Dynamical modeling of drug effect using hybrid systems
 Xiangfang Li^{1}Email author,
 Lijun Qian^{2} and
 Edward R Dougherty^{1, 3, 4}
https://doi.org/10.1186/16874153201219
© Li et al.; licensee Springer. 2012
Received: 30 April 2012
Accepted: 12 November 2012
Published: 26 December 2012
Abstract
Drug discovery today is a complex, expensive, and timeconsuming process with high attrition rate. A more systematic approach is needed to combine innovative approaches in order to lead to more effective and efficient drug development. This article provides systematic mathematical analysis and dynamical modeling of drug effect under gene regulatory network contexts. A hybrid systems model, which merges together discrete and continuous dynamics into a single dynamical model, is proposed to study dynamics of the underlying regulatory network under drug perturbations. The major goal is to understand how the system changes when perturbed by drugs and give suggestions for better therapeutic interventions. A realistic periodic drug intake scenario is considered, drug pharmacokinetics and pharmacodynamics information being taken into account in the proposed hybrid systems model. Simulations are performed using MATLAB/SIMULINK to corroborate the analytical results.
Keywords
Introduction
The ultimate goal of drug therapy is to modulate the phenotypic behavior of cells by altering the behavior of the gene and protein components of the cell [1]. This approach is possible because the phenotypic behavior of the cell reflects the dynamics of the gene and proteinbased regulatory network. When it comes to drug therapeutics and disease modeling, the major goal is to understand how the system changes when perturbed and how to modify the system to achieve a desired outcome. To understand and exploit the complicated mapping between genome and phenome, especially in the context of drug discovery, it is critical to evaluate the regulatory interactions between the genes and proteins that form the gene regulatory network (GRN). To date, the hope of the rapid translation of “genes to drugs” has foundered on the reality that disease biology is complex and drug development must be driven by insights into biological responses [2]. A systems approach is crucial for moving biology from a descriptive to a predictive science [3, 4]. This calls for appropriate modeling to establish a functional understanding of disease–drug interaction, in order to better predict drug effects and make drug discovery a faster and more systematic process.
Pharmacokinetics (PK) is the study of what the body does to the drug, i.e., the absorption, distribution, metabolism, and excretion of the drug, and pharmacodynamics (PD) seeks to study what the drug does to the body. A salient challenge is to link a drug’s PK information with PD characteristics to provide a better understanding of the time course of drug effect (PK/PD) after drug administration [5]. Modeling and simulation tools are required to integrate PK and PD data and optimize drug regimens.
A salient problem is finding a dosing regimen of a drug candidate that is both efficacious and safe [6]. Traditionally, drugs have been administered on an experimental basis, but it is virtually impossible to optimize dosing regimens using strictly empirical methods, especially since different patients may respond differently to the same drug dosage [7]. Moreover, traditionally designing the dosing regimen to achieve some desired target goal such as relatively constant serum concentration may not be optimal because of underlying dynamic biological networks. For example, Shah et al. [8] demonstrate that BCR–ABL inhibitor dasatinib, which has greater potency and a short halflife, can achieve deep clinical remission in CML patients by achieving transient potent BCR–ABL inhibition, while traditionally approved tyrosine kinase inhibitors usually have prolonged halflives that result in continuous target inhibition. A similar study of whether short pulses of higher dose or persistent dosing with lower doses has the most favorable outcomes has been carried out by Amin et al. [9] in the setup of inactivation of HER2–HER3 signaling. Finding an optimal dosing regimen based on the dynamics of biological systems and relevant PK/PD information is critically important.
System modeling is emerging as a valuable tool in therapeutics to address these challenges [3, 10–12]. The process begins with building a quantitative model of a biological system. Consequences of particular perturbations, such as optimal dosing regimens, optimal drug targets, or combinational therapy, can be simulated in time courses using such models. In this study, we propose a hybrid systems model for GRNs and incorporate a drug’s PK and PD information by using a statespace approach. We first study drug effect assuming the drug target to be a gene or protein in the proposed drug perturbation model using dynamical system theory, considering the case of periodic drug intake and analytically deriving the conditions for the drug to be effective. We extend the analysis to the 2gene case and then to the case of a network with multiple coupled genes and positive feedback loops. Simulations are performed using MATLAB/SIMULINK to supplement our analytical results.
Model formulation
While discrete modeling leaves out many details, continuous modeling includes so many details that computational demands preclude their applications to many larger systems. Hybrid systems, which aim to merge ideas from both continuous and discrete modeling into one paradigm, are appealing for GRN modeling under drug perturbations because biological systems are naturally nonlinear, have highly varied regulatory requirements, and possess a wide range of control strategies for meeting their needs. While some simple, local, feedback control methods can provide sufficient regulation of many moreorless continuous cellular processes, the regulation of discontinuous processes possessing the character of computational decision making requires more elaborate regulatory methods [13]. In particular, some genes display regulation in a thresholded switchlike manner [14].
Hybrid systems include a broad space of models and systems. Several hybrid systems models have been developed for biological networks. Some of these have been used to perform reachability analysis to elucidate biologically meaningful properties. For example, the Lac operon system has been well studied both experimentally and using continuous models [15, 16]. A hybrid model and use of a reachability algorithm were validated by comparison with experimental data and continuous models [17]. Other biological hybrid systems analyzed in similar ways include the DeltaNotch decision process [18, 19], GRNs of carbon starvation [20], and nutritional stress response [21] in Escherichia coli. As far as we know, the only hybrid systems modeling concerning treatment or drug effects is contained in our earlier work [22].
