Analyzing blood cell concentration as a stochastic process.
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Animals↗
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Kinetics of single cells: observation and modeling of a stochastic process.
The successive generation times for single cells of Escherichia coli K-12 were measured as described by A. Elfwing, Y. LeMarc, J. Baranyi, and A. Ballagi (Appl. Environ. Microbiol. 70:675-678, 2004), and the histograms they generated were used as empirical distributions to simulate growth of the population as the result of the multiplication of its single cells. This way, a stochastic birth model in which the underlying distributions were measured experimentally was simulated. To validate the model, analogous bacterial growth curves were generated by the use of different inoculum levels. The agreement with the simulation was very good, proving that the growth of the population can be predicted accurately if the distribution of the first few division times for the single cells within that population is known. Two questions were investigated by the simulation. (i) To what extent can we say that the distribution of the detection time, i.e., the time by which a single-cell-generated subpopulation reaches a detectable level, can be identified with that of the lag time of the original single cell? (ii) For low inocula, how does the inoculum size affect the lag time of the population?
Cell Division↗
Modelling predation as a capped rate stochastic process, with applications to fish recruitment.
Many mathematical models use functions the value of which cannot exceed some physically or biologically imposed maximum value. A model can be described as 'capped-rate' when the rate of change of a variable cannot exceed a maximum value. This presents no problem when the models are deterministic but, in many applications, results from deterministic models are at best misleading. The need to account for stochasticity, both demographic and environmental, in models is therefore important but, as this paper shows, incorporating stochasticity into capped-rate models is not trivial. A method using queueing theory is presented, which allows randomness and spatial heterogeneity to be incorporated rigorously into capped rate models. The method is applied to the feeding and growth of fish larvae.
Animals↗
Sensitivity of membranes to their environment. Role of stochastic processes.
Ionic flow through biomembranes often exhibits a sensitivity to the environment, which is difficult to explain by classical theory, that usually assumes that the free energy available to change the membrane permeability results from the environmental change acting directly on the permeability control mechanism. This implies, for example, that a change delta V in the trans-membrane potential can produce a maximum free energy change, delta V X q, on a gate (control mechanism) carrying a charge q. The analysis presented here shows that when stochastic fluctuations are considered, under suitable conditions (gate cycle times rapid compared with the field relaxation time within a channel), the change in free energy is limited, not by the magnitude of the stimulus, but by the electrochemical potential difference across the membrane, which may be very much greater. Conformational channel gates probably relax more slowly than the field within the channel; this would preclude appreciable direct amplification of the stimulus. It is shown, however, that the effect of impermeable cations such as Ca++ is to restore the amplification of the stimulus through its interaction with the electric field. The analysis predicts that the effect of Ca++ should be primarily to affect the number of channels that are open, while only slightly affecting the conductivity of an open channel.
Animals↗
Nonparametric estimation of the mean function of a stochastic process with missing observations.
In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right-censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.
Acquired Immunodeficiency Syndrome↗
Models of the interaction of mortality and the evolution of risk factor distribution: a general stochastic process formulation.
Generally analyses of longitudinal studies of chronic disease risks do not directly model the change with time of risk factor values and the interactions of those changes with risk levels. Failure to account for such process characteristics can lead to incorrect inferences about the specific effects of risk factors on mortality, the inability to accurately forecast the future risk of the cohort, and inaccurate statements about the effects of specific risk factor interventions on mortality. We present a model which does describe such a process and show how it can be estimated from longitudinal studies. We also illustrate the effects of certain risk factor process features on the evolution of disease risk data from males in the Framingham, Massachusetts study.
Actuarial Analysis↗
Changes in self-renewal potential of human leukemic cells (K562): a bidirectional stochastic process.
Daughter cells arising from a single cell division in the leukemic cell line K562 have equivalent self-renewal potential with respect to their ability to form clones in semisolid medium. However, individual cells isolated from these clones in sequence have vastly different abilities in their self-renewal potentials. Thus, cells originating from a clone with any particular self-renewal potential exhibit the full range of self-renewal potentials--from highly renewing to none renewing, cells. These results show that self-renewal potential in the K562 cell line is a random, reversible and partially noninherited characteristic. It is suggested that the stochastic variability of the intraclonal self-renewal potential of K562 progeny cells either reflects the initial expression of a differentiation program or the expression of the predeterministic portion of the normal myelopoietic differentiation pathway.
Cell Differentiation↗
Stochastic processes influence stationary-phase decisions in Bacillus subtilis.
