State vector models of the cell cycle. III: Continuous time cell cycle models

In this paper the elements of the matrix of the Hahn cell-cycle model are identified with the infinitesimal transition probabilities of a Markov process, and as a limiting process a differential equation analogue is derived. The probability density function of the discrete time model is derived and used to obtain the density function for transit times of the continuous time model. It is shown that the mean transit time remains constant and that the variances of the discrete and continuous time models are the same to the order of the time increment. Finally, it is shown how to derive the Takahashi model from the continuous time Hahn model.     

A Stochastic Regression Model for General Trend Analysis of Longitudinal Continuous Data

A predictive continuous time model is developed for continuous panel data to assess the effect of time‐varying covariates on the general direction of the movement of a continuous response that fluctuates over time. This is accomplished by reparameterizing the infinitesimal mean of an Ornstein–Uhlenbeck processes in terms of its equilibrium mean and a drift parameter, which assesses the rate that the process reverts to its equilibrium mean. The equilibrium mean is modeled as a linear predictor of covariates. This model can be viewed as a continuous time first‐order autoregressive regression model with time‐varying lag effects of covariates and the response, which is more appropriate for unequally spaced panel data than its discrete time analog. Both maximum likelihood and quasi‐likelihood approaches are considered for estimating the model parameters and their performances are compared through simulation studies. The simpler quasi‐likelihood approach is suggested because it yields an estimator that is of high efficiency relative to the maximum likelihood estimator and it yields a variance estimator that is robust to the diffusion assumption of the model. To illustrate the proposed model, an application to diastolic blood pressure data from a follow‐up study on cardiovascular diseases is presented. Missing observations are handled naturally with this model.     

Approximation of continuous fermentation by semicontinuous cultures

Semicontinuous fermentations, in which a fraction of a culture is replaced with fresh media at regular intervals, have been previously used as a means of approximating continuous growth. In most cases deviations from continuous operation were erroneously estimated using Fencl's model, which is only valid when the specific growth rate is independent of the substrate concentration. An approach to modeling Semicontinuous growth that incorporates the same kinetics followed in batch and continuous growth was developed and tested for Monod's expression for the specific growth rate. A dimensionless form of the model was used to simulate Semicontinuous fermentations for comparison to continuous growth. Differences between Semicontinuous and continuous growth were found to depend on three dimensionless variables: feed concentration, replacement rate, and time between replacements. For given values of the dimensionless feed concentration and time between replacements, a range of dimensionless replacement rates can be determined over which semi-continuous cultures are approximately continuous.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

Some interrelationships among the regression coefficient estimates arising in a class of models appropriate to response-time data.

Interrelationships among three response-time models which incorporate covariate information are explored. The most general of these models is the logistic-exponential in which the log odds of the probability of responding in a fixed interval is assumed to be a linear function of the covariates; this model includes a parameter W for the width of discrete time intervals in which responses occur. As W leads to O this model is equivalent to a continuous time exponential model in which the log hazard is linear in the covariates. As W leads to infininity it is equivalent to a continuous time exponential model in which the hazard itself is a linear function of the covariates. This second model was fitted to the data used in an earlier publication describing the logistic exponential model, and very close agreement of the estimates of the regression coefficients is demonstrated.     

On the Lacker-Peskin model for muscular contraction

We study Lacker and Peskin's version of A.F. Huxley's model of muscular contraction. The model, which presumes a continuous (and independent) distribution of sites of attachment for a myosin cross-bridge, is presumed to apply to uniformly excited skeletal muscles under conditions of maximal overlap of fibres. We show that given velocity as a function of time there exists a unique force as a function of time, and that the velocity-force relation is continuous and one-to-one. We also suggest minor modifications of the model.     

A simple mathematical model of gradual Darwinian evolution: emergence of a Gaussian trait distribution in adaptation along a fitness gradient

We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.     

A continuous time version and a generalization of a Markov-recapture model for trapping experiments

Wileyto et al. [E.P. Wileyto, W.J. Ewens, M.A. Mullen, Markov-recapture population estimates: a tool for improving interpretation of trapping experiments, Ecology 75 (1994) 1109] propose a four-state discrete time Markov process, which describes the structure of a marking-capture experiment as a method of population estimation. They propose this method primarily for estimation of closed insect populations. Their method provides a mark-recapture estimate from a single trap observation by allowing subjects to mark themselves. The estimate of the unknown population size is based on the assumption of a closed population and a simple Markov model in which the rates of marking, capture, and recapture are assumed to be equal. Using the one step transition probability matrix of their model, we illustrate how to go from an embedded discrete time Markov process to a continuous time Markov process assuming exponentially distributed holding times. We also compute the transition probabilities after time t for the continuous time case and compare the limiting behavior of the continuous and discrete time processes. Finally, we generalize their model by relaxing the assumption of equal per capita rates for marking, capture, and recapture. Other questions about how their results change when using a continuous time Markov process are examined.     