with ${\Xi}_{i}^{j}$ defined similarly. ${\Omega}_{i}^{j}$ and ${\Xi}_{i}^{j}$ may be set to 0 or 1, or different forms when appropriate threshold values are chosen. For example, ${\Omega}_{1}^{1}={s}^{+}({x}_{1},{0}^{}){s}^{}({x}_{1},\infty )=1$ and ${\Omega}_{1}^{2}={s}^{+}({x}_{2},{0}^{}){s}^{}({x}_{2},{\theta}_{1}^{2})={s}^{}({x}_{2},{\theta}_{2}^{1})$. ${\Omega}_{i}^{u}$ and ${\Xi}_{i}^{u}$ describe how the drug u affect gene i. ${\beta}_{i}^{u}\ge 0$ and ${\gamma}_{i}^{u}\ge 0$ are the synthesis and degradation factors of the drug on gene i. ${\beta}_{i}^{u}{\Omega}_{i}^{u}$ and ${\gamma}_{i}^{u}{\Xi}_{i}^{u}{x}_{i}$ are used when the drug is activating or repressing certain genes, respectively. Since most drugs are used to repress genes, only ${\gamma}_{i}^{u}{\Xi}_{i}^{u}{x}_{i}$ is considered in the examples of this article. Note that γ^{ u } is defined as a drugeffect factor, which is closely related to the drug pharmacology model discussed in the following section.
It should be kept in mind that the focus of this article is studying the effect of dosing, in particular, dosing regimens, on the expression of genes involved in a pathology by using hybrid systems theory. Whereas the simpler Equation (1) is widely accepted, it does not contain drugeffect terms. Equation (2) extends Equation (1) by including such terms. While the structure is intuitively reasonable and somewhat general, the actual details of the drugeffect terms are unknown. Finding the specific form of Equation (2) for a specific disease is a system identification problem, which is quite distinct from the analysis problem addressed in this article. We are addressing optimization of treatment intervention, given the system. The details of our analysis might change when the details of Equation (2) are clarified, but we expect that the hybrid systems approach taken in the article will go through with appropriate modifications in the mathematical details.
Pharmacology model
The basis of clinical pharmacology is the fact that the intensities of many pharmacological effects are functions of the amount of drug in the body and, more specifically, the concentration of drug at the effectsite [5]. Historically, PK and PD were considered as separate disciplines; however, the information provided by these disciplines is limited if regarded in isolation [25]. A drugeffect factor γ^{ u } is included in our proposed model (Equation 2), which is related to drug’s PD characteristic (concentration–response) and its PK information (dose–concentration). In order to describe the time course of drug effect in response to different dosing regimens, the integrated PK/PD model is indispensable because it builds the bridge between these two classical disciplines of pharmacology [25]. Following each dosing regimen, instead of a twodimensional PK and PD relationship, the proposed approach enables a description of a threedimensional dose–concentration–effect relationship. Specifically, PK and PD are linked through γ^{ u } by a statespace approach to facilitate the description and prediction of the time course of drug effects resulting from different drug administration regimens.
Drug concentration–response curve: PD model
where ${q}_{1}^{u}$ is the ratio between the drugeffect factor ${\gamma}_{1}^{u}$ and the effective drug concentration $u{\theta}_{1}^{\underset{\u203e}{u}}$ in the linear range. This reflects the fact that the drug only starts to take effect when its concentration level is above a lower threshold ${\theta}_{1}^{\underset{\u203e}{u}}$ and its effect saturates when its concentration level exceeds an upper threshold ${\theta}_{1}^{\overline{u}}$. Note that the sigmoidal E_{max} model can be well approximated by the proposed PD model. By taking the derivative of E with respect to C and evaluating it at E C_{50}, we obtain the slope as ${q}_{1}^{u}=\frac{m{E}_{max}}{4E{C}_{50}}$. The upper and lower bounds should satisfy ${q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})={E}_{max}$. An example of the sigmoidal E_{max} model when m=4 and our proposed PD model are plotted together in Figure 8, where the proposed model closely resembles the sigmoidal E_{max} model.
Periodic drug intake: PK model
Drug concentration at the effectsite is critical for its pharmacological effect. Currently, plasma drug concentrations are markers that serve as surrogates for drug concentration at the effectsite for beneficial and adverse effects; however, markers not grounded on a sound theoretical foundation and therapeutic mechanismbased intervention can limit the usefulness of PK/PD modeling to drug development. For example, it has been demonstrated that the intracellular PK of a drug is quite different from plasma drug concentration [29, 30]. As observed in the study by Kuh et al. [29], the intracellular concentration of a drug will exponentially increase as the drug is absorbed after each drug intake. The drug concentration may change very slowly (in our model, we approximate that as a flat curve) when the intracellular and extracellular drug concentration approach equilibrium. In time, drug concentration will exponentially decrease as the rate at which it is eliminated is more than the rate at which it enters the effectsite and, as a result, effects diminish.
Mathematical analysis of drug effect
In this section, we study the time course of drug effect for different dosage and schedule arrangements where the drug is designed to repress a “target gene”. The case with a special PK profile (drug concentration only has exponential decay) was analytically studied in our previous work [22]. In this study, we extend the analysis considering a general PK profile given in Figure 9 and PD model given in Figure 8. Closedform analytical solution is provided and simulations are performed to validate theoretical analysis. In later sections, we show that the same methodology can be applied to interactive genes, where not only will the drug affect the gene expression level, but the target gene is also coupled with other genes.
where for k τ_{ i } ≤ t ≤ (k+1)τ_{ i } and i = 1,2,… denoting the index of different dosing regimens, ${q}_{1}^{{u}_{i}}={q}_{1}^{u}$, ${\theta}_{1}^{{\underset{\u203e}{u}}_{i}}={\theta}_{1}^{\underset{\u203e}{u}}$, ${\theta}_{1}^{{\overline{u}}_{i}}={\theta}_{1}^{\overline{u}},\phantom{\rule{2.77695pt}{0ex}}$ for any i, since we assume that the same drug is taken in different dosage and schedule settings. ζ_{ i } is the highest concentration level reached after taking the drug.
Statespace analysis
When the state transits in each period under periodic drug intake, it may pass through different domains (depending on the changes of drug concentration along time). During the transit time through domains D_{5} and D_{3}, the gene expression level is pushed lower (to the left), while the driving strength will depend on the drug’s PD characteristic. During the transit time through D_{1}, the expression level will rise (to the right), since the drug concentration is lower than ${\theta}_{1}^{\underset{\u203e}{u}}$. For the drug to be effective, the reduction of the expression level in D_{5} and D_{3} has to be larger than the increase of the expression level in D_{1}. In summary, we should have x_{1}((k+1)τ) < x_{1}(k τ), so that after each treatment the expression level x_{1} will decrease.
State trajectory analysis
We analyze the drug effect considering the scenario shown in Figure 10, where ${\zeta}_{i}>{\theta}_{1}^{\overline{u}}$. The same methodology can be applied to a simpler scenario where ${\theta}_{1}^{\underset{\u203e}{u}}<{\zeta}_{i}<{\theta}_{1}^{\overline{u}}$. We divide the state trajectory in a period k τ_{ i } ≤ t ≤ (k+1)τ_{ i } into stages a, b, c, d, e, f, and g as marked in Figure 10 and examine the drug effect stagebystage. The time notations used in the derivation are given by