It has recently been proposed that phenotypic variation in clonal populations of bacterial species results from intracellular "noise," i.e., random fluctuations in levels of cellular molecules, which would be predicted to be insensitive to selective pressure. To test this notion, we propagated five populations of Bacillus subtilis for 5,000 generations with selection for one phenotype: the decision to sporulate. In support of the noise hypothesis, we report that none of the populations responded to selection by improving their efficiency of sporulation, indicating that intracellular noise is independent of heritable genotype.
Bacillus subtilis↗
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.
Algorithms↗
Application of stochastic processes to participant recruitment in clinical trials.
The allocation of sufficient time for participant recruitment is one of the fundamental aspects in planning a clinical trial. This paper illustrates how a Poisson process can be used to determine an optimal period of time for participant recruitment by simulating Poisson arrival into a clinical trial. The simulation study provides the means to generate of an empirical probability density function for the recruitment time based on time-dependent changes in the accrual rate. From this empirical distribution, a clinical trial recruitment period can be planned to provide a high level of confidence (e.g., 90% probability) of enrolling the sample size within the planned amount of time given the simulation assumptions.
Clinical Trials as Topic↗
Strategy and model building in the fourth dimension: a null model for genotype x age interaction as a Gaussian stationary stochastic process.
BACKGROUND: Using univariate and multivariate variance components linkage analysis methods, we studied possible genotype x age interaction in cardiovascular phenotypes related to the aging process from the Framingham Heart Study. RESULTS: We found evidence for genotype x age interaction for fasting glucose and systolic blood pressure. CONCLUSIONS: There is polygenic genotype x age interaction for fasting glucose and systolic blood pressure and quantitative trait locus x age interaction for a linkage signal for systolic blood pressure phenotypes located on chromosome 17 at 67 cM.
Adult↗
Stochastic processes defining sensitivity and variability of internally calibrated quantitative NASBA-based viral load assays.
For quantitative assessment of virus particles in patient plasma samples various assays are commercially available. Typical performance characteristics for such assays are sensitivity, precision and the range of linearity. In order to assess these properties it is common practice to divide the range of inputs into subranges in order to apply different statistical models to evaluate these properties separately. We developed a general statistical model for internally calibrated amplification based viral load assays that combines these statistical properties in one powerful analysis. Based on the model an unambiguous definition of the lower limit of the linear range can be given. The proposed method of analysis was illustrated by a successful application to data generated by the NucliSens EasyQ HIV-1 assay.
HIV-1↗
A spatiotemporal stochastic process model for species spread.
We use a spatiotemporal Markov process to model the spread of an ecological population through its environment over time. Available habitat is divided into sites, and a parametric function of spatial variables is used to model the probability that one site is colonized from another. This allows us both to make predictions about the future spread of a population, and to determine which are the important factors governing colonizations. The model evolves in discrete time, allowing the population distribution to change seasonally in accordance with breeding patterns. Discrete time formulations are natural for ecological populations, but are problematic due to difficulties of fitting and predicting over irregular time intervals. The model described here can accommodate years of missing data and can therefore fit and predict at irregular intervals. Two methods of approximating the likelihood are described and applied to ornithological survey data for the woodlark, Lullula arborea, from Thetford Forest in the U.K.
Animals↗
Generalized approach for modeling minimally invasive surgery as a stochastic process using a discrete Markov model.
Minimally invasive surgery (MIS) involves a multidimensional series of tasks requiring a synthesis between visual information and the kinematics and dynamics of the surgical tools. Analysis of these sources of information is a key step in defining objective criteria for characterizing surgical performance. The Blue DRAGON is a new system for acquiring the kinematics and the dynamics of two endoscopic tools synchronized with the endoscopic view of the surgical scene. Modeling the process of MIS using a finite state model [Markov model (MM)] reveals the internal structure of the surgical task and is utilized as one of the key steps in objectively assessing surgical performance. The experimental protocol includes tying an intracorporeal knot in a MIS setup performed on an animal model (pig) by 30 surgeons at different levels of training including expert surgeons. An objective learning curve was defined based on measuring quantitative statistical distance (similarity) between MM of experts and MM of residents at different levels of training. The objective learning curve was similar to that of the subjective performance analysis. The MM proved to be a powerful and compact mathematical model for decomposing a complex task such as laparoscopic suturing. Systems like surgical robots or virtual reality simulators in which the kinematics and the dynamics of the surgical tool are inherently measured may benefit from incorporation of the proposed methodology.
Computer Simulation↗
A Bayesian compound stochastic process for modeling nonstationary and nonhomogeneous sequence evolution.
