- Open Access
Influence of parameter perturbations on the reachability of therapeutic target in systems with switchings
© The Author(s) 2017
- Published: 18 August 2017
Examination of physiological processes and the influences of the drugs on them can be efficiently supported by mathematical modeling. One of the biggest problems is related to the exact fitting of the parameters of a model. Conditions inside the organism change dynamically, so the rates of processes are very difficult to estimate. Perturbations in the model parameters influence the steady state so a desired therapeutic goal may not be reached. Here we investigate the effect of parameter deviation on the steady state in three simple models of the influence of a therapeutic drug on its target protein. Two types of changes in the model parameters are taken into account: small perturbations in the system parameter values, and changes in the switching time of a specific parameter. Additionally, we examine the systems response in case of a drug concentration decreasing with time.
The models which we analyze are simplified, because we want to avoid influences of complex dynamics on the results. A system with a negative feedback loop is the most robust and the most rapid, so it requires the largest drug dose but the effects are observed very quickly. On the other hand a system with positive feedback is very sensitive to changes, so small drug doses are sufficient to reach a therapeutic target. In systems without feedback or with positive feedback, perturbations in the model parameters have a bigger influence on the reachability of the therapeutic target than in systems with negative feedback. Drug degradation or inactivation in biological systems enforces multiple drug applications to maintain the level of a drug’s target under the desired threshold. The frequency of drug application should be fitted to the system dynamics, because the response velocity is tightly related to the therapeutic effectiveness and the time for achieving the goal.
Systems with different types of regulation vary in their dynamics and characteristic features. Depending on the feedback loop, different types of therapy may be the most appropriate, and deviations in the model parameters have different influences on the reachability of the therapeutic target.
- Biological model
- Piece-wise continuous nonlinear models
A significant part of biological processes proceed in a switch-like manner. A typical example of such step changes in biological systems is related to application of a therapeutic drug; after drug administration, especially through injection, specific processes are activated and intracorporeal conditions change in a sudden, step-like manner. When a drug is administered to a patient (orally or by a drip-bag), the number of drug molecules in the body increases rapidly and drug-induced processes start suddenly. The level of active drug cannot be maintained at a steady level due to processes such as transport of drug molecules through the organism, import and export by cells, binding to other molecules, and drug degradation or inactivation . Administration of a drug is intended to enforce specific processes such as inactivation or degradation of a protein target or blockade of inter- or extra-cellular transport . Consequently, the activity of a drug should be included in a model in a multiplicative manner: drug molecules influence the rate of the processes related to the number of protein or other molecular targets. Instead of considering the drug application as an input to the model, we propose considering it as an abrupt change of the parameter values; in this case the drug accelerates the process of protein degradation. Other examples of biological processes which proceed in a switch-like manner are gene activation and deactivation  and transmembrane transport through (divided by) ion channels . These and many other biological processes can be effectively modeled as systems with switches in the parameter values.
The intra- and inter-cellular environments are very complex, containing many different types of molecules both non-organic and organic such as carbohydrates, proteins, lipids, and nucleic acids. Because of intra- and extra-cellular processes such as export and import, production and degradation of proteins and their modification by phosphorylation, ubiquitylation, or glycosylation, the chemical and physical conditions inside the cell change dynamically . Similarly, cell growth during the cell cycle influences the concentrations of all the substances inside the cell and thereby the rate of all processes. Consequently, the rates of cellular processes are dynamically changing . The building of mathematical models requires exact determination of model parameters. The parameters for biological processes are calculated or estimated using the results of biological experiments and thus depend highly on the biological, chemical and physical conditions in cells. Sensitivity analysis of models is usually employed to test their vulnerability to small perturbations of parameters , but when a change of the parameter values is large the model outputs may be significantly different. In systems with switching, the problem of parameter perturbation can cause even greater inaccuracy due to cumulation of two effects: first, the initial state before the switch can be significantly altered, and second, the parameters in the model after a switch are biased. Consequently, the dynamics of such models can be significantly different from the real dynamics. This factor plays an essential role, especially in situations when therapy with minimized doses of a drug is considered.
In our preliminary study we neglect the pharmacodynamics and pharmacokinetics and assume that the concentration of a drug in the organism is maintained at a stable level. This approach is useful when examining the internal properties of the system, although in real-life systems the problem of estimating drug level and its influence on the organism play an essential role. The problem of drug pharmacodynamics and pharmacokinetics has been widely examined and discussed (see e.g. [7–9]). Mechanism-based Pharmacokinetic/Pharmacodynamic (PK-PD) models contain expressions to quantitatively characterize processes such as drug distribution at the target site and drug binding, activation, and inactivation or excretion. Moreover, there are many other factors which can influence the therapeutic or toxic activity of a drug such as the problem of drug-drug interactions which affect the pharmacokinetics , by influencing gastro-intestinal absorption, occupying binding sites on a protein, or competing for drug-metabolizing enzyme systems are described exhaustively by Kristensen . Nevertheless, an interesting problem of pharmacokinetics arises in a system with multiple drug dosage. Clinical research on drug administration supported by mathematical modeling is useful in determining the efficacy and safety of proposed therapies . Model parameters should be fitted for a specific drug, as in our previous work where based on the results published by Zhang et al.  we created model equations describing the active and total concentrations for the experimental antitumour drug Nutlin . In the present work we do not consider any specific drug, so we use the previously published equations and slightly modify the parameters to obtain desirable dynamics. The kinetics of the protein degradation is not described by the Michaelis–Menten kinetics, because the number of active molecules of a drug is comparable to the number of protein target molecules, so the process of increased degradation of a target protein by a drug can be described by the mass-action law.
