\chapter{Future work}
Using C16:0 sphingolipid \emph{de novo} biosynthesis as a case study, the concept, development, application, and verification of the proposed approach in using a comparator model to reverse engineer homeostasis in a highly regulated metabolic pathway have been discussed in the previous chapters. So far, the model results, i.e., predicted steady-state feedback to capture homeostatic pathway regulation, are encouraging and should warrant at least some further investigation in this area. This dissertation is an initial study of what will hopefully become a longer program in cellular control. In this chapter, I discuss limitations of the proposed comparator model and existing experimental data well as potential biomedical applications of the proposed approach.
\section{Model and data limitations}
\subsection{Modeling nonlinear pathway dynamics}
Using generalized mass action (GMA) to simplify system dynamics in this case study, a linear system of first-order ODEs is used to describe what could very possibly be nonlinear pathway dynamics. To account for nonlinear pathway dynamics, additional modeling parameters in terms of the plant and comparator dynamics could be written as follows:
\begin{eqnarray}
\dot{\vec{x}}(t) & = & \mathbf{A}\vec{x}(t) + \vec{b}[u(t) + \vec{\alpha}(t)^{T}\Phi(\vec{x}(t))]\\
u(t) & = & \vec{h}_x(t)^{T}\vec{x}(t) + h_r(t)r(t) - \hat{\vec{\alpha}}(t)^{T}\Phi(\vec{x}(t)) \\
\dot{\hat{\vec{\alpha}}}(t) & = & \Gamma_\alpha\Phi(\vec{x}(t))\vec{e}(t)^{T}\mathbf{P}\vec{b}
\end{eqnarray}
where $\Phi(\vec{x}(t))$ is a set of predefined nonlinear parameters, $\vec{\alpha}(t)$ is the true contribution of these parameters to the underlying pathway dynamics, and $\hat{\vec{\alpha}}(t)$ approximates the true contribution of these terms.
Particular nonlinear terms could be derived from the experimental data \cite{schmidt}, although such approaches for parameter estimation in dynamic nonlinear systems do not rely on any prior knowledge of the system. Instead, model 'building blocks' that are generic mathematical functions that are not specific to the pathway biochemistry are tested for goodness-of-fit to the experimental data. As a result, such approaches to estimating nonlinear parameters may not provide any biological basis to support the parameters. Nonetheless, although it is beyond the scope of this dissertation, such a "model-free", or "top-down", approach to parameter estimation may still be useful to uncover radically new descriptions of the dynamics of metabolic pathways in the long term by essentially rewriting the current laws of biochemistry.
At the same time, to avoid data over-fitting, i.e., where model parameters may be added indiscriminately for the sole purpose of reducing error between model results and experimental data, measures of goodness-of-fit such as the Akaike information criterion \cite{akaike,hurvich} may be useful to quantify the tradeoffs between precision and complexity of the model. For any proposed model to be of interest and utility to the field of biology at present, mathematical terms to model pathway dynamics must be informed and justified by an understanding of the underlying pathway biochemistry and dynamics.
\subsection{Exploring control models: additional data requirements}
The proposed comparator model is only a basic component of standard, as well as more advanced, controlled systems. In this case, its function is simple: to compute the difference between the \emph{plant} and \emph{reference} system outputs, which is feedback to the plant system so as to guide the plant to converge to the reference. However, in more advanced controllers, the comparator output is examined more carefully in significantly greater detail such that how the primary issue of interest lies in how the comparator output is, or may be, dealt with in terms of \emph{feedback}. Consequently, the objective in engineering control design is to specify the appropriate feedback to achieve the desired system response.
Still, improved biological data acquisition is required in order to better meet, and indeed embrace, the theoretical and practical challenges of "[using] existing techniques in well-developed areas of control theory to analyze problems of interest to biologists" \cite{sontag}. Representative modes of (feedback) control are summarized in Table 7, where key features of these various modes of control also suggest what experimental data may be useful to explore and integrate, in the broad sense, the proposed approach using control theory to reverse engineer homeostasis in biochemical systems more fruitfully. In other words, additional experimental data is needed to properly calibrate such models of homeostasis based on control theory. Regardless, to achieve any real success, modelers surely need to work more closely with experimentalists.
In particular, for example, because the key feature of adaptive control is that the plant system is time-varying, i.e., where state-space parameters change with time, this suggests one of two possible demands on experimental data: \emph{either} the pathway must be sampled more frequently, i.e., to gather time-series data, \emph{or} over a contiguous period of experiment, the plant system should be subject to a sequence of different experimental treatments and such that each treatment affects the pathway dynamics differently. In the former case, the contribution of more time-series data to modeling is to enable detection (and derivation) of changes in reaction rate constants in real-time, i.e., to improve state-space parameter estimation with significantly improved temporal resolution. In the latter case, changes in the plant system as a result of a sequence of different experimental treatments also requires, and results in, different parameters to describe pathway dynamics following each treatment. In both cases, the effect of these changes in experimental procedure or data acquisition is that the key feature of adaptive control can be satisfied, i.e., a time-varying system where the state-space parameters change with time can be properly defined.