t_{1}: the traveling time from the initial state to the boundary between D_{3} and D_{1}.

t_{2}: from initial state to boundary between D_{5} and D_{3}.

t_{3}: from initial state to the end of stage c, t_{3} = k τ_{ i }+p_{1}.

t_{4}: time at which the drug concentration starts to decrease, t_{4} = k τ_{ i }+p_{1}+p_{2}.

t_{5}: from the initial state to the end of stage e.

t_{6}: from the initial state to the end of stage f.
For k τ_{ i } ≤ t ≤ (k+1)τ_{ i }, where i is the index for different dosing regimens, the corresponding equations and solutions for each stage are given by:

Stage ( a )  D_{1} (k τ_{ i } ≤ t ≤ t_{1}):$\begin{array}{l}{\stackrel{\phantom{\rule{2.3em}{0ex}}\u0307}{\phantom{\rule{2em}{0ex}}x}}_{1}={\beta}_{1}{\gamma}_{1}{x}_{1},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {\phantom{\rule{2em}{0ex}}x}_{1}\left(t\right)=\frac{{\beta}_{1}}{{\gamma}_{1}}+\left[{x}_{1}\left(k{\tau}_{i}\right)\frac{{\beta}_{1}}{{\gamma}_{1}}\right]{e}^{{\gamma}_{1}(tk{\tau}_{i})},\\ {\phantom{\rule{2em}{0ex}}u}_{i}\left(t\right)={e}^{{\lambda}_{a}(tk{\tau}_{i})}1.\end{array}$(10)

Stage ( b )  D_{3} (t_{1}≤ t ≤ t_{2}):$\begin{array}{l}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}[{\gamma}_{1}+{q}_{1}^{u}(u{\theta}_{1}^{\underset{\u203e}{u}}\left)\right]{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\frac{{x}_{1}\left({t}_{1}\right)A\left({t}_{1}\right)}{A\left(t\right)}\\ \phantom{\rule{3em}{0ex}}+\frac{1}{A\left(t\right)}\underset{{t}_{1}}{\overset{t}{\int}}{\beta}_{1}{e}^{\left[\right({q}_{1}^{u}(1+{\theta}_{1}^{\underset{\u203e}{u}}){\gamma}_{1})\sigma \frac{{q}_{1}^{u}}{{\lambda}_{a}}{e}^{{\lambda}_{a}(\sigma k{\tau}_{i})}]}\mathit{d\sigma},\\ A\left(t\right)={e}^{\left[\right({q}_{1}^{u}(1+{\theta}_{1}^{\underset{\u203e}{u}}){\gamma}_{1})t\frac{{q}_{1}^{u}}{{\lambda}_{a}}{e}^{{\lambda}_{a}(tk{\tau}_{i})}]},\\ {u}_{i}\left(t\right)={e}^{{\lambda}_{a}(tk{\tau}_{i})}1.\end{array}$(11)

Stage ( c )  D_{5} (t_{2} ≤ t ≤ t_{3} = k τ_{ i }+p_{1}):$\begin{array}{l}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right]{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})}\\ \phantom{\rule{3.5em}{0ex}}+\left[{x}_{1}({t}_{2})\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\underset{\u203e}{u}})}\right]\\ \phantom{\rule{3.5em}{0ex}}\times {e}^{[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right](t{t}_{2})},\\ {u}_{i}\left(t\right)={e}^{{\lambda}_{a}(tk{\tau}_{i})}1\end{array}$(12)

Stage ( d )  D_{5} (t_{3} ≤ t ≤ t_{4} = k τ_{ i } + p_{1} + p_{2}):$\begin{array}{l}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right]{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})}\\ \phantom{\rule{3.5em}{0ex}}+\left[{x}_{1}\left({t}_{3}\right)\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})}\right]\\ \phantom{\rule{3.5em}{0ex}}\times {e}^{[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right](t{t}_{3})},\\ {u}_{i}\left(t\right)={u}_{i}^{max}={\zeta}_{i}\end{array}$(13)

Stage ( e )  D_{5} (t_{4} ≤ t ≤ t_{5}):$\begin{array}{l}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right]{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})}\\ \phantom{\rule{3.5em}{0ex}}+\left[{x}_{1}\left({t}_{4}\right)\frac{{\beta}_{1}}{{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})}\right]\\ \phantom{\rule{3.5em}{0ex}}\times {e}^{[{\gamma}_{1}+{q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}}\left)\right](t{t}_{4})},\\ {u}_{i}\left(t\right)={\zeta}_{i}{e}^{{\lambda}_{d}(t{t}_{4})}.\end{array}$(14)

