Transitional states in a chemostat culture of Candida utilis

Transient states of the chemostat Candida utilis 1668-3-37 culture were studied when its growth was limited by ethanol and an abrupt acidification of the medium from pH 5.0 to 2.2 was done or when the dilution rate was rapidly changed from D = 0.1 to 0.3 h-1 and back to 0.07 h-1. The pH shock was found to cause stronger oscillations in a number of parameters (the weight of dry biomass, the content of residual ethanol, the content of RNA in the cells) than a change in the dilution rate. In the latter case the population density changed more smoothly than the content of RNA did. DNA content remained at one and the same level in all of the experiments. All of the oscillations were observed only in the first generation after a shock; there upon, the culture remained for a long time (7 to 10 generations) in a very stable state typical of chemostat cultures. The oscillations induced by the unfavourable pH of the medium were compared with those caused by an abrupt change in the dilution rate. The pH shock brought about multiple damping oscillations of the parameters whereas a change in the dilution rate resulted, most often, in a merely one oscillation.     

An analysis of the growth of Gluconobacter oxydans in chemostat cultures

Gluconobacter oxydans was grown successively in glucose and nitrogen-limited chemostat cultures. Construction of mass balances of organisms growing at increasing dilution rates in glucose-limited cultures, at pH 5.5, revealed a major shift from extensive glucose metabolism via the pentose phosphate pathway to the direct pathway of glucose oxidation yielding gluconic acid. Thus, whereas carbon dioxide production from glucose accounted for 49.4% of the carbon input at a dilution rate (D)=0.05 h^-1, it accounted for only 1.3% at D=0.26 h^-1. This decline in pentose phosphate pathway activity resulted in decreasing molar growth yields on glucose. At dilution rates of 0.05 h^-1 and 0.26 h^-1 molar growth yields of 19.5 g/mol and 3.2 g/mol, respectively, were obtained. Increase of the steady state glucose concentration in nitrogen-limited chemostat cultures maintained at a constant dilution rate also resulted in a decreased flow of carbon through the pentose phosphate pathway. Above a threshold value of 15–20 mM glucose in the culture, pentose phosphate pathway activity almost completely inhibited. In G. oxydans the coupling between energy generation and growth was very inefficient; yield values obtained at various dilution rates varied between 0.8–3.4 g/cells synthesized per 0.5 mol of oxygen consumed.     

Population dynamics in river networks: analysis of steady states

Journal of Mathematical Biology - We study the population dynamics of an aquatic species in a river network. The habitat is viewed as a binary tree-like metric graph with the population density...     

An analysis for transient states with application to tumor shrinkage

The evaluation of therapies for chronic diseases is often based on the frequency and/or the duration of improvement. Treated separately, these endpoints may give contradictory impressions of the efficacy of the therapy. We propose a more unified method of summarizing improvement-related data--the probability of being in response, i.e., improved, as a function of time. Although improvement is not the only endpoint considered in most trials and this function will not always provide a clear answer to the question of which treatment has better improvement-related characteristics, it does combine the information on several endpoints usually considered separately into a single easily interpreted item. This function is estimated using the method of maximum likelihood on a distribution-free stochastic model of times to improvement and failure. Censored observations are taken into account. A detailed example using data from a cancer clinical trial is presented.     

Establishment of the steady state in glucose-limited chemostat cultures of Klebsiella pneumoniae

To investigate the relationship between growth rate and concentration of the nutrient that limits growth, 'Klebsiella aerogenes' NCTC 418 (K. pneumoniae) was grown in a glucose-limited chemostat. The actual time required to establish a steady-state glucose concentration exceeded that expected theoretically. Apparently, there is a long-term adaptation of the cells to nutrient limitation. As yet, it is not clear whether this has a phenotypic or genetic origin. In the final steady state, the dependence of the growth rate on glucose concentration could be mathematically described equally well by a hyperbolic and by a logarithmic function.     

A reaction-diffusion coupled system with multiple steady states

A nonlinear constitutive relation between the chemical affinity and the chemical reaction rate is given. It is demonstrated that the steady state reaction rate profiles exhibit spatial oscillations within the membrane or the reaction column and that there exists a multiplicity of such solutions as well. Both the features are seen to be immediate consequences of the nonlinearity, since neither appears in the treatment of facilitated diffusion by Katchalsky &; Oster (1969).     

