\chapter{Introduction: Robustness in metabolic systems}
Recent advancements in high-throughput biotechnology enable, and in fact necessitate, matching developments in the field of biology. Specifically, studies in genomics, proteomics, and metabolomics require comparable methods for data mining in order to handle vast volumes of biological data that can otherwise prove unwieldy. Furthermore, recognizing the opportunity to use this wealth of data to "understand biology at the system level" \cite{kitano}, '-omic' technology has also led to the emergence of "systems biology", which is focused on the abstraction of data, or modeling \cite{quo-c}.
As an engineer-in-training, it is my intuition that "control" is integral to a system-level understanding of biology. More precisely, the premise of control in biology, in particular systems biology, is that robust behaviors in biological systems resembles stabilized dynamics in engineering systems under control. Consequently, control theory is essential to, and should complement, the study of systems biology.
A number of opportunities and challenges may emerge from systems biology for control theory, which can be broadly classified in four categories \cite{sontag}. They are: (a) the role of control and signal processing techniques in bioinstrumentation, (b) the use of existing techniques in well-developed areas of control theory to analyze problems of interest to biologists, (c) the abstraction of new ideas for control engineering from biology, and (d) the formulation of new theoretical problems in control theory.
This dissertation is focused on the second category of problems. In particular, using lipidomic data on a metabolic pathway that is reported to be highly regulated in nature, a comparator is developed and applied to model the differences between treated and wild type cells, under the assumption that treated cells approach the same steady-state as the wild type in terms of intracellular lipid amounts. Thus, in this case, the effect of the applied comparator is to predict pathway feedback that may be responsible to maintain pathway stability under a specific treatment condition.
\section{Mechanisms of metabolic robustness: emergence and design}
Biological robustness refers to the persistence of living organisms to maintain certain traits or behaviors under changing, and often times unfavorable, conditions \cite{kitano-c,lesne}. Specifically, phenotypic robustness refers to the ability of living organisms to adapt, reproduce, and evolve to changes in the environment over time as a result of genetic mutation, while component robustness refers to the persistence of metabolic function in living organisms under dynamic and relatively short-lived conditions in the micro-environment. The distinction between these sub-definitions occurs in terms of scale in space and time, where phenotypic robustness spans the levels of cell to population over multiple generation times while component robustness is generally limited to the cell level within a single generation. Component robustness, or robustness in biochemical systems, is the focus of this section.
Robustness in biochemical systems is an increasingly popular research topic for two reasons. First, it is a ubiquitous property, i.e., it is readily observed, e.g., in glycolysis \cite{gehrmann}, energy metabolism \cite{cloutier}, protein synthesis \cite{bar}, mitochondrial apoptosis \cite{huber}, and chemotaxis \cite{baker,yi-a,hansen}. Second, with sufficient understanding of the underlying mechanisms, it may be manipulated to serve the prevailing needs of society, e.g., for disease treatment \cite{araujo,kitano-d,menolascina,szalay}, in particular drug development \cite{hellerstein,hopkins,kitano-a,luni,luni-a}. For the latter reason, the public literature on this topic to date comprises mostly studies on representative mechanisms of robustness such as hierarchy \cite{stelling}, modularity and redundancy \cite{lorenz,min}, stochasticity \cite{rao}, connectivity (and topology) \cite{lesne}, and more recently, feedback (and feedforward) control \cite{goulian,wolkenhauer}. Yet, insofar as such mechanisms are well-documented, there is comparatively little question (and even fewer answers) on the conception of robustness; furthermore, on how underlying assumptions on conception may bias the choice of one apparent mechanism, or few combined, over others to explain and perhaps in future to enable the robustness property in biochemical systems. Although not often discussed explicitly, a review of the public literature suggests that robustness in biochemical systems may be attributed to one of two notions: emergence and design.
In the context of systems theory, 'emergence' refers to the 'coming out' of more complex behaviors from simpler parts, especially where such 'higher' properties cannot be reasonably deduced from its 'lower' component interactions (see Bedau \cite{bedau} for a more rigorous, and philosophical, treatment as well as the difference between strong and weak emergence). In other words, emergence is 'bottom-up'. In many instances, emergence is synonymous with synergy, where the oft-encountered phrase "the whole is greater than the sum of its parts" is an acceptable though simplified description of the phenomena. From the aforementioned list, mechanisms of robustness in biochemical systems that are implied to arise as a result of emergence include hierarchy, modularity and redundancy, stochasticity, and connectivity and topology.
