Statistical estimation of parameters in a phylogenetic tree using a dynamic model of the substitutional process

Using a modified version of the substitutional process proposed by Neyman, we estimate the parameters of the phylogenetic tree made up of three species (or groups of species). The parameters estimated are the rate of substitution of amino acids along a protein and the ratio of the times of divergence of the species (or group of species). A method is given for determining the tree structure when it is not known. Both the maximum likelihood and Bayes methods are used in the estimation. The basic model of the substitutional process within the proteins is validated by showing that the estimates of the ratio of the times of divergence of three species computed from two different protein molecules (haemoglobin α and fibrinopeptides) are within one standard deviation of each other. Next we consider the construction of the correct phylogenetic tree made up of three or more taxonomic categories like phyla or class utilizing the structure of the various types of protein molecules of the species in the three categories. The generalization of the procedure for the construction of the entire phylogenetic tree is also indicated. The main advantage of this method of tree construction over the traditional method is that the latter method can use the information of only one type of protein (for example cytochrome c) while the method of this paper can use all the available data from the different molecules. We also discuss the recent controversy over the constancy of the molecular clock.     

Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings

The dengue virus is a vector-borne disease transmitted by mosquito Aedes aegypti and the incidence is strongly influenced by temperature and humidity which vary seasonally. To assess the effects of temperature on dengue transmission, mathematical models are developed based on the population dynamics theory. However, depending on the hypotheses of the modelling, different outcomes regarding to the risk of epidemics are obtained. We address this question comparing two simple models supplied with model's parameters estimated from temperature-controlled experiments, especially the entomological parameters regarded to the mosquito's life cycle in different temperatures. Once obtained the mortality and transition rates of different stages comprising the life cycle of mosquito and the oviposition rate, we compare the capacity of vector reproduction (the basic offspring number) and the risk of infection (basic reproduction number) provided by two models. The extended model, which is more realistic, showed that both mosquito population and dengue risk are situated at higher values than the simplified model, even that the basic offspring number is lower.     

Simulation-based estimation of stochastic process parameters in tumor growth

Models are generally developed at the micro level. Data are generally gathered at the macro level. Obtaining the macromodel which is the natural consequence of the underlying micro model is generally not feasible. SIMEST gives a means whereby the micromodel is used to generate, for a given assumed set of parameters, simulated sets of macro data. These data are compared with the actual clinical macro data. The parameters are then adjusted to obtain concordance with the clinical data. In this manner, simulation gives us a means of parameter estimation without the necessity of generating the macro model.     

Validation and estimation of parameters for a general probabilistic model of the PCR process.

Earlier work rigorously derived a general probabilistic model for the PCR process that includes as a special case the Velikanov-Kapral model where all nucleotide reaction rates are the same. In this model, the probability of binding of deoxy-nucleoside triphosphate (dNTP) molecules with template strands is derived from the microscopic chemical kinetics. A recursive solution for the probability function of binding of dNTPs is developed for a single cycle and is used to calculate expected yield for a multicycle PCR. The model is able to reproduce important features of the PCR amplification process quantitatively.With a set of favorable reaction conditions, the amplification of the target sequence is fast enough to rapidly outnumber all side products. Furthermore, the final yield of the target sequence in a multicycle PCR run always approaches an asymptotic limit that is less than one. The amplification process itself is highly sensitive to initial concentrations and the reaction rates of addition to the template strand of each type of dNTP in the solution. This paper extends the earlier Saha model with a physics based model of the dependence of the reaction rates on temperature, and estimates parameters in this new model by nonlinear regression. The calibrated model is validated using RT-PCR data.     

Conflicting biomedical assumptions for mathematical modeling: the case of cancer metastasis