Mechanics of a mosquito bite with applications to microneedle design

The mechanics of a fascicle insertion into the skin by a mosquito of the type aedes aegypti has been studied experimentally using high-speed video (HSV) imaging, and analytically using a mathematical model. The fascicle is a polymeric microneedle composed of a ductile material, chitin. It has been proposed that the mosquito applies a non-conservative follower force component in addition to the Euler compressive load in order to prevent buckling and penetrate the skin. In addition, the protective sheath surrounding the fascicle (labium) provides lateral support during insertion. The mechanics model presented approximates the fascicle as a slender column supported on an elastic foundation (labium) subjected to non-conservative (Beck) and conservative Euler loads simultaneously at the end. Results show that the lateral support of the fascicle provided by the labium is essential for successful penetration by increasing the critical buckling load by a factor of 5. The non-conservative follower force application increases the buckling load by an additional 20% and may or may not be necessary for successful penetration. Experimental results showing the importance of the labium have been cited to validate the model predictions, in addition to the video observations presented in this work. This understanding may be useful in designing painless needle insertion systems as opposed to miniaturized hypodermic needles.     

Estimation in the general incomplete block design with applications to the analysis of mating structures

Summary Models for analysing data for two commonly used sampling schemes involving inbreeding and environmental effects is considered. The general incomplete block model is utilized where mating types are equated to incomplete blocks. The models allow for varying degrees of inbreeding within mating types and between mating types. The analysis is such as to remove some of the confounding of inbreeding and environmental effects. The models are formulated in matrix notation utilizing pseudoinverses of matrices. Intrablock and interblock estimates of environmental effects or treatment effects are obtained with a suggestion as how one might combine the two estimates. An example is presented to demonstrate the computations.     

On the number of steady states in a multiple futile cycle

The multisite phosphorylation-dephosphorylation cycle is a motif repeatedly used in cell signaling. This motif itself can generate a variety of dynamic behaviors like bistability and ultrasensitivity without direct positive feedbacks. In this paper, we study the number of positive steady states of a general multisite phosphorylation–dephosphorylation cycle, and how the number of positive steady states varies by changing the biological parameters. We show analytically that (1) for some parameter ranges, there are at least n + 1 (if n is even) or n (if n is odd) steady states; (2) there never are more than 2n − 1 steady states (in particular, this implies that for n = 2, including single levels of MAPK cascades, there are at most three steady states); (3) for parameters near the standard Michaelis–Menten quasi-steady state conditions, there are at most n + 1 steady states; and (4) for parameters far from the standard Michaelis–Menten quasi-steady state conditions, there is at most one steady state.      

Identification of all steady states in large networks by logical analysis

The goal of generalized logical analysis is to model complex biological systems, especially so-called regulatory systems, such as genetic networks. This theory is mainly characterized by its capacity to find all the steady states of a given system and the functional positive and negative circuits, which generate multistationarity and a cycle in the state sequence graph, respectively. So far, this has been achieved by exhaustive enumeration, which severely limits the size of the systems that can be analysed. In this paper, we introduce a mathematical function, called image function, which allows the calculation of the value of the logical parameter associated with a logical variable depending on the state of the system. Thus the state table of the system is represented analytically. We then show how all steady states can be derived as solutions to a system of steady-state equations. Constraint programming, a recent method for solving constraint satisfaction problems, is applied for that purpose. To illustrate the potential of our approach, we present results from computer experiments carried out on very large randomly-generated systems (graphs) with hundreds, or even thousands, of interacting components, and show that these systems can be solved using moderate computing time. Moreover, we illustrate the approach through two published applications, one of which concerns the computation times of all steady states for a large genetic network.     

Periodicities of open linear systems with positive steady states

Systems of two, three, and four linear non-homogeneous differential equations are examined with a view toward determining whether they can possibly serve as mathematical models to describe periodicities in the concentrations of substances which enhance or inhibit each other's rate of production (or dissipation). The nature of the model demands that the solutions of the differential equations be non-negative at all times, i.e., that all the steady states be positive. Conditions for periodicity and for positive steady states are derived, and it is shown that these conditions are not always compatible with each other. In particular it is shown that certain three- and four-hormone models proposed to account for the periodicities observed in the menstrual cycle cannot satisfy the above conditions for any values of the parameters and hence are inadequate.     