On the other hand, 'design' refers to the specification of a particular construct, or set of constructs, to accomplish specific objectives under certain constraints \cite{savageau}. In a well-designed system, various components are integrated to perform a well-defined function. In other words, design is 'top-down'. The apparently seamless combination of parts to serve a single purpose in a well-designed system has sparked a lively discussion of irreducible complexity throughout the years \cite{behe,miller}, which is also implicated in the debate on evolution versus intelligent design. The point being that robustness of biochemical systems by 'design' implies, or in some perspectives necessitates, the existence of a designer (an argument better left for the interested reader to explore separately) or an almost incredible confluence of events. To reconcile this rather awkward implication with mainstream support for the theory of evolution, it is worthwhile to consider that perhaps 'welding points' in robust biochemical systems may have 'eroded' over time such that prior steps in evolution are not detectable, which makes robustness by design the best explanation. As a recently proposed mechanism of robustness, feedback (and feedforward) control may be attributed to design.
Here, the question of whether the property of robustness in biochemical systems is conceived from emergence or design is a basis to discuss the strength and implications of a decision to pursue one resultant mechanism, or few combined, over others to explain the phenomenon, not a mirror of the debate on evolution versus intelligent design. Although not often acknowledged, the starting point and subsequent path of research on robustness in biochemical systems is naturally affected by the assumption, indeed acceptance, of either emergence or design as its basis.
What follows in this section is a survey of the robustness property in biochemical systems in terms of the aforementioned mechanisms, i.e., hierarchy, modularity and redundancy, stochasticity, connectivity and topology, and feedback (and feedforward) control; their assumed basis in emergence or design, and theoretical tradeoffs in implementing these mechanisms leading to the intriguing 'robust yet fragile' feature of such systems \cite{alderson,doyle,kitano-e,kitano-g}. To facilitate discussion, robustness of biochemical systems in this context shall refer to the ability of a cell signaling system to function reliably despite changes in effective molecular concentrations, i.e., to produce a desired output response if and only if there is an appropriate input stimulus. Finally, based on knowledge of these representative mechanisms, the future of research in biochemical robustness and its applications is discussed.
\subsection{Hierarchy}
Hierarchy refers to the stratification of component molecules into different biochemical levels of organization, e.g., genes (nucleotides), enzymes (proteins), and metabolites (lipids and carbohydrates) \cite{tsankov}. Specifically, order is enforced in the hierarchy where 'higher-order' molecules govern the action of 'lower-order' molecules, e.g., genes code for enzymes that act in turn on metabolites. Furthermore, signals from higher-order molecules are amplified in lower-order levels, e.g., a single strand of messenger RNA can be translated repeatedly to produce multiple enzymes. This cascade effect in cell signaling systems also implies that it is easier for signal transmission to occur in one direction from higher-order to lower-order molecules than the opposite.
Based on these features of stratification, ordering, and cascade effect, hierarchy as a mechanism ensures system robustness in two ways. First, by effectively compartmentalizing the functional molecules, weaknesses and failures are contained within particular levels when they occur. Second, by enforcing order and directional bias, variations in amounts of lower-order molecules, which occur most frequently, are not as easily propagated to higher-order molecules. In addition, because of the amplification cascade, hierarchy can also aid in minimizing the cost of cell signaling where energy needed for transcription is higher than for translation. The bow-tie architecture observed in cell signaling systems illustrates these features of hierarchy as a mechanism of robustness in biochemical systems \cite{kitano-a}, where the knot in the middle represents higher-order molecules while the ends of the bow represents lower-order molecules.
At the same time, there are tradeoffs in implementing hierarchy as a mechanism of system robustness. First, in demarcating clearly stratified levels, there is an equivalent need to develop protocols to communicate between the corresponding levels \cite{csete}. For example, transcription and translation are protocols to facilitate communication between DNA, RNA, and enzymes. It follows that cell signaling, which traverses various biochemical levels, relies heavily on the efficacy of these protocols. Thus, while necessary, such protocols are points of weakness and fragility in the system. Second, because higher-order molecules govern lower-order molecules, additional fragility and risk is introduced to the system when perturbations can occur at the level of higher-order molecules, e.g., in virus infections where foreign DNA is introduced into the host cell and mistakenly replicated. Perturbations at this level are also more readily propagated and amplified because of the cascade effect. Third, also because of the cascade effect, sensitivity and signaling speed are lower because of an effective directional bias against signaling from lower-order to higher-order molecules. For instance in sensing, i.e., where signals are initiated by lower-order molecules, a higher stimulus threshold is required to generate an output response, e.g., in terms of sustained receptor activation over time or simultaneous activation of multiple receptors.