Computational models in biomedicine rely on biological and clinical assumptions. The selection of these assumptions contributes substantially to modeling success or failure. Assumptions used by experts at the cutting edge of research, however, are rarely explicitly described in scientific publications. One can directly collect and assess some of these assumptions through interviews and surveys. Here we investigate diversity in expert views about a complex biological phenomenon, the process of cancer metastasis. We harvested individual viewpoints from 28 experts in clinical and molecular aspects of cancer metastasis and summarized them computationally. While experts predominantly agreed on the definition of individual steps involved in metastasis, no two expert scenarios for metastasis were identical. We computed the probability that any two experts would disagree on k or fewer metastatic stages and found that any two randomly selected experts are likely to disagree about several assumptions. Considering the probability that two or more of these experts review an article or a proposal about metastatic cascades, the probability that they will disagree with elements of a proposed model approaches 1. This diversity of conceptions has clear consequences for advance and deadlock in the field. We suggest that strong, incompatible views are common in biomedicine but largely invisible to biomedical experts themselves. We built a formal Markov model of metastasis to encapsulate expert convergence and divergence regarding the entire sequence of metastatic stages. This model revealed stages of greatest disagreement, including the points at which cancer enters and leaves the bloodstream. The model provides a formal probabilistic hypothesis against which researchers can evaluate data on the process of metastasis. This would enable subsequent improvement of the model through Bayesian probabilistic update. Practically, we propose that model assumptions and hunches be harvested systematically and made available for modelers and scientists.     

Learning with incomplete information and the mathematical structure behind it

We investigate the problem of learning with incomplete information as exemplified by learning with delayed reinforcement. We study a two phase learning scenario in which a phase of Hebbian associative learning based on momentary internal representations is supplemented by an ‘unlearning’ phase depending on a graded reinforcement signal. The reinforcement signal quantifies the success-rate globally for a number of learning steps in phase one, and ‘unlearning’ is indiscriminate with respect to associations learnt in that phase. Learning according to this model is studied via simulations and analytically within a student–teacher scenario for both single layer networks and, for a committee machine. Success and speed of learning depend on the ratio λ of the learning rates used for the associative Hebbian learning phase and for the unlearning-correction in response to the reinforcement signal, respectively. Asymptotically perfect generalization is possible only, if this ratio exceeds a critical value λ _c , in which case the generalization error exhibits a power law decay with the number of examples seen by the student, with an exponent that depends in a non-universal manner on the parameter λ. We find these features to be robust against a wide spectrum of modifications of microscopic modelling details. Two illustrative applications—one of a robot learning to navigate a field containing obstacles, and the problem of identifying a specific component in a collection of stimuli—are also provided.     

A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

Some mathematical aspects of the medical diagnostic process. I: a general mathematical model

Making a medical diagnosis consists of correlating knownpatterns of disease with the various classes of clinical data elicited from the history, physical examination, and batteries of tests relative to the diagnostic dynamics symbolized by atree branching into the various possible diagnostic decisions. In this paper a relational mathematical model of the reasoning aspects of the conventional medical diagnostic process is suggested as a way of extracting a general, formal concept of medical diagnosis. Computer implementation of the model is discussed briefly.     

Some mathematical aspects of the diagnostic process. II: A mathematical model of electrocardiographic diagnosis

In a previous paper (Bartholomay, 1971), a general mathematical model of the medical diagnostic process was described. The present paper amounts to a relization of that process in terms of conventional 12-lead electrocardiographic diagnosis as enunciated by Dr. Harold D. Levine (1966) in the course of a collaborative study by Dr. Levine and the present author at the Peter Bent Brigham Hospital of the Harvard Medical School between 1963 and 1966. The main details of the cognitive component of that model are described in detail here. The model has been programmed onto a computer system consisting of an analog-digital converter and general purpose digital computer and amounts to a simulation of Dr. Levine’s electrocardiographic analysis procedure.     

Parameter estimation for a mathematical model of the cell cycle in frog eggs.

Parameter values for a kinetic model of the nuclear replication-division cycle in frog eggs are estimated by fitting solutions of the kinetic equations (nonlinear ordinary differential equations) to a suite of experimental observations. A set of optimal parameter values is found by minimizing an objective function defined as the orthogonal distance between the data and the model. The differential equations are solved by LSODAR and the objective function is minimized by ODRPACK. The optimal parameter values are close to the guesstimates of the modelers who first studied this problem. These tools are sufficiently general to attack more complicated problems, where guesstimation is impractical or unreliable.     

Simultaneous estimation of all the parameters of a stepwise mutation model.

