, survival models are also a valuable tool to assess the safety of prepared foods, especially those made of raw meats, poultry, and eggs, where surviving pathogens can be a public health issue.The heat resistance of a bacterium, or any other microorganism, is almost always determined from a set of its isothermal survival curves, recorded at several lethal temperatures. The kinetic models, which define the heat resistance parameters, may vary, but the calculation procedure itself is usually the same. First, the experimental isothermal survival data are fitted with what is known as the “primary model.” Once fitted, the temperature dependence of this primary model''s coefficients is described by what is known as the “secondary model.” When combined with a temperature profile expression, T(t), and incorporated into the inactivation rate equation, the result is a “tertiary model,” which enables its user to predict the organism''s survival curve under any static or dynamic (i.e., nonisothermal) conditions.The traditional log-linear (“first-order kinetic”) model is the best-known primary survival model, and it is still widely used in sterility calculations in the food, pharmaceutical, and other industries. Traditionally, it has been assumed that the D value calculated with this model has a log-linear temperature dependence or, alternatively, that the temperature effect on the exponential rate constant, k, the D value''s reciprocal, follows the Arrhenius equation. However, accumulating experimental evidence in recent years indicates that bacterial heat inactivation only rarely follows the first-order kinetics and that there is no reason that it should (3, 18, 29). Nonlinear survival curves can be described by a variety of mathematical models (6). Perhaps the most frequently used in recent years is the Weibullian model, of which the traditional log-linear model is a special case—see below.Regardless of the log-linearity issue, none of the above-mentioned models accounts for adaptation, the ability of certain bacterial cells to adjust their metabolism in response to stress in order to increase their survivability (2, 10, 26, 27, 28). A notable example is Escherichia coli. Its cells can produce “heat shock proteins,” which help them to survive mild heat treatments (1, 11). Other organisms, Salmonella enterica and Bacillus cereus among them, can also develop defensive mechanisms that help them to survive in an acidic environment (8, 9, 13). Whether adaptation allows the cells to avoid injury or to repair damage once it has occurred, or both, should not concern us here. (Injury and recovery, although related, are a separate issue, one which is amply discussed in the literature. Their quantitative aspects and mathematical modeling are discussed elsewhere [5].)The cells'' ability to augment their resistance is not unlimited, and it takes time for the cells to activate the protective system and synthesize its chemical elements (10, 12). Consequently, the effect of heat adaptation on an organism''s survival pattern becomes measurable only at or at slightly above what''s known as the “sublethal” temperature range. Under dynamic conditions, therefore, adaptation can be detected only when the heating rate is sufficiently low to allow the cells to respond metabolically to the heat stress prior to their destruction.Several investigators have reported and discussed the quantitative aspects of adaptation (25, 27, 28). When it occurs, adaptation is noticed as a gap between survival curves determined at low heating rates and those predicted by kinetic models whose parameters had been determined at high lethal temperatures (7, 8, 9, 27, 28). The question is how to modify the inactivation kinetic model so that it can properly account for adaptation at low heating rates while maintaining its predictive ability at high rates and clearly lethal temperatures. Stasiewicz et al. (25) have recently given a partial answer to this question. They started with the Weibullian inactivation model (see below) and assumed that its rate parameter''s temperature dependence follows a modified version of the Arrhenius equation. Using this model and experimental data for Salmonella bacteria, they showed that a “pathway-dependent model” is more reliable than a “state-dependent model.”The objectives of our work were to develop a variant of the Weibullian-log logistic (WeLL) inactivation model to account for dynamic adaptation and to demonstrate its applicability with reported adaptive survival patterns exhibited by Escherichia coli and Listeria monocytogenes, two organisms of food safety concern.     

Fides: Reliable trust-region optimization for parameter estimation of ordinary differential equation models

Ordinary differential equation (ODE) models are widely used to study biochemical reactions in cellular networks since they effectively describe the temporal evolution of these networks using mass action kinetics. The parameters of these models are rarely known a priori and must instead be estimated by calibration using experimental data. Optimization-based calibration of ODE models on is often challenging, even for low-dimensional problems. Multiple hypotheses have been advanced to explain why biochemical model calibration is challenging, including non-identifiability of model parameters, but there are few comprehensive studies that test these hypotheses, likely because tools for performing such studies are also lacking. Nonetheless, reliable model calibration is essential for uncertainty analysis, model comparison, and biological interpretation.We implemented an established trust-region method as a modular Python framework (fides) to enable systematic comparison of different approaches to ODE model calibration involving a variety of Hessian approximation schemes. We evaluated fides on a recently developed corpus of biologically realistic benchmark problems for which real experimental data are available. Unexpectedly, we observed high variability in optimizer performance among different implementations of the same mathematical instructions (algorithms). Analysis of possible sources of poor optimizer performance identified limitations in the widely used Gauss-Newton, BFGS and SR1 Hessian approximation schemes. We addressed these drawbacks with a novel hybrid Hessian approximation scheme that enhances optimizer performance and outperforms existing hybrid approaches. When applied to the corpus of test models, we found that fides was on average more reliable and efficient than existing methods using a variety of criteria. We expect fides to be broadly useful for ODE constrained optimization problems in biochemical models and to be a foundation for future methods development.     