Mixtures of varying coefficient models for longitudinal data with discrete or continuous nonignorable dropout

The analysis of longitudinal repeated measures data is frequently complicated by missing data due to informative dropout. We describe a mixture model for joint distribution for longitudinal repeated measures, where the dropout distribution may be continuous and the dependence between response and dropout is semiparametric. Specifically, we assume that responses follow a varying coefficient random effects model conditional on dropout time, where the regression coefficients depend on dropout time through unspecified nonparametric functions that are estimated using step functions when dropout time is discrete (e.g., for panel data) and using smoothing splines when dropout time is continuous. Inference under the proposed semiparametric model is hence more robust than the parametric conditional linear model. The unconditional distribution of the repeated measures is a mixture over the dropout distribution. We show that estimation in the semiparametric varying coefficient mixture model can proceed by fitting a parametric mixed effects model and can be carried out on standard software platforms such as SAS. The model is used to analyze data from a recent AIDS clinical trial and its performance is evaluated using simulations.     

Capacity optimization and scheduling of a multiproduct manufacturing facility for biotech products

A general mathematical framework has been proposed in this work for scheduling of a multiproduct and multipurpose facility involving manufacturing of biotech products. The specific problem involves several batch operations occurring in multiple units involving fixed processing time, unlimited storage policy, transition times, shared units, and deterministic and fixed data in the given time horizon. The different batch operations are modeled using state‐task network representation. Two different mathematical formulations are proposed based on discrete‐ and continuous‐time representations leading to a mixed‐integer linear programming model which is solved using General Algebraic Modeling System software. A case study based on a real facility is presented to illustrate the potential and applicability of the proposed models. The continuous‐time model required less number of events and has a smaller problem size compared to the discrete‐time model. © 2014 American Institute of Chemical Engineers Biotechnol. Prog., 30:1221–1230, 2014     

Axial dispersion in plate-pulsed columns

The residence time distribution (RTD) of the phase under continuous flow and dispersed flow was measured in a plate-pulsed column under countercurrent liquid-liquid flow, and modelled using (i) the axial dispersion model and (ii) the noninteger ideal stage cascade model. The axial mixing in the continuous phase was found to be significantly higher than in the dispersed phase. The model parameters were related to the operating conditions and system variables.     

Modeling the Residence Time Distribution of Integrated Continuous Bioprocesses

In response to the biopharmaceutical industry advancing from traditional batch operation to continuous operation, the Food and Drug Administration (FDA) has published a draft for continuous integrated biomanufacturing. This draft outlines the most important rules for establishing continuous integration. One of these rules is a thorough understanding of mass flows in the process. A computer simulation framework is developed for modeling the residence time distribution (RTD) of integrated continuous downstream processes based on a unit‐by‐unit modeling approach in which unit operations are simulated one‐by‐one across the entire processing time, and then combined into an integrated RTD model. The framework allows for easy addition or replacement of new unit operations, as well as quick adjustment of process parameters during evaluation of the RTD model. With this RTD model, the start‐up phase to reach steady state can be accelerated, the effects of process disturbances at any stage of the process can be calculated, and virtual tracking of a section of the inlet material throughout the process is possible. A hypothetical biomanufacturing process for an antibody was chosen for showcasing the RTD modeling approach.     

Path integration of head direction: updating a packet of neural activity at the correct speed using neuronal time constants

A key question in understanding the neural basis of path integration is how individual, spatially responsive, neurons may self-organize into networks that can, through learning, integrate velocity signals to update a continuous representation of location within an environment. It is of vital importance that this internal representation of position is updated at the correct speed, and in real time, to accurately reflect the motion of the animal. In this article, we present a biologically plausible model of velocity path integration of head direction that can solve this problem using neuronal time constants to effect natural time delays, over which associations can be learned through associative Hebbian learning rules. The model comprises a linked continuous attractor network and competitive network. In simulation, we show that the same model is able to learn two different speeds of rotation when implemented with two different values for the time constant, and without the need to alter any other model parameters. The proposed model could be extended to path integration of place in the environment, and path integration of spatial view.     

Effect of mixing on the washout and steady-state performance of continuous cultures

The effects of mixing on the critical mean holding time for washout and the steady state performance of growth processes in continuous flow reactors are investigated. Macromixing, micromixing, and cell recycle arc considered. The tanks-in-series model composed of N completely mixed flow reactors, the dispersion model, the plug flow model, and a combined model composed of a plug flow reactor and a continuous stirred tank flow reactor connected in series arc used to represent the macro-mixing or residence time distribution. The extreme cases of micromixing, namely, complete segregation and maximum mixedness, as well as intermediate states of micromixing are investigated to determine their effects on washout and the occurence of multiple steady states. A technique for predicting the maximum mixedness washout condition from a knowledge of the residence time distribution is presented and used to determine the washout condition for the dispersion model under maximum mixedness conditions.     

