A multi-algorithm, multi-timescale method for cell simulation

MOTIVATION: Many important problems in cell biology require the dense nonlinear interactions between functional modules to be considered. The importance of computer simulation in understanding cellular processes is now widely accepted, and a variety of simulation algorithms useful for studying certain subsystems have been designed. Many of these are already widely used, and a large number of models constructed on these existing formalisms are available. A significant computational challenge is how we can integrate such sub-cellular models running on different types of algorithms to construct higher order models. RESULTS: A modular, object-oriented simulation meta-algorithm based on a discrete-event scheduler and Hermite polynomial interpolation has been developed and implemented. It is shown that this new method can efficiently handle many components driven by different algorithms and different timescales. The utility of this simulation framework is demonstrated further with a 'composite' heat-shock response model that combines the Gillespie-Gibson stochastic algorithm and deterministic differential equations. Dramatic improvements in performance were obtained without significant accuracy drawbacks. A multi-timescale demonstration of coupled harmonic oscillators is also shown.     

Unstructured models for suspension cultures of Taxus media cells in a bioreactor under substrate-sufficient conditions

For a better understanding of the simulation, optimization, and process control in cell cultures, good kinetic models are necessary for large scale plant cell culture. In this paper, the systematic kinetics of taxol production by Taxus media cell suspension cultures in a stirred 15-L bioreactor under substrate-sufficient conditions and the absence of inducer intervention were studied. A kinetic model of cell growth was established by logistic equation, and kinetic unstructured models of substrate consumption, product synthesis and rheological behavior were constituted, which incorporated energy spilling. These models were verified by comparing the simulation results with those obtained experimentally. These results showed that energy spilling was a key factor that must be considered in constructing unstructured kinetic models of Taxus media cell suspension cultures in a stirred bioreactor under substrate-sufficient conditions. Besides, an optimized operation measure of decreasing energy spilling was proposed. An increase of 17.64% in cell biomass and 14.88% in taxol concentration were obtained when the strategy of limiting added carbon several times was experimentally implemented in a 15-L bioreactor. Results demonstrated that these established models should be helpful in the process prediction and operation optimization to guide the production and amplification of Taxus media cell suspension cultures in a bioreactor.     

Structured modelling of animal cells

Recent advances in computer technology have promoted the design and use of detailed, computer-based models for biological systems. For many non-biological systems, the complexity of such simulations may be considered inappropriate and unwieldy, but in biological systems, and more specifically in animal cell culture, this level of complexity simply mimics what is only beginning to be understood about metabolic processs. With this in mind, we contend that complex, structured models are vital tools in the investigation of fundamental biological processes. An example of such a simulation, which describes the commercial production of therapeutic proteins by animal cell cultures, is considered.     

New direct dynamic models of protein interactions coupled to photosynthetic electron transport reactions

This review covers the methods of computer simulation of protein interactions taking part in photosynthetic electron transport reactions. A direct multiparticle simulation method that simulates reactions describing interactions of ensembles of molecules in the heterogeneous interior of a cell is developed. In the models, protein molecules move according to the laws of Brownian dynamics, mutually orient themselves in the electrical field, and form complexes in the 3D scene. The method allows us to visualize the processes of molecule interactions and to calculate the rate constants for protein complex formation reactions in the solution and in the photosynthetic membrane. Three-dimensional multiparticle computer models for simulating the complex formation kinetics for plastocyanin with photosystem I and cytochrome bf complex, and ferredoxin with photosystem I and ferredoxin:NADP⁺-reductase are considered. Effects of ionic strength are featured for wild type and mutant proteins. The computer multiparticle models describe nonmonotonic dependences of complex formation rates on the ionic strength as the result of long-range electrostatic interactions.     

A Monte Carlo simulation of plasmid replication during the bacterial division cycle

Plasmids have cell cycle replication patterns that need to be considered in models of their replication dynamics. To compare current theories for control of plasmid replication with experimental data for timing of plasmid replication with the cell cycle, a Monte Carlo simulation of plasmid replication and partition was developed. High-copy plasmid replication was simulated by incorporating equations previously developed from the known molecular biology of ColE1-type plasmids into the cell-cycle simulation. Two types of molecular mechanisms for low-copy plasmid replication were tested: accumulation of an initiator protein in proportion to cell mass and binding of the plasmid origin to the cell membrane. The low-copy plasmids were partitioned actively, with a specific mechanism to mediate the transfer from mother to daughter cells, whereas the high-copy plasmids were partitioned passively with cell mass.The simulation results and experimental data demonstrate cell-cycle-specific replication for the low-copy F plasmid and cell-cycle-independent replication for the high-copy pBR322, ColBM, and R6K plasmids. The simulation results indicate that synchronous replication at multiple plasmid origins is critical for the cell-cycle-specific pattern observed in rapidly growing cells. Variability in the synchrony of initiation of multiple plasmid origins give rise to a cell-cycle-independent pattern and is offered as a plausible explanation for the controversy surrounding the replication pattern of the low-copy plasmids. A comparison of experimental data and simulation results for the low-copy F plasmid at several growth rates indicates that either initiation mechanism would be sufficient to explain the timing of replication with the cell cycle. The simulation results also demonstrate that, although cell-cycle-specific and cell-cycle independent replication patterns give rise to very different gene-expression patterns during short induction periods in age-selected populations, long-term expression of genes encoded on low-copy and high-copy plasmids in exponentially growing cells have nearly the same patterns. These results may be important for the future use of low-copy plasmids as expression vectors and validate the use of simpler models for high-copy plasmids that do not consider cell-cycle phenomena. (c) 1996 John Wiley & Sons, Inc.     