Group mating among Norway rats II. The social dynamics of copulation: Competition,cooperation, and mate choice

The social interactions within groups of Norway rats (Rattus norvegicus) had a strong impact on the individual pattern of copulation which, in turn, affects sperm precedence and the probability of implantation in this species. Males alternated uninterrupted ejaculatory series, augmenting each others' copulatory investment. Females took turns mating after receiving an intromission, collectively potentiating the males' copulatory behaviour; increasing the number of oestrous females increased the number of intromissions and ejaculations achieved by each male but did not affect the amount of copulation experienced by each female. These turn-taking patterns within each sex provided the opportunity to change partners and permitted the emergence of different sex-typical patterns of copulation. Furthermore, the dominant male contributed more intromissions and tended to give each female more ejaculations than the subordinates did. Dominant males were also more likely to inhibit the subordinates' sperm transport. Females competed among themselves for the opportunity to mate with a male as he approached ejaculation and were likely to protect more of the dominant male's sperm transport than the subordinate male's.     

Chemical mutagens: Dosimetry,Haber's rule and linear systems

We present an overall scheme for chemical mutagen risk assessment, which leads naturally to consideration of Haber's rule, a classical concept of toxicology. A rationale is given for considering compartmental models based on mammalian anatomy and physiology as the most reasonable and practical conceptual framework for risk assessment. Haber's rule is extended to the area of chemical dosimetry, defined in terms of our compartmental models. Then it is proved that Haber's rule holds for any system of linear ordinary differential equations with constant coefficients which is physically realizable. Finally, we comment on non-linearities and on the Blum-Druckey model for time-to-occurrence of tumors.     

Species recognition in Darwin's finches (Geospiza, Gould). III. Male responses to playback of different song types,dialects and heterospecific songs

The songs of the six different species of Darwin's ground finches (Geospiza) on the Galápagos Islands are difficult to distinguish unambiguously because of high levels of intraspecific variation and interspecific similarity in some cases. We recorded the responses of males on five islands to playback of (a) the two main conspecific song types, A and B, (b) local conspecific and heterospecific song, and (c) local and foreign dialects. Males reacted equally strongly to different conspecific song types (A and B), but responded significantly more strongly to local conspecific song than to either heterospecific song or foreign dialect. These results are inconsistent with earlier suggestions that song types subdivide Geospiza populations and that Geospiza song lacks species-distinctness because of loss-of-contrast or character convergence. The apparent paradox of low song specificity and well-developed acoustic discrimination is discussed in the light of other data showing that close-range species recognition also depends on visual cues.     

The effects of intramolecular and intermolecular electrostatic repulsions on the stability and aggregation of NISTmAb revealed by HDX‐MS,DSC, and nanoDSF

The stability and aggregation of NIST monoclonal antibody (NISTmAb) were investigated by hydrogen/deuterium exchange mass spectrometry (HDX‐MS), differential scanning calorimetry (DSC), and nano‐differential scanning fluorimetry (nanoDSF). NISTmAb was prepared in eight formulations at four different pHs (pH 5, 6, 7, and 8) in the presence and absence of 150 mM NaCl and analyzed by the three methods. The HDX‐MS results showed that NISTmAb is more conformationally stable at a pH near its isoelectric point (pI) in the presence of NaCl than a pH far from its pI in the absence of NaCl. The stabilization effects were global and not localized. The midpoint temperature of protein thermal unfolding transition results also showed the C_H2 domain of the protein is more conformationally stable at a pH near its pI. On the other hand, the onset of aggregation temperature results showed that NISTmAb is less prone to aggregate at a pH far from its pI, particularly in the absence of NaCl. These seemingly contradicting results, higher conformational stability yet higher aggregation propensity near the pI than far away from the pI, can be explained by intramolecular and intermolecular electrostatic repulsion using Lumry‐Eyring model, which separates folding/unfolding equilibrium and aggregation event. The further a pH from the pI, the higher the net charge of the protein. The higher net charge leads to greater intramolecular and intermolecular electrostatic repulsions. The greater intramolecular electrostatic repulsion destabilizes the protein and the greater intermolecular electrostatic repulsion prevents aggregation of the protein molecules at pH far from the pI.     