Hierarchy as a mechanism for robustness due to emergence may be supported by the endosymbiotic theory \cite{kitano-f}. On this basis, it follows that to establish hierarchy as the mechanism of robustness for a given biochemical system it is critical to (1) identify and establish order between groups of functionally dissimilar or at least non-interchangeable molecules, and (2) discern the molecular protocols for communication between levels.
\subsection{Modularity and redundancy}
Modularity refers to the autonomy of component molecules, usually in self-contained functional groups, to perform specific actions that contribute towards the overall system objective. The trivial definition is that an individual enzymatic reaction is a module in a series of reactions that accomplishes a specific metabolic function. Modules in a given system may also be ordered, i.e., hierarchical, but not necessarily so. Modularity alone does not guarantee robustness and must be complemented by redundancy \cite{min,wang}, i.e., many modules exist to perform the same functions in a system. Thus, taken together, these features of modularity and redundancy ensure robustness in biochemical systems.
Modularity and redundancy are similar to hierarchy but key differences exist. First, modularity resembles hierarchy in terms of autonomy but modules are not necessarily hierarchical. The feature of being 'self-contained' ensures that failed modules are functionally removed from the system and do not interfere with overall system performance. At the same time, redundancy ensures that there are other modules that accomplish the same function to fill the void. Second, redundancy differs from the cascade effect in that the extent of redundancy is the same for all modules whereas in the cascade effect, duplication occurs only for some components, which are also largely lower-order molecules. As a result, redundancy does not also facilitate signal amplification unlike the cascade effect.
There are also two main issues in implementing modularity and redundancy as a mechanism for robustness in biochemical systems. First, in maintaining redundancy, it is costly because resources must be dedicated to create more than one component to perform the same function. Second, to ensure modularity also implies separating defined modules, which may difficult to do in the crowded molecular environment of the cytoplasm. As a result, the fragility of this mechanism for robustness is that biochemically similar molecules from one functional module may interfere or compete with molecules in another, leading to unintended system performance. These issues may be resolved where if modularity can be defined and implemented efficiently, then modules that perform functions common to several systems may also be shared. The sharing of such common modules may then balance the additional costs required to maintain redundancy. In these considerations, functionality is the basis on which modularity and redundancy as a mechanism for robustness in biochemical systems may be attributed to emergence \cite{lorenz}. Thus, the challenge in research on modularity and redundancy as a mechanism for robustness in biochemical systems is to clearly define appropriate functional units while allowing for the possibility of duplication.
\subsection{Stochasticity}
Stochasticity in biochemical systems refers to the probabilistic nature of biochemical reactions as a result of the unpredictable motion of molecules \cite{rao,rocco}. As a mechanism for robustness, stochasticity depends on maintaining threshold concentrations of molecules so that rates of reaction in a biochemical system remain effective to accomplish system function despite low odds of collision for individual molecules. At the same time, the stochastic nature of biochemical reactions also minimizes the probability of 'rogue' reactions as a result of perturbation by biochemically similar but functionally different molecules, i.e., by decreasing sensitivity.
Because of its probabilistic nature, stochasticity as a mechanism for robustness in biochemical systems suggests some readily observable features. First, such systems must be relatively simple. The low odds of reaction implies that theoretically, there is a limit on the number of reactions for a biochemical system to be robust via stochasticity. Second, to increase the odds of reaction in an already crowded cytoplasm, requisite molecules may be sequestered to maintain optimal concentrations. This may manifest in terms of increased number of vesicles in the cytoplasm or extensive use of existing organelles for compartmentalization. Third, the 'induced fit' theory of enzymatic action suggests that enzyme specificity may be relaxed for reactions in such systems to ensure overall system function. So, enzymes involved in such robust systems by stochasticity may be less specific in substrate recognition compared to other enzymes. Last, cofactor binding or positive cooperativity in enzymatic action may also increase the odds of reaction, which is another empirical feature.
From these features, it follows that stochasticity as a mechanism for robustness in biochemical systems can be attributed to emergence from component enzymatic reactions. Interestingly, the probabilistic nature of biochemical reactions also confers a degree of fragility as a result of a few 'stiff' combinations of parameters \cite{daniels}, which may require other considerations to overcome, e.g., by including other structural elements \cite{ao}.