Regulation of protein production
Modeling of biological systems demands a special approach due to their very high complexity. The great majority of these systems are modeled by deterministic algorithms, although in some cases a stochastic approach is more suitable especially when a low number of reacting molecules is involved such as in gene switching processes. Parameter fluctuations may be also considered as stochastic changes of parameter values  or modeled as a bounded stochastic process [15–17]. However, here we use a deterministic approach despite the existence of gene states in our models and we postpone investigation of stochastic models, since the majority of published models are still deterministic.
The use of a switched system for modeling biological processes is not a new approach, although to our knowledge the achievability of a therapeutic target in systems with parameter perturbation has not been considered. There have been attempts to use piecewise linear systems to model highly non-linear processes, in which the dependencies between molecules are described by the switching rules which enforce changes in the model structure or parameter values [18–20]. Similar methods were presented by de Jong and co-workers, who proposed qualitative simulation of regulatory networks using piecewise linear models [21, 22]. A different application of switched system was presented by Parise et al. , where a stochastic model of a controlled gene expression system is discussed focussing on the achievability region for the protein produced from the gene by solving an optimization problem.
In this work we investigate the problem of the influence of parameter perturbations on a medical therapy designed to decrease the number of active molecules of a target protein. In the first part of this paper three models are considered which all consist of three variables: G—the gene state, R—the number of mRNA molecules transcribed from the gene, and A—the number of molecules of the protein A. In the second part the model is expanded by two more equations, which describe the total and active drug levels. The model without a feedback loop is a well-known model of protein production, and the model with a negative feedback loop was published in . Additionally, we create a model with a positive feedback loop to compare the influence of different types of autoregulation. The parameter values are taken partially from our previous work [24, 25], where we presented some preliminary results. Here the positive feedback loop is created in a different way, and we further examine the case when a drug is degraded or inactivated. The number of target protein molecules in the steady state before drug application is established arbitrarily at 160,000 molecules and the therapeutic goal is to reduce this level to below 70,000 molecules.
Values of the model parameters
Spontaneous gene deactivation (negative feedback)
Protein induced gene deactivation (negative feedback)
Spontaneous gene deactivation (positive feedback)
Protein induced gene activation (positive feedback)
Protein degradation dependent on drug
Number of drug molecules
Number of alleles
Values of the steady states in models with or without drug application (drug dose 140,000 molecules)
All the simulations were performed by a deterministic algorithm, specifically a Runge–Kutta 4th order algorithm. The results obtained cannot be treated as time courses of the number of molecules in one cell, because we do not take into account the stochastic nature of the processes. However, the deterministic approach enables prediction of the mean behavior of cells, which can be considered as the response of a large cell population like a whole organism. The simple control objective is to keep the protein level \(A(t)<Th\), where Th is the required threshold. To ensure effective therapy, the protein level must be maintained under the threshold until the end of the simulation. The time when this target is achieved (when A goes below Th) is not specified, but a faster therapeutic effect is preferable.
The values of the parameters for model with drug degradation
Dose conversion factor
Drug dose rate
Equilibrium association rate
Concentration of protein binding sites
Drug elimination rate
Drug import to cell
Drug export from cell
Results for models without drug degradation
Time of the switch change
Drug degradation and inactivation
Systems with switching form a quite common and well investigated group of automatic control problems [27, 28]. However, application of methods known from control theory to real biological systems with stepwise changes of parameter values is quite a new approach. In the present work abrupt changes of parameter values for a target protein are used to model the results of application of a therapeutic drug. In the case of the drug degradation multiple drug dosage case can be treated as a multiple switch in the system. This property creates a variety of possible applications, for example cell disorders like carcinogenesis which induce many different changes in cellular processes can be modeled as switches in the structure of the system. All the models presented in this work, even those with feedback, are very simplified, and we do not consider any specific target protein. In biological systems, dependencies between proteins and other molecules create very complex regulatory networks which can include multiple different feedbacks, both positive and negative. Moreover, in more complicated systems we can observe delays in signal transduction so that the dynamics can be even more complicated. It is even possible that complicated dynamics of a real system will filter out the effect of parameter perturbations so that no effect will not be observed at all. Analysis of real systems can therefore be challenging, but in our opinion it is worth performing. However, in this work we neglect the complicated real-life intercellular dynamics to focus on the problem of the effects of parameter perturbations in the system. Efficient therapy usually requires the application of several drugs. The pharmacodynamics and pharmacokinetics of particular drugs, especially their removal from and inactivation by the organism, enforce frequent drug application to maintain the desired effect and due to drug toxicity and side effects, the cumulative dose should be minimized. In the case of similar target protein levels in the steady state, the internal feedback can result in decreased or increased sensitivity to the drug (Fig. 2). The effects observed here emphasize the importance of careful investigations of the regulatory networks inside the cell. An optimal