Simulation of the diffusion-controlled reaction between superoxide and superoxide dismutase. II. Detailed models

A two-stage Brownian dynamics simulation method is used to study the diffusion-influenced bimolecular reaction between superoxide and superoxide dismutase (SOD). The crystal structure of the dimeric enzyme is used in constructing detailed topographical and electrostatic models. Several electrostatic models are considered. In the most realistic, the excluded volume of the protein, which is impermeable to penetration by mobile ions, is assigned a dielectric constant of 2 and the surrounding “solvent” is assigned a value of 78. A finite difference method is used to solve the linearized Poisson–Boltzmann equation. For native SOD, the simulations reproduce the pronounced salt dependence of the rate constant observed experimentally. This salt dependence is attributed to electrostatic interactions between enzyme and substrate that are inherently attractive and amplified by the low dielectric constant of the protein interior. The simulation method is also applied to a modified enzyme, acylated SOD.     

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.     

Application of theoretical models in ecology.

Certain general prey-predator models are compared with a particular ecosystem simulation. The adequacy of the models is considered in relation to the structure of the equations; to the representation of ecological features by terms in these equations; and to the need to incorporate resource equations in such models.     

Cluster growth by mitosis

A stochastic mitosis growth model is introduced. The colony enlarge by random mitosis of a mother cell which is replaced by two daughter cells. Unlike growth models based on a grid, the present model is isotropic. Using simulation of colonies of 10,000 cells, it is found that the density is constant and the boundary is fractal.     

Estimation of non-linear growth models by linearization: a simulation study using a Gompertz function

A method based on Taylor series expansion for estimation of location parameters and variance components of non-linear mixed effects models was considered. An attractive property of the method is the opportunity for an easily implemented algorithm. Estimation of non-linear mixed effects models can be done by common methods for linear mixed effects models, and thus existing programs can be used after small modifications. The applicability of this algorithm in animal breeding was studied with simulation using a Gompertz function growth model in pigs. Two growth data sets were analyzed: a full set containing observations from the entire growing period, and a truncated time trajectory set containing animals slaughtered prematurely, which is common in pig breeding. The results from the 50 simulation replicates with full data set indicate that the linearization approach was capable of estimating the original parameters satisfactorily. However, estimation of the parameters related to adult weight becomes unstable in the case of a truncated data set.     

Asymmetric regression models with limited responses with an application to antibody response to vaccine

We develop regression models for limited and censored data based on the mixture between the log‐power‐normal and Bernoulli‐type distributions. A likelihood‐based approach is implemented for parameter estimation and a small‐scale simulation study is conducted to evaluate parameter recovery, with emphasis on bias estimation. The main conclusion is that the approach is very much satisfactory for moderate and large sample sizes. A real data example, the safety and immunogenecity study of measles vaccine in Haiti, is presented to illustrate how different models can be used to fit this type of data. As shown, the asymmetric models considered seem to present the best fit for the data set under study, revealing significance of the explanatory variable sex, which is not found significant with the log‐normal model.     

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model.     

Analysis of oscillations in yeast continuous cultures by a new simplified model

The autonomous oscillations in yeast continuous cultures are investigated analytically and related to the behaviour of the single cell by means of a suitable modified version of Monod’s classical chemostat model. Two main cell phases or states are considered to account for the experimentally observed changes occurring in the cell growth course: the budded phase and the unbudded one. Thus, a sort of two compartment structure is given to the total biomass. The model so far obtained allows one to analyse the local properties of the predicted steady states under various assumptions, both on the yield coefficients and the specific growth rates. Necessary conditions for the local instability are derived and the existence of stable limit cycles is shown by computer simulation. With respect to the qualitative changes in the metabolic parameters, this analysis agrees with the results obtained by simulation of complex structured and segregated models. However, the oscillation period is too long compared with the experimental one and this fact may be mainly due to the strong simplifying assumptions on the dynamic evolution of the transfer rates between the two compartments. The model’s usefulness seems until now restricted to the identification of the relationships between the cell cycle regulation and the oscillation triggering.     

INDISIM-YEAST: an individual-based simulator on a website for experimenting and investigating diverse dynamics of yeast populations in liquid media

Simulation modelling can be used to capture and mimic real-world microbial systems that, unlike the real-world, can then be experimented upon as a new kind of experimental milieu. Individual-based models, in which individuals interact dynamically with each other as structural elements in the model world, exemplify this view of simulation modelling. These models are more difficult to analyze, understand and communicate than traditional analytical models. It is good practice to provide executable versions that perform simulation results. INDISIM-YEAST, developed to deal with yeast populations in liquid media, models the evolution of a set of yeasts by setting up rules of behavior for each individual cell according to its own biological regulations and characteristics. The aim of this work is to develop and present a website from which INDISIM-YEAST is accessible, and how to carry out yeast simulations to further the skills associated with the use of this individual-based simulator. A good and useful way to analyze this yeast simulator is to experiment and explore the manner in which it reacts to changes in parameter values, initial conditions or assumptions. The application results in a very versatile program that could be used in controlled simulation experiments via the Internet.     