Stage ( f )  D_{3} (t_{5} ≤ t ≤ t_{6}):$\begin{array}{l}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}[{\gamma}_{1}+{q}_{1}^{u}(u{\theta}_{1}^{\underset{\u203e}{u}}\left)\right]{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\left[{x}_{1}\left({t}_{5}\right){e}^{[\frac{{q}_{1}^{u}}{{\lambda}_{d}}{\theta}_{1}^{\overline{u}}+({q}_{1}^{u}{\theta}_{1}^{\underset{\u203e}{u}}{\gamma}_{1}\left){t}_{5}\right]}\right.\\ \phantom{\rule{3.5em}{0ex}}\left(\right)close="]">+\underset{{t}_{5}}{\overset{t}{\int}}{\beta}_{1}{e}^{[\frac{{q}_{1}^{u}}{{\lambda}_{d}}{\theta}_{1}^{\overline{u}}{e}^{{\lambda}_{d}(\sigma {t}_{5})}+({q}_{1}^{u}{\theta}_{1}^{\underset{\u203e}{u}}{\gamma}_{1}\left)\sigma \right]}\mathrm{d\sigma}\end{array}\phantom{\rule{3.5em}{0ex}}\times {e}^{[\frac{{q}_{1}^{u}}{{\lambda}_{d}}{\theta}_{1}^{\overline{u}}{e}^{{\lambda}_{d}(t{t}_{5})}+({q}_{1}^{u}{\theta}_{1}^{\underset{\u203e}{u}}{\gamma}_{1}\left)t\right]},\\ {u}_{i}\left(t\right)={\theta}_{1}^{\overline{u}}{e}^{{\lambda}_{d}(t{t}_{5})}.$(15)

Stage ( g )  D_{1} (t_{6} ≤ t ≤ (k+1)τ_{ i }):$\begin{array}{l}\phantom{\rule{1.3em}{0ex}}{\stackrel{\u0307}{x}}_{1}={\beta}_{1}{\gamma}_{1}{x}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\Rightarrow \\ {x}_{1}\left(t\right)=\frac{{\beta}_{1}}{{\gamma}_{1}}+\left[{x}_{1}\left({t}_{6}\right)\frac{{\beta}_{1}}{{\gamma}_{1}}\right]{e}^{{\gamma}_{1}(t{t}_{6})},\\ {u}_{i}\left(t\right)={\theta}_{1}^{\underset{\u203e}{u}}{e}^{{\lambda}_{d}(t{t}_{6})}.\end{array}$(16)
For the drug to be effective, we need the disease gene expression level to decrease following each period of drug intake. Hence, we can express x_{1}((k+1)τ) < x_{1}(k τ) in terms of dosage and frequency schedule and derive the region where the drug is effective using the above listed equations.
Results and analysis
Analysis of 2gene networks
The above equation has an important biological interpretation: when the degradation of x_{1} due to the strength of the drug is faster than the increase of x_{1} due to the positive feedback loop, both eigenvalues are negative, the system is stable and x_{1} will experience exponential decay; on the other hand, if the effect of the positive feedback loop is dominant, then one of the eigenvalues will be positive and x_{1} will increase exponentially.
Now with the baseline analysis of the secondorder system, we provide detailed state trajectory analysis by taking into account the practical form of PK/PD (${\gamma}_{1}^{u}$) when the drug is taken periodically.
State trajectory analysis
We analyze the drugeffect following the same framework given in the subsection “State trajectory analysis” under the main section “Mathematical analysis of drug effect”. For k τ_{ i } ≤ t ≤ (k+1)τ_{ i }, i = 1,2,…, the corresponding equations and solutions for each stages are given as follows:

Stage ( a )  D_{1} (k τ_{ i } ≤ t ≤ t_{1}):$\begin{array}{ccc}{\stackrel{\u0307}{x}}_{1}\hfill & =\hfill & {\beta}_{1}+{\eta}_{1}{x}_{2}{\gamma}_{1}{x}_{1}\hfill \\ {\stackrel{\u0307}{x}}_{2}\hfill & =\hfill & {\beta}_{2}+{\eta}_{2}{x}_{1}{\gamma}_{2}{x}_{2}\hfill \\ {u}_{i}\left(t\right)\hfill & =\hfill & {e}^{{\lambda}_{a}(tk{\tau}_{i})}1.\hfill \end{array}$(34)

The solution of x_{1}(t) is given by Equation (29), with k_{1} and k_{2} given by Equations (32) and (33), and t_{0} = k τ_{ i }.