algorithm for drug application is still a problem which has not been fully examined. High drug doses can be toxic for patients, and moreover they are very quickly removed from the body due to the nonlinear process of drug excretion. Side effects can be minimized by application of smaller drug doses with higher frequency, but the dynamics of the process is very important. The time to reach the therapeutic goal can be a crucial factor in selection of a therapy and in the case of fast and insensitive systems such as those with negative feedback studied here, more frequent drug application allows the therapeutic target to be reached much earlier than with a schedule of less frequent applications with the same cumulative dose. Inaccuracies in parameter determination and times of drug administration could have a significant impact on the results of therapy, and thus should be always taken into consideration. Attempts have been made to estimate the influence of parameter deviations on the reachability of the therapeutic goal using the analytical approach . Parameter perturbation will be especially crucial for systems where switching depends on the system’s state. If the switches are used for modeling the internal features, for example step-like activation of a protein by other macromolecules, perturbations in the model parameters will influence not only the numbers of molecules but also the time of the switch, and as a result such models will have significantly different dynamics. In such case the problem of parameter perturbation will be even more interesting and thus worth focussing on. In this paper, despite the existence of genes in our models we have focused only on the deterministic approach. The reason was that the majority of published models is still deterministic, and we want to point out some possible problems related to the determination of parameter values (and thus their fluctuations) when systems with switchings (such as drug application) are considered. This leads us to results which may be considered as the expected values realized in a real stochastic system. One should remember, however, that the real response of a single cell to a drug may be more strongly affected by the intrinsic noise resulting from stochastic gene switching than by parameter deviations, and this possibility will be the subject of our future studies. The other interesting phenomena related to stochastic gene switching and drug efficiency was pointed out in our previous work  where we show that the drug dose required to achieve the therapeutic goal may be significantly higher when a system is modeled by the stochastic approach than by the deterministic approach. In real biological systems, differences in parameter values can be mitigated by the very complex dynamics of non-linear systems and the stochasticity of processes such as gene activation. Nevertheless, numerous models presented in the literature were created as deterministic systems, so the problem discussed in this paper is applicable to them.
Creation of models of biological systems can be very useful and thus becomes a more and more popular method for examination of biological processes. One of the biggest problems in model development is determination of the parameter values. Some parameter values are calculated based on biological experiments, which are often very expensive, time consuming and gives only approximate results. However, the values of many biological parameters can not be calculated directly and therefore must be estimated. Moreover, the rates of biological processes are dynamically changing due to the physical, chemical and biological conditions inside the cell or organism. Taking all these factors into account, the problem of perturbations in model parameters is a significant issue. In this work we focus on the influence of parameter deviations on the reachability of a therapeutic goal in quite simple models of a therapeutic drug targeting a specific protein in vivo, and we examine the general properties of these models. Negative feedback in the model decreases the sensitivity to a switch (in this case the application of the drug) but also decreases the sensitivity to parameter perturbations. This model has very rapid dynamics and its response shows overregulation. On the other hand, a model with positive feedback is characterized by very slow changes. It is very sensitive to the schedule of drug application, but perturbations in the model parameters result in significant difference in the output and, moreover, changes in the parameter values can make the model unstable. Considering changes in the level of active drug due to its degradation, multiple drug application is optimal. More frequent drug application allows for using a smaller total dose, so that the therapy can be effective and has lower toxicity. A high drug dose results in a very large and rapid jump of drug concentration which could cause serious damage to the whole body. The frequency of drug application has a significant influence on the time to reach the therapeutic target. In our opinion, analysis of dynamics and sensitivity for the models of such processes plays a crucial role in their application
AS suggested the problem, MO and KP developed the models, MO performed numerical simulations, MO and KP analyzed the data, AS, KP and MO wrote the paper. All authors read and approved the final manuscript.
The authors wish to thank Prof. Ronald Hacock for his assistance in preparation of the revised version of the manuscript.
Calculations were performed on the Ziemowit computational cluster (http://www.ziemowit.hpc.polsl.pl)created in the EU Innovative Economy Operational Programme (POIG).02.01.00-00-166/08 project (BIO-FARMA) and expanded in the POIG.02.03.01-00-040/13 project (Syscancer).
The authors declare that they have no competing interests.
The research presented here was partially supported by the National Science Centre in Poland grants with decision numbers DEC-2012/05/D/ST7/02072 (for KP) and DEC-2014/13/B/ST7/00755 (for AS and MO). Publication costs were funded by Silesian University of Technology grant with decision number BK/213/RAU1/2016.
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This article has been published as part of BioMedical Engineering OnLine Volume 16 Supplement 1, 2017: Selected articles from the 4th International Work-Conference on Bioinformatics and Biomedical Engineering-IWBBIO 2016. The full contents of the supplement are available online at https://biomedical-engineering-online.biomedcentral.com/articles/supplements/volume-16-supplement-1.
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