DeepCME: A deep learning framework for computing solution statistics of the chemical master equation

Stochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov’s forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability distribution of the random state-vector representing the copy-numbers of all the reacting species. The size of this system is given by the number of states that are accessible by the chemical system, and for most examples of interest this number is either very large or infinite. Moreover, approximations that reduce the size of the system by retaining only a finite number of important chemical states (e.g. those with non-negligible probability) result in high-dimensional ODE systems, even when the number of reacting species is small. Consequently, accurate numerical solution of the CME is very challenging, despite the linear nature of the underlying ODEs. One often resorts to estimating the solutions via computationally intensive stochastic simulations. The goal of the present paper is to develop a novel deep-learning approach for computing solution statistics of high-dimensional CMEs by reformulating the stochastic dynamics using Kolmogorov’s backward equation. The proposed method leverages superior approximation properties of Deep Neural Networks (DNNs) to reliably estimate expectations under the CME solution for several user-defined functions of the state-vector. This method is algorithmically based on reinforcement learning and it only requires a moderate number of stochastic simulations (in comparison to typical simulation-based approaches) to train the “policy function”. This allows not just the numerical approximation of various expectations for the CME solution but also of its sensitivities with respect to all the reaction network parameters (e.g. rate constants). We provide four examples to illustrate our methodology and provide several directions for future research.     

Photometric assays of cell shrinkage-the resolution of a conflict

Two groups have induced the shrinkage of yeast cells and measured its influence on visible light transmittance. Interestingly, they observed opposite effects; Bryant, Latimer and Seiber found shrinkage to increase suspension transmittance, Bussey found it to decrease transmittance. Each group corroborated its findings with light scattering theory. Bryant et al. used their own theoretical method while Bussey used an equation of Koch. We now find that the opposite effects of shrinkage on suspension transmittance were probably caused by differences in yeast cell size and in the designs of the photometer optical systems. All observed effects are found to be predicted by our method in terms of particle size and photometer geometry. The cited agreement of Bussey's findings with Koch's equation is fortuitous since the experiments were outside the proper domain of that approximate equation—the yeast cells were too large as was the acceptance angle of the photometer.     

On Finding and Using Identifiable Parameter Combinations in Nonlinear Dynamic Systems Biology Models and COMBOS: A Novel Web Implementation

Parameter identifiability problems can plague biomodelers when they reach the quantification stage of development, even for relatively simple models. Structural identifiability (SI) is the primary question, usually understood as knowing which of P unknown biomodel parameters p ₁,…, p_i,…, p_P are-and which are not-quantifiable in principle from particular input-output (I-O) biodata. It is not widely appreciated that the same database also can provide quantitative information about the structurally unidentifiable (not quantifiable) subset, in the form of explicit algebraic relationships among unidentifiable p_i. Importantly, this is a first step toward finding what else is needed to quantify particular unidentifiable parameters of interest from new I–O experiments. We further develop, implement and exemplify novel algorithms that address and solve the SI problem for a practical class of ordinary differential equation (ODE) systems biology models, as a user-friendly and universally-accessible web application (app)–COMBOS. Users provide the structural ODE and output measurement models in one of two standard forms to a remote server via their web browser. COMBOS provides a list of uniquely and non-uniquely SI model parameters, and–importantly-the combinations of parameters not individually SI. If non-uniquely SI, it also provides the maximum number of different solutions, with important practical implications. The behind-the-scenes symbolic differential algebra algorithms are based on computing Gröbner bases of model attributes established after some algebraic transformations, using the computer-algebra system Maxima. COMBOS was developed for facile instructional and research use as well as modeling. We use it in the classroom to illustrate SI analysis; and have simplified complex models of tumor suppressor p53 and hormone regulation, based on explicit computation of parameter combinations. It’s illustrated and validated here for models of moderate complexity, with and without initial conditions. Built-in examples include unidentifiable 2 to 4-compartment and HIV dynamics models.     

Protein induction process and stochastic nature of cell commitment to proliferation and differentiation

A differential equation model is constructed to describe competition between trees with overlapping crowns. The model is based upon Thornley's mechanistic plant growth model which divides plant biomass into three components corresponding to storage material, degradable structural tissue and non-degradable structural tissue. The available incident radiation is partitioned among trees according to their sizes, with large trees intercepting more light than smaller neighbours. Analysis of the dynamic stability of the model reveals that suppression occurs over a wide range of parameter space. Typically, as canopy overlap increases and competition for light becomes intense, some trees are suppressed by their neighbours. The suppression-dominance phenomenon occurs even in stands of trees with identical parameter values. Model simulations are compared with data on the growth of Pinus radiata.