Stage ( b )  D_{3}(t_{1} ≤ t ≤ t_{2}):$\begin{array}{ccc}\phantom{\rule{7em}{0ex}}{\stackrel{\u0307}{x}}_{1}\hfill & =\hfill & {\beta}_{1}+{\eta}_{1}{x}_{2}{\gamma}_{1}{x}_{1}{\gamma}_{1}^{u}{x}_{1}\hfill \\ \phantom{\rule{7em}{0ex}}{\stackrel{\u0307}{x}}_{2}\hfill & =\hfill & {\beta}_{2}+{\eta}_{2}{x}_{1}{\gamma}_{2}{x}_{2}\hfill \\ \phantom{\rule{8.5em}{0ex}}\Rightarrow \hfill \\ \phantom{\rule{.7em}{0ex}}{\stackrel{\u0308}{x}}_{1}+a{\stackrel{\u0307}{x}}_{1}+b{x}_{1}\hfill & =\hfill & d\hfill \\ \phantom{\rule{6em}{0ex}}{u}_{i}\left(t\right)\hfill & =\hfill & {e}^{{\lambda}_{a}(tk{\tau}_{i})}1\hfill \end{array}$(35)

where a, b, d are defined as before. When incorporating the practical form of ${\gamma}_{1}^{u}={q}_{1}^{u}(u{\theta}_{1}^{\underset{\u203e}{u}})$ and $u={e}^{{\lambda}_{a}(tk{\tau}_{i})}1$, the above secondorder ODE has no closedform solution. In this case, the solution can be obtained numerically.

Stage ( c )  D_{5} (t_{2} ≤ t ≤ t_{3} = k τ_{ i } + p_{1}): The set of equations are the same as in Stage (b) except that ${\gamma}_{1}^{u}={q}_{1}^{u}({\theta}_{1}^{\overline{u}}{\theta}_{1}^{\underset{\u203e}{u}})$. Since ${\gamma}_{1}^{u}$ does not depend on $u={e}^{{\lambda}_{a}(tk{\tau}_{i})}1$ explicitly, x_{1} has a closedform solution given by Equation (29).

Stage ( d )  D_{5} (t_{3} ≤ t ≤ t_{4} = k τ_{ i } + p_{1} + p_{2}): The solution of x_{1} is the same as that in Stage (c) except the start and end times, and the equation of u, which is ${u}_{i}\left(t\right)={u}_{i}^{max}={\zeta}_{i}$ in this stage.

Stage ( e )  D_{5} (t_{4} ≤ t ≤ t_{5}): The solution of x_{1} is the same as in Stage (c) except the start and end times, and the equation of u, which now is ${u}_{i}\left(t\right)={\zeta}_{i}{e}^{{\lambda}_{d}(t{t}_{4})}$.

Stage ( f )  D_{3} (t_{5} ≤ t ≤ t_{6}): The solution of x_{1} is the same as in Stage (b) except the start and end times, and the equation of u, which now is ${u}_{i}\left(t\right)={\theta}_{1}^{\overline{u}}{e}^{{\lambda}_{d}(t{t}_{5})}$.