Variations of nucleotidic composition affect phylogenetic inference conducted under stationary models of evolution. In particular, they may cause unrelated taxa sharing similar base composition to be grouped together in the resulting phylogeny. To address this problem, we developed a nonstationary and nonhomogeneous model accounting for compositional biases. Unlike previous nonstationary models, which are branchwise, that is, assume that base composition only changes at the nodes of the tree, in our model, the process of compositional drift is totally uncoupled from the speciation events. In addition, the total number of events of compositional drift distributed across the tree is directly inferred from the data. We implemented the method in a Bayesian framework, relying on Markov Chain Monte Carlo algorithms, and applied it to several nucleotidic data sets. In most cases, the stationarity assumption was rejected in favor of our nonstationary model. In addition, we show that our method is able to resolve a well-known artifact. By Bayes factor evaluation, we compared our model with 2 previously developed nonstationary models. We show that the coupling between speciations and compositional shifts inherent to branchwise models may lead to an overparameterization, resulting in a lesser fit. In some cases, this leads to incorrect conclusions, concerning the nature of the compositional biases. In contrast, our compound model more flexibly adapts its effective number of parameters to the data sets under investigation. Altogether, our results show that accounting for nonstationary sequence evolution may require more elaborate and more flexible models than those currently used.
Animals↗
General statistics of stochastic process of gene expression in eukaryotic cells.
Thousands of genes are expressed at such very low levels (< or =1 copy per cell) that global gene expression analysis of rarer transcripts remains problematic. Ambiguity in identification of rarer transcripts creates considerable uncertainty in fundamental questions such as the total number of genes expressed in an organism and the biological significance of rarer transcripts. Knowing the distribution of the true number of genes expressed at each level and the corresponding gene expression level probability function (GELPF) could help resolve these uncertainties. We found that all observed large-scale gene expression data sets in yeast, mouse, and human cells follow a Pareto-like distribution model skewed by many low-abundance transcripts. A novel stochastic model of the gene expression process predicts the universality of the GELPF both across different cell types within a multicellular organism and across different organisms. This model allows us to predict the frequency distribution of all gene expression levels within a single cell and to estimate the number of expressed genes in a single cell and in a population of cells. A random "basal" transcription mechanism for protein-coding genes in all or almost all eukaryotic cell types is predicted. This fundamental mechanism might enhance the expression of rarely expressed genes and, thus, provide a basic level of phenotypic diversity, adaptability, and random monoallelic expression in cell populations.
Animals↗
An elastically tethered viscous load imposes a regular gait on the motion of myosin-V. Simulation of the effect of transient force relaxation on a stochastic process.
Myosin-V is a processive molecular motor that moves membrane vesicles along actin tracks. In the simple model for motor and cargo motion investigated here, an elastic connection between motor and cargo transiently absorbs the abrupt mechanical transitions of the motor, and allows smooth relaxation of the cargo to a new position. We use a stochastic description to model motor stepping, with kinetics that depends on the instantaneous force exerted on the motor through the elastic connection. Tether relaxation is modelled as a continuous process, in which the rate is determined by the viscous drag of the cargo and the stiffness profile of the connection. Quantitative combined stochastic-continuous simulation of the dynamics of this system shows that bulky loads can impose a highly regular gait on the motor. If the characteristics of the elastic connection are similar to those of the myosin-II coiled-coil domain, the myosin-V motor, tether and cargo form a true escapement, in which the motor only escapes from its current position after one or more force thresholds have been crossed. Multiple thresholds limit the variation in tether length to values below that of the total step size.
Actins↗
Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models.
A distinctive signature of living systems is Darwinian evolution, that is, a propensity to generate as well as self-select individual diversity. To capture this essential feature of life while describing the dynamics of populations, mathematical models must be rooted in the microscopic, stochastic description of discrete individuals characterized by one or several adaptive traits and interacting with each other. The simplest models assume asexual reproduction and haploid genetics: an offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take a mutation step to new trait values; selection follows from ecological interactions among individuals. Here we present a rigorous construction of the microscopic population process that captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by the trait values of each individual, and interactions between individuals. A by-product of this formal construction is a general algorithm for efficient numerical simulation of the individual-level model. Once the microscopic process is in place, we derive different macroscopic models of adaptive evolution. These models differ in the renormalization they assume, i.e. in the limits taken, in specific orders, on population size, mutation rate, mutation step, while rescaling time accordingly. The macroscopic models also differ in their mathematical nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. These models include extensions of Kimura's equation (and of its approximation for small mutation effects) to frequency- and density-dependent selection. A novel class of macroscopic models obtains when assuming that individual birth and death occur on a short timescale compared with the timescale of typical population growth. On a timescale of very rare mutations, we establish rigorously the models of "trait substitution sequences" and their approximation known as the "canonical equation of adaptive dynamics". We extend these models to account for mutation bias and random drift between multiple evolutionary attractors. The renormalization approach used in this study also opens promising avenues to study and predict patterns of life-history allometries, thereby bridging individual physiology, genetic variation, and ecological interactions in a common evolutionary framework.
Algorithms↗