\subsection{Connectivity and topology}
Connectivity is a concept of graph theory that refers in general to the minimum number of nodes or edges to be removed before a network is disconnected. A more robust graph, or network, requires more network elements to be removed before it is disconnected. As a measure of robustness in graph theory, connectivity is also a mechanism of robustness in biochemical systems because of the nature of linkage in biochemical systems, e.g., gene, metabolic, and protein-protein interactions, specifically in terms of topology \cite{almaas,lesne,wunderlich}. In particular, the scale-free topology of biochemical systems is directly linked to robustness in biochemical systems \cite{aldana,balaji,sweetlove}.
At its core, connectivity in biochemical systems is measured in terms of the frequency of nodes that are associated with specific number of edges, i.e., degree distribution. In random networks, the degree distribution resembles a Poisson distribution that peaks at the average degree. This means that because the edges are placed randomly in a random network, most nodes have similar numbers of edges. However, in real networks, e.g., biochemical networks, the degree distribution resembles a power-law distribution that deviates significantly from the Poisson distribution. This means that a small number of nodes are connected to many nodes, i.e., hubs, while a large number of nodes are connected to fewer nodes.
Assuming an equal chance of failure for individual nodes, robustness in biochemical systems is ensured as a result of scale-free topology because in this topology there are significantly fewer hubs compared to other less connected nodes. Thus, the basis of robustness in biochemical systems is emergence from the nature of linkage, or topology, between component molecules. It follows that the effect of connectivity to ensure robustness is appreciable only in large systems or networks where the ratio of hubs to nodes is non-trivial, e.g., the World Wide Web, the Internet, and cells \cite{albert-a}. At the same time, while scale-free topology confers high error tolerance, i.e., robustness to high failure rates of nodes in general but is vulnerable to attacks, i.e., when a hub breaks down or is removed \cite{albert-b,jing}. It remains to be seen if other topologies, i.e., connectivity in terms of different degree distributions, may also facilitate robustness in biochemical systems.
\subsection{Feedback and feedforward control}
Feedback control, also known as 'closed-loop' control, refers to the use of current output to regulate future output \cite{bechhoefer,wolkenhauer-a}. To do this, sensor units sample the system output at the present time, which is relayed to actuator units to guide or regulate system output at the next time, thus 'closing the loop'. Although well developed in control theory, the concept of feedback control entered mainstream research only recently in the context of robustness in biochemical systems \cite{joslyn,sontag,wellstead}. This may be in part because of difficulties in reconciling its basis by (intelligent) design with support for the theory of evolution; the corollary is that there has been more attention on mechanisms of robustness based on emergence than on design as seen in the literature. Since then the concept of feedback control has been readily developed in the context of biochemical systems, e.g., in terms of concentration and buffering, \cite{schmitt,shinar}, homeostasis \cite{davis}, and gene regulation \cite{el-samad-b,shin}.
The feature of feedback control in biochemical systems is that sensor and actuator units are used to sample and incorporate the current output into the system dynamics to guide future output. This system architecture forms a closed loop between current and future outputs, which ensures robustness by guiding the output to converge to the desired state. Specifically, based on the difference between current output and the desired state from the sensor unit, the actuator unit regulates the system to generate future output that is closer to the desired state. This cycle terminates when the current output and desired state are the same. Thus, the effectiveness, and corresponding fragility, of feedback control as a mechanism for robustness depends on the design of sensor and actuator units \cite{kim}.
There are various different ways to implement feedback control in biochemical systems, e.g., proportional, integral, and derivative (PID) feedback in energy metabolism \cite{cloutier}, integral feedback in bacterial chemotaxis \cite{alon,yi}, and multiple feedback loops in E. coli tryptophan regulation \cite{bhartiya}. Consequently, it can be difficult to determine the exact form that is responsible for robustness in a given biochemical system. This issue arises because, from the perspective of design, the various forms of feedback control are sufficient but not necessary to ensure robustness. Nonetheless, in the research of feedback control as a mechanism of robustness in biochemical systems, this challenge can still be overcome by the process of elimination, i.e., to first assume a specific form of feedback control and then test for particular dynamic features that are associated with it \cite{ma}.
In addition to feedback control, feedforward control is also a form of control that ensures robustness in biochemical systems. In feedforward control, the input stimulus is not only received by the immediate system module, e.g., receptor, but is also relayed ahead to one or more subsequent modules. In other words, the input stimulus triggers a reaction in more than one module in the system. Such a mechanism is robust to ensure system output in response to input stimulus in case of intermediate module failure, amplifies the effects of the input stimuli, and also increases the response time of the system, e.g., in the use of a transcriptional factor to trigger the bacterial heat shock response \cite{el-samad}.