Variability in a single compartment system: A note on S. R. Bernard's model

In this note we examine a continuous time version of a compartmental model introduced in a discrete time setting by S. R. Bernard. The model allows for more than one particle to leave the system at any time. This introduces additional randomness into the system, over the pure death system and this is reflected in the variance function.     

Stochastic continuous time neurite branching models with tree and segment dependent rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(γ,t) for finding a tree γ at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number μ(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.     

Time required for gene frequency change in a deterministic model of gene-culture coevolution, with special reference to the lactose absorption problem

The time required for gene frequency change under natural selection in a deterministic model of gene-culture coevolution is investigated. A discrete generations model is formulated, and its continuous time approximation is derived. In passing to the continuous time limit, it is assumed that the frequency of the culturally transmitted trait does not change under oblique (between generations) transmission. The system of ordinary differential equations thus obtained are solved, and the dependence on the parameters of horizontal (within generations) transmission and natural selection is examined. The time required is found to be substantially longer when the determination of a phenotypic difference subject to natural selection is partly cultural rather than completely genetic. The predictions are relevant to the possibility of the coevolution of lactose absorbers and milk drinkers in some human populations. Alternative hypotheses are briefly discussed in the light of the theoretical results.     

How parasitism affects critical patch-size in a host-parasitoid model: application to the forest tent caterpillar

Habitat structure has broad impacts on many biological systems. In particular, habitat fragmentation can increase the probability of species extinction and on the other hand it can lead to population outbreaks in response to a decline in natural enemies. An extreme consequence of fragmentation is the isolation of small regions of suitable habitat surrounded by a large region of hostile matrix. This scenario can be interpreted as a critical patch-size problem, well studied in a continuous time framework, but relatively new to discrete time models. In this paper we present an integrodifference host-parasitoid model, discrete in time and continuous in space, to study how the critical habitat-size necessary for parasitoid survival changes in response to parasitoid life history traits, such as emergence time. We show that early emerging parasitoids may be able to persist in smaller habitats than late emerging species. The model predicts that these early emerging parasitoids lead to more severe host outbreaks. We hypothesise that promoting efficient late emerging parasitoids may be key in reducing outbreak severity, an approach requiring large continuous regions of suitable habitat. We parameterise the model for the host species of the forest tent caterpillar Malacosoma disstria Hbn., a pest insect for which fragmented landscape increases the severity of outbreaks. This host is known to have several parasitoids, due to paucity of data and as a first step in the modelling we consider a single generic parasitoid. The model findings are related to observations of the forest tent caterpillar offering insight into this host-parasitoid response to habitat structure.     

Truly continuous low pH viral inactivation for biopharmaceutical process integration

Continuous virus inactivation (VI) has received little attention in the efforts to realize fully continuous biomanufacturing in the future. Implementation of continuous VI must assure a specific minimum incubation time, typically 60 min. To guarantee the minimum incubation time, we implemented a packed bed continuous viral inactivation reactor (CVIR) with narrow residence time distribution (RTD) for low pH incubation. We show that the RTD does not broaden significantly over a wide range of linear flow velocities—which highlights the flexibility and robustness of the design. Prolonged exposure to acidic pH has no impact on bed stability, assuring constant RTD throughout long term operation. The suitability of the packed bed CVIR for low pH inactivation is shown with two industry-standard model viruses, that is xenotropic murine leukemia virus and pseudorabies virus. Controls at neutral pH showed no system-induced VI. At low pH, significant VI is observed, even after only 15 min. Based on the low pH inactivation kinetics, the continuous process is equivalent to traditional batch operation. This study establishes a concept for continuous low pH inactivation and, together with previous reports, highlights the versatility of the packed bed reactor for continuous VI, regardless of the inactivation method.     

Hydrolysis of liquefied corn starch in a membrane reactor

The objective of this study was to develop a continuous hydrolysis process for the enzymatic saccharification of liquefied corn starch using a membrane reactor. A residence time distribution study confirmed that the membrane reactor could be modeled as a simple continuous stirred tank reactor (CSTR). Kinetic studies indicated that the continuous reactor operated in the first-order region with respect to substrate concentration at substrate concentrations greater than 200 g/L. At a residence time of 1 h and an enzyme concentration of 1 g/L, the maximum reaction velocity (V(m)) was 3.86 g glucose/L min and the apparent Michaelis constant (K(m) (')) was 562 g/L. The K(m) (') value for the continuous reactor was 2-7 times greater than that obtained in a batch reactor.Kinetic data were fit to a model based on the Michaelis-Menten rate expression and the design equation for a CSTR. Application of the model at low reactor space times was successful. At space times of 6 min or less, the model predicted the reactor's performance reasonably well. Additional work on the detection and quantitation of reversion products formed by glucoamylase is required. Isolation, detection, and quantitation of reversion products by HPLC was difficult. Detailed analysis on the formation of these reversion products could lead to better reactor designs in the future.