For robust control, apart from a fully known system, a range of inputs is required for the estimation of controller parameters in order to establish limits of input perturbation where it can be claimed that pathway homeostasis may indeed be described as such. Of the representative control modes summarized in Table 7, the data requirements for robust control may be the easier to meet, where the key factor to be varied could simply be the magnitude of pathway input, e.g., precursor molecule for the metabolic pathway of interest. Even so, depending on the resolution of the measurement instrumentation, sufficiently different levels of pathway input that still lead to the same steady-state behavior may still be challenging to discern.
Another example: to apply optimal control as a model of homeostasis, where the desired system performance as well as system dynamics must be fully specified, substantial \emph{a priori} knowledge of the metabolic pathway or system of interest will be useful, if not absolutely necessary, to define these key features. Optimal control implies direction, either internal or external, where system performance is driven towards achieving some known objective. Under heavily-regulated conditions for externally-driven objectives, e.g., ensuring pre-defined levels of particular metabolites for specific bioengineering projects, optimal control could be a viable model of regulation (but not homeostasis) \cite{fossas,hasty}. On the other hand, it is challenging to define "desired" system performance for homeostasis in metabolic pathways without controversy, given that biological variance all but ensures a range of acceptable (and healthy) behaviors in the wild type. To do so will require conclusive evidence that deviation of plant system response necessarily leads to "destructive" outcomes that are detrimental to cell survival. Still, in such cases, data on cancerous tumor cells could provide some support to modeling homeostasis in metabolic pathways based on optimal control.
Finally, for proportional, integral, and derivative (PID) control to be an effective model of homeostasis in metabolic pathways, a fully known system and coherent set of inputs are required. At the same time however, because it can be directly applied to scalar representations of pathway systems, PID control may be advantageous over other control modes from a modeling perspective in that even single-variable measurements, still sufficiently resolved in time, may suffice to support model development. For instance, PID control (specifically integral control) is reported to be an adequate model of regulation in bacterial chemotaxis \cite{yi,yi-a}. Nonetheless, it should also be noted that where some terms of PID control may seem more abstract than others, e.g., how integral control is defined in bacterial chemotaxis, \emph{a priori} knowledge of the metabolic system goes a long way to illustrate how specific forms of PID control may be implemented via some of the known underlying molecular mechanisms.
\clearpage
\begin{landscape}
\begin{table}[p]
\caption{Representative control modes}
\begin{center}
\begin{tabular}{lp{4.2cm}p{5.4cm}p{6.6cm}}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Mode & Key feature(s) & Prior knowledge & Result\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Adaptive & Time-varying system & Not required & Real-time control response\\
\noalign{\smallskip}
\noalign{\smallskip}
& & & Guaranteed system stability\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Robust & Fully known system and known range of inputs & Some required, to determine limit of change in parameters & Pre-determined control response, i.e., does not react to unexpected disturbance\\
\noalign{\smallskip}
\noalign{\smallskip}
& & & Guaranteed system stability\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Optimal & Fully known system and desired performance objectives & Fully required, to specify objective function for optimization & Pre-determined control response, i.e., does not respond to unexpected disturbance\\
\noalign{\smallskip}
\noalign{\smallskip}
& & & Guaranteed system stability\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
PID & Fully known system and inputs & Some required, to tune control response & Pre-determined control response, i.e., does not respond to unexpected disturbance\\
\noalign{\smallskip}
\noalign{\smallskip}
& & & System stability may not be guaranteed\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\end{tabular}
\end{center}
\begin{flushleft}
\end{flushleft}
\end{table}
\end{landscape}
\clearpage
\section{Biomedical applications}
\subsection{Studying molecular mechanisms}
The predicted steady-state feedback indicates which specific metabolites may be involved in potential homeostatic pathway interactions, but does not suggest how these homeostatic interactions may be accomplished \emph{in vitro} in terms of observable molecular mechanisms, e.g., in terms of intracellular location, transport, or biochemistry of enzymatic reactions. More precisely, for instance in this case study, a particular molecular mechanism that may be of interest to biologists is how treated cells, i.e., SPT overexpressing cells, may sense that increased amounts of sphingolipid precursor PalCoA are present to determine if the balance of sphingolipid \emph{de novo} biosynthesis and turnover is indeed upset. Could this be accomplished at the molecular level via the rate at which extracellular palmitate is taken up by the cells and converted to PalCoA?
Specific to this case study, because intracellular location is critical to sphingolipid metabolism, molecular imaging techniques, e.g., using molecular beacons, may be the immediate methods of choice to first determine if particular sphingolipid metabolites are indeed co-localized so that subsequent enzymatic reactions could possibly occur and be detected. Ultimately, unraveling such mechanisms at the molecular levels will go a long way to lend substantial experimental support to the use of control theory to reverse engineer homeostasis in metabolic pathways.