The MYpop toolbox: Putting yeast stress responses in cellular context on single cell and population scales

Systems biology holds the promise to integrate multiple sources of information in order to build ever more complete models of cellular function. To do this, the field must overcome two significant challenges. First, the current strategy to model average cells must be replaced with population based models accounting for cell‐to‐cell variability. Second, models must be integrated with each other and with basic cellular function. This requires a core model of cellular physiology as well as a multiscale simulation platform to support large‐scale simulation of culture or tissues from single cells. Here, we present such a simulation platform with a core model of yeast physiology as scaffold to integrate and simulate SBML models. The software automates this integration helping users simulate their model of choice in context of the cell division cycle. We benchmark model merging, simulation and analysis by integrating a minimal model of osmotic stress into the core model and analyzing it. We characterize the effect of single cell differences on the dynamics of osmoadaptation, estimating when normal cell growth is resumed and obtaining an explanation for experimentally observed glycerol dynamics based on population dynamics. Hence, the platform can be used to reconcile single cell and population level data.     

Diffusion approximation of the stochastic process of microtubule assembly

Microtubules are protein polymers that guide intracellular motility. Stochastic switching of a microtubule between states of elongation, shortening, and pause is described in detail by the dynamic instability (DI) model. Recently we have described the dynamics of microtubules phenomenologically as generalized diffusion of their ends. Genesis of the diffusion dynamics and accuracy of diffusion model are studied in this work. It is shown that wandering of the end of a microtubule undergoing DI asymptotically approaches the Wiener diffusion process. Accuracy of the diffusion approximation is evaluated by comparing its predictions with results of simulation of DI. Stationary distributions of microtubule length and lifetime that are predicted by both models differ qualitatively between two cell types considered. However, predictions of the diffusion model are in each case practically identical to predictions of the DI model being also consistent with experimental data. The peculiar stochastic process of microtubule assembly thus converges at cell scale to a kind of widespread-in-nature diffusion process. This result is considered an example of qualitative change in dynamical properties in transition from the molecular to cellular level of biological organization. Additionally, it suggests employment of diffusion process theory in studying functions of microtubules in the cell.     

Effect of joint laxity on polyethylene wear in total knee replacement

Experimental simulator studies are frequently performed to evaluate wear behavior in total knee replacement. It is vital that the simulation conditions match the physiological situation as closely as possible. To date, few experimental wear studies have examined the effects of joint laxity on wear and joint kinematics and the absence of the anterior cruciate ligament has not been sufficiently taken into account in simulator wear studies.The aim of this study was to investigate different ligament and soft tissue models with respect to wear and kinematics.A virtual soft tissue control system was used to simulate different motion restraints in a force-controlled knee wear simulator.The application of more realistic and sophisticated ligament models that considered the absence of anterior cruciate ligament lead to a significant increase in polyethylene wear (p=0.02) and joint kinematics (p<0.01). We recommend the use of more complex ligament models to appropriately simulate the function of the human knee joint and to evaluate the wear behavior of total knee replacements. A feasible simulation model is presented.     

Stochastic models for cell motion and taxis

Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in response to a stimulus, termed taxis, has previously been modeled at a phenomenological level using the Keller-Segel diffusion equation. The Keller-Segel model cannot distinguish certain modes of taxis, and this motivates the introduction of a richer class of models which is nevertheless still amenable to statistical analysis. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation via maximum likelihood for a range of applicable models. One particular experimental situation, involving the effect of an electric field on cell behavior, is considered in detail. In this case, an Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model.     

Segregated mathematical model for growth of anchorage-dependent MDCK cells in microcarrier culture

To describe the growth behavior of anchorage-dependent mammalian cells in microcarrier systems, various approaches comprising deterministic and stochastic single cell models as well as automaton-based models have been presented in the past. The growth restriction of these often contact-inhibited cells by spatial effects is described at levels with different complexity but for the most part not taking into account their metabolic background. Compared to suspension cell lines these cells have a comparatively long lag phase required for attachment and start of proliferation on the microcarrier. After an initial phase of exponential growth only a moderate specific growth rate is achieved due to restrictions in space available for cell growth, limiting medium components, and accumulation of growth inhibitors. Here, a basic deterministic unstructured segregated cell model for growth of Madin Darby Canine Kidney (MDCK) cells used in influenza vaccine production is described. Four classes of cells are considered: cells on microcarriers, cells in suspension, dead cells, and lysed cells. Based on experimental data, cell attachment and detachment is taken explicitly into account. The model allows simulation of the overall growth behavior in microcarrier culture, including the lag phase. In addition, it describes the time course of uptake and release of key metabolites and the identification of parameters relevant for the design and optimization of vaccine manufacturing processes.