Oscillations and multiple steady states in a cyclic gene model with repression

In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.This research was supported in part by the Air Force Office of Scientific Research under Contract #AFOSR-84-0376 and by the US Army Research Office under Contract #DAAG29-84-K-0082     

Adaptive torsion-angle quasi-statics: a general simulation method with applications to protein structure analysis and design

MOTIVATION: The cost of molecular quasi-statics or dynamics simulations increases with the size of the simulated systems, which is a problem when studying biological phenomena that involve large molecules over long time scales. To address this problem, one has often to either increase the processing power (which might be expensive), or make arbitrary simplifications to the system (which might bias the study). RESULTS: We introduce adaptive torsion-angle quasi-statics, a general simulation method able to rigorously and automatically predict the most mobile regions in a simulated system, under user-defined precision or time constraints. By predicting and simulating only these most important regions, the adaptive method provides the user with complete control on the balance between precision and computational cost, without requiring him or her to perform a priori, arbitrary simplifications. We build on our previous research on adaptive articulated-body simulation and show how, by taking advantage of the partial rigidification of a molecule, we are able to propose novel data structures and algorithms for adaptive update of molecular forces and energies. This results in a globally adaptive molecular quasi-statics simulation method. We demonstrate our approach on several examples and show how adaptive quasi-statics allows a user to interactively design, modify and study potentially complex protein structures.     

Principles of the light-limited chemostat: theory and ecological applications

Light is the energy source that drives nearly all ecosystems on planet Earth. Yet, light limitation is still poorly understood. In this paper, we present an overview of the principles of the light-limited chemostat. The theory for light-limited chemostats differs considerably from the standard theory for substrate-limited chemostats. In particular, photons cannot be mixed by vigorous stirring, so that phototrophic organisms experience the light-limited chemostat as a heterogeneous environment. Similar to substrate-limited chemostats, however, light-limited chemostats do reach a steady state. This allows the study of phototrophic microorganisms under well-controlled light conditions, at a constant specific growth rate, for a prolonged time. The theory of the light-limited chemostat is illustrated with several examples from laboratory experiments, and a variety of ecological applications are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Asymmetry of fluctuation with respect to time reversal in steady states of biological systems

The asymmetry of fluctuation with respect to lime reversal which is expected in an energy-consuming steady state is discussed with special attention to biological systems. The necessary condition for asymmetry of fluctuation of an observed quantity is given. To show the usefulness of the experimental analysis of asymmetry of fluctuation, some calculations are carried out on two simple examples of three-state reactions. In one of them, the two-point time correlation function of the observed quantity has an oscillatory component, while in the other the function is nearly exponential, but in both cases, the fluctuation has a pronounced asymmetry. A method to estimate the degree of asymmetry of fluctuation is proposed, and the application of the present method to investigation of the molecular events in biological systems such as muscle is discussed.     

An adaptive single-step FDR procedure with applications to DNA microarray analysis

The use of multiple hypothesis testing procedures has been receiving a lot of attention recently by statisticians in DNA microarray analysis. The traditional FWER controlling procedures are not very useful in this situation since the experiments are exploratory by nature and researchers are more interested in controlling the rate of false positives rather than controlling the probability of making a single erroneous decision. This has led to increased use of FDR (False Discovery Rate) controlling procedures. Genovese and Wasserman proposed a single-step FDR procedure that is an asymptotic approximation to the original Benjamini and Hochberg stepwise procedure. In this paper, we modify the Genovese-Wasserman procedure to force the FDR control closer to the level alpha in the independence setting. Assuming that the data comes from a mixture of two normals, we also propose to make this procedure adaptive by first estimating the parameters using the EM algorithm and then using these estimated parameters into the above modification of the Genovese-Wasserman procedure. We compare this procedure with the original Benjamini-Hochberg and the SAM thresholding procedures. The FDR control and other properties of this adaptive procedure are verified numerically.