Stage ( g )  D_{1} (t_{6} ≤ t ≤ (k+1)τ_{ i }): The solution of x_{1} is the same as in Stage (a) except the start and end times, and the equation of u, which now is ${u}_{i}\left(t\right)={\theta}_{1}^{\underset{\u203e}{u}}{e}^{{\lambda}_{d}(t{t}_{6})}$.
We can deduce the necessary and sufficient condition for the effectiveness of the drug by expressing the inequality x_{1}((k+1)τ) < x_{1}(k τ) in terms of dosing period τ and unit dose. In the 2gene case, no explicit closedform expression can be deduced for the solutions in stages (b) and (f) and numerical methods have to be applied. However, through such analysis, it is observed that the same methodology for analyzing drug effect can be extended to GRNs with multiple interactive genes, although the mathematics involved will become more complicated and sometimes numerical methods must be applied when there is no closedform solution.
Simulation results and analysis
Extension and discussion
In previous sections, we have considered the drugeffect on onegene and a 2gene case. In this section, we will consider the drugeffect on a target gene in a more sophisticated GRN context.
3gene network with multiple feedback loops
 1.Drug response is related to disease stage. Simulations are performed with different initial target gene expression level (x _{1}(0)). Figure 16a–c shows the system responses with x _{1}(0) = 20, which is not too high (corresponding to early disease state). As shown in Figure 16a, x _{1} expression level reduces to the range [7.7, 8.4] under periodic drug intake, while x _{2} and x _{3}, the two other interactive genes settle at 1.0 and 4.0, respectively. The system reaches a new steady state (a semistable limit cycle, to be exact), with ${x}_{1}^{s}=\frac{{\beta}_{1}+{\eta}_{1}{x}_{3}^{s}}{{\gamma}_{1}^{u}}=\frac{{\beta}_{1}+{\eta}_{1}{\beta}_{3}/{\gamma}_{3}}{{\gamma}_{1}^{u}}$, ${x}_{2}^{s}=\frac{{\beta}_{2}}{{\gamma}_{2}}$, and ${x}_{3}^{s}=\frac{{\beta}_{3}}{{\gamma}_{3}}$, where x _{1} is well controlled. The trajectories of x _{1} vs. u and x _{1} versus x _{3} are given in Figure 16b,c, respectively. The semistable limit cycle is shown in Figure 16b.
System responses with x_{1}(0) = 40 (corresponding to late disease state) are shown in Figure 16d–f for comparison. Although the other parameter settings are exactly the same, the drug will not repress the disease gene x_{1} (Figure 16d) owing to the interaction between the disease gene x_{1} and gene x_{3}. When ${x}_{1}\left(0\right)=20<{\theta}_{1}^{3}=21$, Equation (36) becomes ${\stackrel{\u0307}{x}}_{3}={\beta}_{3}{s}^{}({x}_{2},{\theta}_{2}^{3}){\gamma}_{3}{x}_{3}$, and thus x_{3} is negative regulated by x_{1} and converge to ${x}_{3}^{s}=\frac{{\beta}_{3}}{{\gamma}_{3}}$. However, when initial condition ${x}_{1}\left(0\right)=40>{\theta}_{1}^{3}=21$, Equation (36) becomes ${\stackrel{\u0307}{x}}_{3}={\beta}_{3}{s}^{}({x}_{2},{\theta}_{2}^{3})$, and thus x_{3} is positively regulated by x_{2} and its expression level will keep increasing. As a result, x_{1} will keep increasing as well, and a positive feedback loop is formed between x_{1} and x_{3}. This is confirmed by the trajectories of x_{1} versus u and x_{1} versus x_{3} given in Figure 16e,f, respectively.
 2.
Under certain conditions, single drug perturbation may not be enough. A drug is usually designed to a specific target. In this example, the drug tries to provide negative feedback to the regulation of x _{1} (tries to repress x _{1}); however, since the target gene is interactive (or, in a more general setting, pathways have crosstalk), only repressing the target gene (or blocking the signal of one pathway) may not prevent the target gene from expressing itself through interactions with other genes (or through interconnected pathways). In our case, x _{1} is interactive with x _{3}. To continue with previous simulation (results shown in Figure 16d–f, we try to increase the drug dosage tenfold from u(k τ) = 24 to u(k τ) = 240 with the same dosing period τ = 8 trying to bring down the expression level of x _{1}. However, from system responses shown in Figure 17a–c, it is observed that the drug is not effective although the dosage is increased tenfold. One step further, not only we increase dosage to u(k τ) = 240, but also to increase the dosing frequency (dosing period is decreased from τ = 8 to τ = 2), systems responses are shown in Figure 18a–d, where Figure 18c shows the left part of the trajectory shown in Figure 18b. It can be observed that although the drug perturbation is very strong, and the drug concentration is always staying in domain D_{5}, drug is still not effective.