\subsection{Impacting drug discovery and development}
By identifying specific pathway metabolites that may be involved in homeostatic pathway interactions based on predicted steady-state feedback, model results from the proposed approach may also contribute to the process of drug discovery and development, e.g., in terms of identifying key pathway metabolites that may be targeted biochemically. Drug therapy is a common treatment for chronic disease where homeostasis is disrupted but remains viable \cite{kitano-a}. However, drug-induced metabolic activity often conflict with homeostatic activities, leading to undesirable side effects. This problem affects drug discovery and development pipelines where drug candidates often fail in development because of latent toxicity.
The drug discovery and development process is a pipeline that involves different phases. In the discovery phase, disease targets and drug candidates are identified. \emph{In silico} methods currently used for target identification focus on algorithms for pattern matching sequences and motifs, functional annotation, and mining expression datasets \cite{leon}. These methods tend to identify singular targets, where less emphasis is placed on assessing the targets in the context of metabolic pathways. Drug candidates are subsequently designed and evaluated for potency towards these targets, commonly to knock out disease-specific pathways. In the development phase, drug candidates are tested for toxicity in animal and human trials. In current pipelines, the separation of drug discovery and development into distinct phases has led to more failures than successes because "with discovery now driven primarily by chemistry and high-throughput screening, the biological effects and, in particular, the toxicity of new compounds is largely not appreciated until a compound enters development" \cite{ulrich}. This results in a very low turnover rate of drug candidates from concept to clinical deployment.
The prevalent strategy to improve the current situation is to try and increase the number of drug candidates discovered such that the number of drugs that are safe for market is not affected significantly. However, this does not improve the turnover rate and critically, it does not alleviate the burden of costs involved in the process. The recent trend of corporate mergers and acquisitions in the pharmaceutical industry reflects the growing financial strain on drug discovery and development as a result. An alternative solution is to consider toxicity earlier upstream in the discovery phase instead of doing so only in the development phase.
Thus, the proposed approach using a comparator model to reverse engineer homeostasis in metabolic pathways provides an analytical framework to identify and interpret aberrant metabolic effects in terms of interference with the dynamics of homeostasis in healthy pathways. In particular, in pathways affected by disease or drug toxicity, the clinical symptoms can be viewed as the result of sub-par pathway feedback that leads the metabolic system to undesirable, and ultimately unstable, points of equilibrium. It follows that a viable strategy for drug intervention in such aberrant pathways may be to try and complement or leverage the action of these inferior "homeostatic" feedback, instead of trying to knock out the pathway entirely. Consequently for drug discovery and development, drug toxicity may be addressed by working with, not against, abnormal homeostatic activity in the context of diseased pathways.
\subsection{Analyzing case/control studies}
Finally, the proposed approach using a comparator model to reverse engineer homeostasis in metabolic pathways is a knowledge-based method, where the pathway dynamics of wild type cells is defined as a reference response for the treated cells to approach. In fact, this \emph{plant-reference} paradigm is motivated by conventional experimental design in the field of biology where effects of experimental treatment are commonly studied using stable cell lines in comparison with the wild type, i.e., traditional case/control studies. Naturally, in such studies, there is only a single difference as a result of experimental treatment, e.g., single-gene mutation, between the treated cells and wild type.
Where dynamics of the metabolic pathway(s) of interest do not evolve beyond the wild type dynamics, such a plant-reference framework is justified where treated cells can be assumed to follow the wild type behavior. Consequently, together with its demonstrated generality in this dissertation, the proposed approach using a comparator model from engineering control to reverse engineer homeostasis in metabolic pathways can be a useful research tool to complement the analysis of data from traditional case/control studies in the field of biology.
Furthermore, taking a broader perspective on this subject, in applied research clinical data from individual patients in times of health may be similarly used as a \emph{reference} to shortlist options for molecular intervention, e.g., drug therapy, in times of disease. Data collected from the patient in times of health may be used to build and calibrate a \emph{reference} system, while data from the same patient in times of disease may be used to estimate a \emph{plant} system. Thereafter, the clinical goal is guide the dynamics of the plant system to return to the dynamics of the reference system. In other words, each patient may (rightfully) provides her own "healthy" data as a reference response, which addresses the problem of biological variance in populations. Thus, if achieved, such a personalized approach to medicine stands in stark contrast to the use of population statistics to define standards of health for the individual patient.
Even remaining within the limits of population (public) health, a reference system could similarly be built and calibrated using data from healthy subjects to test novel \emph{intervention} treatments (as discussed in the previous Section 4.2.2). In other words, intervention therapy could be predicted \emph{in silico} in models of plant systems based on data from diseased patients. Ultimately, such an application is based on the clinical need to unravel the figurative knot of biochemical interactions as a result of increasingly common drug cocktail therapy to manage "lifestyle diseases", because in many cases it is no longer sufficient to merely predict the initial or immediate patient response to particular drug therapies. As a result of such chronic diseases, an increasing number of patients are already subject to specific dosing regimens which may interfere with additional prescribed treatments.