Minisatellite and microsatellite are short tandemly repetitive sequences dispersed in eukaryotic genomes, many of which are highly polymorphic due to copy number variation of the repeats. Because mutation changes copy numbers of the repeat sequences in a generalized stepwise fashion, stepwise mutation models are widely used for studying the dynamics of these loci. We propose a minimum chi-square (MCS) method for simultaneous estimation of all the parameters in a stepwise mutation model and the ancestral allelic type of a sample. The MCS estimator requires knowing the mean number of alleles of a certain size in a sample, which can be estimated using Monte Carlo samples generated by a coalescent algorithm. The method is applied to samples of seven (CA)n repeat loci from eight human populations and one chimpanzee population. The estimated values of parameters suggest that there is a general tendency for microsatellite alleles to expand in size, because (1) each mutation has a slight tendency to cause size increase and (2) the mean size increase is larger than the mean size decrease for a mutation. Our estimates also suggest that most of these CA-repeat loci evolve according to multistep mutation models rather than single-step mutation models. We also introduced several quantities for measuring the quality of the estimation of ancestral allelic type, and it appears that the majority of the estimated ancestral allelic types are reasonably accurate. Implications of our analysis and potential extensions of the method are discussed.SINCE the discovery that a large number of loci with tandemly repeated sequences in human and many eukaryote species are highly polymorphic because of copy number variation of the repeats in different individuals (Jeffreys 1985; Litt and Luty 1989; Weber and May 1989), allele size data from such loci are rapidly becoming the dominant source of genetic markers for genome mapping, forensic testing, and population studies. Loci with repeat sequences longer than 5 bp are generally referred to as minisatellite or variable number tandem repeat loci, and those with repeat sequences between 2 to 5 bp are referred to as microsatellite or short tandem repeat loci (Tautz 1993). Because mutations change the copy number of such loci in a stepwise fashion, rapid accumulation of population samples from minisatellite and microsatellite loci has resurrected the interest of the stepwise mutation model (SMM), which was popular in the 1970s.     

Evaluation of dynamic behavior forecasting parameters in the process of transition rule induction of unidimensional cellular automata

The simulation of the dynamics of a cellular systems based on cellular automata (CA) can be computationally expensive. This is particularly true when such simulation is part of a procedure of rule induction to find suitable transition rules for the CA. Several efforts have been described in the literature to make this problem more treatable. This work presents a study about the efficiency of dynamic behavior forecasting parameters (DBFPs) used for the induction of transition rules of CA for a specific problem: the classification by the majority rule. A total of 8 DBFPs were analyzed for the 31 best-performing rules found in the literature. Some of these DBFPs were highly correlated each other, meaning they yield the same information. Also, most rules presented values of the DBFPs very close each other. An evolutionary algorithm, based on gene expression programming, was developed for finding transition rules according a given preestablished behavior. The simulation of the dynamic behavior of the CA is not used to evaluate candidate transition rules. Instead, the average values for the DBFPs were used as reference. Experiments were done using the DBFPs separately and together. In both cases, the best induced transition rules were not acceptable solutions for the desired behavior of the CA. We conclude that, although the DBFPs represent interesting aspects of the dynamic behavior of CAs, the transition rule induction process still requires the simulation of the dynamics and cannot rely only on the DBFPs.     

Dry-heat destruction of lipopolysaccharide: mathematical approach to process evaluation.

A mathematical model was developed to estimate the number of logarithmic cycles (LDec) of lipopolysaccharide concentration destroyed by a dry-heat sterilization process. The LDec values calculated from the mathematical model agreed well with those obtained from the destruction of lipopolysaccharide by a dry-heat treatment. A discussion of how the mathematical model may be used to evaluate a dry-heat sterilization cycle is presented. This mathematical model and the dry-heat destruction curves indicated existence of a maximum LDec value at each temperature. The implications of this finding are discussed.     

Dry-heat destruction of lipopolysaccharide: mathematical approach to process evaluation.

A mathematical model was developed to estimate the number of logarithmic cycles (LDec) of lipopolysaccharide concentration destroyed by a dry-heat sterilization process. The LDec values calculated from the mathematical model agreed well with those obtained from the destruction of lipopolysaccharide by a dry-heat treatment. A discussion of how the mathematical model may be used to evaluate a dry-heat sterilization cycle is presented. This mathematical model and the dry-heat destruction curves indicated existence of a maximum LDec value at each temperature. The implications of this finding are discussed.