From the nonlinear dynamical system perspective, the equation ${x}_{1}^{s}=\frac{{\beta}_{1}+{\eta}_{1}{x}_{3}^{s}}{{\gamma}_{1}^{u}}=\frac{{\beta}_{1}+{\eta}_{1}{\beta}_{3}/{\gamma}_{3}}{{\gamma}_{1}^{u}}$ represents a semistable limit cycle. If the initial condition is from the inside of the limit cycle, then the system will converge to the limit cycle; however, if the initial condition is from the outside of the limit cycle, then the system will diverge from the limit cycle. Such simulation results demonstrate the heterogeneity of the drug’s responses due to the nonlinearities in complex systems, where multiple inputs affect each output and the underpinning structure may include parallel, redundant, and feedback loop processes, it is likely that some cases will not respond to a single drug perturbation no matter how strong it is. As a result, innovative perturbation methods, such as finding a better target or combinatorial therapy, are necessary.
Simulation of effects of different drugs and a drug combination on N F−κ B pathway
In this article, the models and examples are selected such that they are mathematically tractable and important insights can be obtained, and we can verify the theoretical results with simulation results. For largescale networks and multiple drugs/drug targets, the proposed model is still applicable; however, analytical results may not be attainable even for this simplistic model. In that case, simulations can be carried out casebycase. To illustrate this point of view, we carried out a simulation study of the N F−κ B pathway under two different drugs and each drug with different drug targets.
Conclusions and future work
This article provides systematic mathematical analysis and dynamical modeling of drug effect in the GRN context, where a drug functions as a control input to reduce the elevated target gene expression level. A hybrid systems model is proposed to study the dynamics of the underlying regulatory network under drug perturbation. Drug pharmacology information is incorporated into drug therapeutic response modeling to demonstrate the significant difference in drug effect for different dosing regimens. Considering the complicated nature of gene regulation, this study is a small step towards quantitative modeling of therapeutic effect. We have kept the examples mathematically tractable so that valuable insights and reasonable predictions can be obtained from theoretical analysis.
Compared to our previous work [22], where drug effect was only studied for a specific PK profile (drug concentration only has exponential decay stage) when the drug is targeted to a single gene, three major extensions are provided in this article: (i) we provide analytical results of drug effect under a very general PK profile, where three stages of drug concentration change (increase, equilibrium, and decrease) are considered; (ii) the proposed methodology is applied to interactive genes in a GRN context, with detailed analytical derivations for both onegene and twogene cases; and (iii) we perform extensive simulations for a more complicated GRN setting and explain several interesting observations due to multiple feedback loops and the existence of limit cycles.
It is expected that the theoretical framework proposed in this article, when correlated to real biological networks, can help improve drug development productivity and make drug discovery more systematic. During such process, cross disciplinary effort is indispensable. For example, application of such a framework will require experiments designed to elucidate model parameters, such as protein concentration levels and synthesis and degradation speeds. While some parameters may be relatively easy to obtain, others may be difficult to get based on current techniques and model simplification may be necessary; nonetheless, the basic hybrid systems model and the conclusions drawn from it, such as the nature of DERs and the role of limit cycles, will remain valid, only their particular forms being changed to represent experimental instantiation of the model.
Appendix
Definition of variables
x _{1}  N F − κ B 
x _{2}  I κ B α 
x _{3}  I κ B α : N F − κ B 
x _{4}  N F − κ B_{ n } 
x _{5}  I κ B α _{ n } 
x _{6}  I κ B α_{ n }:N F − κ B_{ n } 
x _{7}  IKK 
x _{8}  I K K:I κ B α 
x _{9}  I K K:I κ B α:N F − κ B 
Parameter values
Parameter  Reaction type  Biochemical reaction  Value  Unit 

