Time scale and dimension analysis of a budding yeast cell cycle model

The progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle. By calculating the eigenvalues of the Jacobian of kinetic differential equations we have determined the stability of the cell cycle trajectories of the Chen model. Based on the sign of the real part of the eigenvalues, the cell cycle can be divided into excitation and relaxation periods. During an excitation period, the cell cycle control system leaves a formerly stable steady state and, accordingly, excitation periods can be associated with irreversible cell cycle transitions like START, entry into mitosis and exit from mitosis. During relaxation periods, the control system asymptotically approaches the new steady state. We also show that the dynamical dimension of the Chen’s model fluctuates by increasing during excitation periods followed by decrease during relaxation periods. In each relaxation period the dynamical dimension of the model drops to one, indicating a period where kinetic processes are in steady state and all concentration changes are driven by the increase of cytoplasmic growth.We apply two numerical methods, which have not been used to analyze biological control systems. These methods are more sensitive than the bifurcation analysis used before because they identify those transitions between steady states that are not controlled by a bifurcation parameter (e.g. cell mass). Therefore by applying these tools for a cell cycle control model, we provide a deeper understanding of the dynamical transitions in the underlying molecular network

The Treatment of Uncertainties in Reactive Pollution Dispersion Models at Urban Scales

The ability to predict NO2 concentrations ([NO¬2]) within urban street networks is important for the evaluation of strategies to reduce exposure to NO2. However, models aiming to make such predictions involve the coupling of several complex processes: traffic emissions under different levels of congestion; dispersion via turbulent mixing; chemical processes of relevance at the street-scale. Parameterisations of these processes are challenging to quantify with precision. Predictions are therefore subject to uncertainties which should be taken into account when using models within decision making. This paper presents an analysis of mean [NO¬2] predictions from such a complex modelling system applied to a street canyon within the city of York, UK including the treatment of model uncertainties and their causes. The model system consists of a micro-scale traffic simulation and emissions model, a Reynolds Averaged turbulent flow model coupled to a reactive Lagrangian particle dispersion model. The analysis focuses on the sensitivity of predicted in-street increments of [NO¬2] at different locations in the street to uncertainties in the model inputs. These include physical characteristics such as background wind direction, temperature and background ozone concentrations; traffic parameters such as overall demand and primary NO2 fraction; as well as model parameterisations such as roughness lengths, turbulent time- and length-scales and chemical reaction rate coefficients. Predicted [NO¬2] is shown to be relatively robust with respect to model parameterisations, although there are significant sensitivities to the activation energy for the reaction NO+O3 as well as the canyon wall roughness length. Under off-peak traffic conditions, demand is the key traffic parameter. Under peak conditions where the network saturates, road-side [NO¬2] is relatively insensitive to changes in demand and more sensitive to the primary NO2 fraction. The most important physical parameter was found to be the background wind direction. The study highlights the key parameters required for reliable [NO¬2] estimations suggesting that accurate reference measurements for wind direction should be a critical part of air quality assessments for in-street locations. It also highlights the importance of street scale chemical processes in forming road-side [NO¬2], particularly for regions of high NOx emissions such as close to traffic queues

Understanding dynamics using sensitivity analysis: caveat and solution

<p>Abstract</p> <p>Background</p> <p>Parametric sensitivity analysis (PSA) has become one of the most commonly used tools in computational systems biology, in which the sensitivity coefficients are used to study the parametric dependence of biological models. As many of these models describe dynamical behaviour of biological systems, the PSA has subsequently been used to elucidate important cellular processes that regulate this dynamics. However, in this paper, we show that the PSA coefficients are not suitable in inferring the mechanisms by which dynamical behaviour arises and in fact it can even lead to incorrect conclusions.</p> <p>Results</p> <p>A careful interpretation of parametric perturbations used in the PSA is presented here to explain the issue of using this analysis in inferring dynamics. In short, the PSA coefficients quantify the integrated change in the system behaviour due to persistent parametric perturbations, and thus the dynamical information of when a parameter perturbation matters is lost. To get around this issue, we present a new sensitivity analysis based on impulse perturbations on system parameters, which is named impulse parametric sensitivity analysis (iPSA). The inability of PSA and the efficacy of iPSA in revealing mechanistic information of a dynamical system are illustrated using two examples involving switch activation.</p> <p>Conclusions</p> <p>The interpretation of the PSA coefficients of dynamical systems should take into account the persistent nature of parametric perturbations involved in the derivation of this analysis. The application of PSA to identify the controlling mechanism of dynamical behaviour can be misleading. By using impulse perturbations, introduced at different times, the iPSA provides the necessary information to understand how dynamics is achieved, i.e. which parameters are essential and when they become important.</p

Quantifying uncertainty, variability and likelihood for ordinary differential equation models

<p>Abstract</p> <p>Background</p> <p>In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.</p> <p>Results</p> <p>The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.</p> <p>Conclusions</p> <p>While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.</p

Parallel chemistry acceleration algorithm with ISAT table-size control in the application of gaseous detonation

In order to improve the computational efficiency of a parallel ISAT (in situ adaptive tabulation)-based chemistry acceleration algorithm in the computations of transient, chemically reacting flows, a control strategy is proposed to maintain the sizes of the data tables in the ISAT computations. The table-size control strategy is then combined with a parallel algorithm to simulate two-dimensional gaseous detonation wave propagation. In the computation of 2H2 + O2 detonation, two sets of tests are conducted to identify the size control strategy. In the first set, the maximum total table size (Mtot) summed over all sub-zones is fixed, while the maximum size of the table on each sub-zone (Msin) is varied. In the second set, a fixed Msin is used for all the tables on the sub-zones while Mtot is varied. A maximum speedup ratio of 4.29 is found in the former tests, while 5.52 is found in the latter. Two parameters, σf and p, are proposed to analyze the load balance and synchronization among table operations in the parallel ISAT computations in the above tests. It is found that both load balance and synchronization have clear influences on the speedup ratio. A parameter pM is defined, and a strategy to choose the optimal maximum table sizes (both Mtot and Msin) based on pM is proposed and is verified to be universal in the computations of both 2H2 + O2 detonation and C2H4 + 3O2 detonation. Finally, the parallel acceleration algorithm enhanced with table-size control is shown to be highly accurate for the detonations in both fuels

Investigation of the Effect of Correlated Uncertain Rate Parameters on a Model of Hydrogen Combustion Using a Generalized HDMR method

The High Dimensional Model Representation (HDMR) method has been applied in several previous studies to obtain global sensitivity indices of uncorrelated model parameters in combustion systems. However, the rate parameters of combustion models are intrinsically correlated and therefore uncertainty analysis methods are needed that can handle such parameters. A generalized HDMR method is presented here, which uses the Rosenblatt transformation on a correlated model parameter sample to obtain a sample of independent parameters. The method provides a full set of both correlated and marginal sensitivity indices. Ignition delay times predicted by an optimizedhydrogen air combustion model in stoichiometric mixtures near the three explosion limits are investigated with this new global sensitivity analysis tool. The sensitivity indices which account for all the correlated effects of the rate parameters are shown to dominate uncertainties in the model output. However these correlated indices mask the individual influence of parameters. The final marginal uncorrelated sensitivity indices for individual parameters better indicate the change of importance of homogeneous gas phase and species wall-loss reactions as the pressure is increased from above the first explosion limit to above the third limit. However, these uncorrelated indices are small and whilst they provide insights into the dominant chemical and physical processes of the model over the range of conditions studied, the correlations between parameters have a very significant effect. The implications of this result on model tuning will be discussed

Investigation of the effect of correlated uncertain rate parameters via the calculation of global and local sensitivity indices

Applications of global uncertainty methods for models with correlated parameters are essential to investigate chemical kinetics models. A global sensitivity analysis method is presented that is able to handle correlated parameter sets. It is based on the coupling of the Rosenblatt transformation with an optimized Random Sampling High Dimensional Model Representation method. The accuracy of the computational method was tested on a series of examples where the analytical solution was available. The capabilities of the method were also investigated by exploring the effect of the uncertainty of rate parameters of a syngas–air combustion mechanism on the calculated ignition delay times. Most of the parameters have large correlated sensitivity indices and the correlation between the parameters has a high influence on the results. It was demonstrated that the values of the calculated total correlated and final marginal sensitivity indices are independent of the order of the decorrelation steps. The final marginal sensitivity indices are meaningful for the investigation of the chemical significance of the reaction steps. The parameters belonging to five elementary reactions only, have significant final marginal sensitivity indices. Local sensitivity indices for correlated parameters were defined which are the linear equivalents of the global ones. The results of the global sensitivity analysis were compared with the corresponding results of local sensitivity analysis for the case of the syngas–air combustion system. The same set of reactions was indicated to be important by both approaches