a _{4}  Complex formation  N F − κ B + I κ B α → N F − κ B : I κ B α  30  μ M^{−1}min^{−1} 
a _{7}  Complex formation  N F − κ B : I κ B α + I K K → N F − κ B : I κ B α : I K K  11.1  μ M^{−1}min^{−1} 
a _{1}  Complex formation  I κ B α + I K K → I κ B α : I K K  1.38  μ M^{−1}min^{−1} 
d _{4}  Dissociation  N F − κ B + I κ B α ← N F − κ B : I κ B α  0.03  min^{1} 
D _{1}  Dissociation  N F − κ B : I κ B α + I K K ← N F − κ B : I κ B α : I K K  0.075  min^{1} 
D _{1}  Dissociation  I κ B α + I K K ← I κ B α : I K K  0.075  min^{1} 
d e g _{1}  Degradation  I κ B α → 0  0.006  min^{1} 
d e g _{4}  Degradation  N F − κ B : I κ B α → N F − κ B  0.0013  min^{1} 
k _{01}  Transport  N F − κ B n → N F − κ B  0.0048  min^{1} 
t p _{2}  Transport  I κ B α n → I κ B α  0.025  min^{1} 
k _{2}  Transport  N F − κ B n : I κ B α n → N F − κ B : I κ B α  0.84  min^{1} 
k _{1}  Transport  N F − κ B → N F − κ B n  5.4  min^{1} 
t p _{1}  Transport  I κ B α → I κ B α n  0.05  min^{1} 
τ  Synthesis (delay)  N F − κ B n → N F − κ B n + I κ B α  40  min 
k _{02}  Inactivation  I K K → 0  0.002  min^{1} 
r_{4}+d_{4}  Catalyzed degradation  N F − κ B : I κ B α : I K K → N F − κ B + I K K  11.1  min^{1} 
r _{1}  Catalyzed degradation  I κ B α : I K K → I K K  2.22  min^{1} 
s _{synthesis}  Synthesis  N F − κ B n → N F − κ B n + I κ B α  0.24  min^{1} 
ODE model of the N F − κ B pathway
Declarations
Acknowledgements
This study was supported in part by the National Cancer Institute (2 R25CA09030106) and the National Science Foundation (NSF1238918).
Authors’ Affiliations
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