\subsection{Themes}
Emergence is 'bottom-up' while design is 'top-down'. In other words, robustness in biochemical systems by emergence or design underscores two contrasting approaches to research on this topic, in particular to model biochemical systems based on modularity and robustness \cite{bruggeman,stelling-a}. The bottom-up approach is appealing because model interactions, namely enzymatic reactions, can be specified from traditional biological knowledge. However, in this approach it is also necessary to identify a large number of kinetic parameters. In most cases, it is possible to estimate but not measure these parameters because of data issues. Even so, parameter estimation is not without its challenges. The top-down approach requires only that a performance objective for the system be specified, namely robustness, so that accuracy of kinetic parameters is not as critical. However, at the present time, the concept of biochemical robustness by design is difficult to reconcile with mainstream support for emergence. So, an interim solution to spur progress on this topic may be to combine top-down and bottom-up approaches so that problems of parameter estimation may be alleviated without the loss of biological knowledge \cite{van-riel}.
At the same time, models of biochemical robustness by design may also play a role to elucidate molecular mechanisms in robust biochemical systems. The logical premise of this application is that because robustness is observed in a system, then an appropriate representation (model) of that system is also robust \cite{morohashi,stelling-a}. For example, feedback control may be simulated by comparing the response of a robust system under perturbation to a baseline (wild type) response \cite{quo}. Then, the resulting feedback may be analyzed to yield insight into the contribution of, or interplay between, component molecules to ensure robustness in the biochemical system. Such an application is an example that illustrates how top-down approaches may be combined with bottom-up approaches to model robustness in biochemical systems.
A persistent theme in research on biochemical robustness is the development of more rigorous definitions of robustness in the context of biochemical systems, e.g., in terms of graphical representations of structure in biochemical systems \cite{goldstein}, model reduction \cite{radulescu}, and mathematical formulation towards a formal theory \cite{kitano-b}. Although robustness is a ubiquitous property, current studies on this topic tend to be reported in different, often qualitative, terms \cite{larhlimi,lesne-a}. As a result, it is not easy to compare study findings to uncover deep, structural commonalities between robustness in different systems. Thus, the development of a standard, possibly analytical, expression for biological robustness is a key challenge, which will surely contribute to the discovery of fundamental principles for robustness.
Finally, robustness is a common, arguably definitive, property of biochemical systems that is manifest and achieved via different mechanisms. In practice it is generally the result of a combination of these mechanisms. For example, in the context of cell signaling, such combinations ensure robustness where some mechanisms may favor initial signal activation, e.g., modularity and redundancy, while others may favor continued signal transmission, e.g., hierarchy. Hence, research on these various mechanisms can be rallied to a common objective, i.e., to elucidate the origin(s), execution \cite{ross}, and rationale \cite{celani,el-samad} of robustness in biochemical systems. However, in both theory and practice, the process to integrate different mechanisms in a biochemical system to achieve robustness is not yet clear and also raises other intriguing questions, e.g., on the controllability of biochemical networks \cite{bianchi,kwon,layek,lombardi}. Thus, the question of how different mechanisms are, or can be, integrated to ensure robustness in biochemical systems will surely be a focus for continuing research.
\clearpage
\section{Metabolic system dynamics and control}
\subsection{Reaction kinetics}
The dynamics of metabolic systems (or pathways) is primarily defined in terms of biochemical, or enzymatic, reaction kinetics. For example, generalized mass action (GMA) and flux balance analysis (FBA) are common approaches to quantify enzymatic reaction kinetics based on the underlying principle of conservation of mass. In engineering terms, these methods provide the equations of motion to describe metabolic system dynamics in terms of enzymatic reactions. They are written in the time domain, and take the form of ordinary differential equations (ODEs):
\begin{equation}
\dot{\vec{x}}=f(\vec{x}(t),t)
\label{eqn:gen_dynamics}
\end{equation}
where $\vec{x}$ represents the variables of interest generally in terms of metabolite amount (concentration) or flux, and $f$ is a function of metabolite amounts over time $t$, which describes the rate of change of these quantities based on an understanding of the enzymatic reactions between said pathway metabolites.
\subsubsection{Generalized mass action}
In generalized mass action (GMA), the metabolic system dynamics is described in terms of metabolite amount (concentration). The rate of change of metabolites is quantified based on the \emph{law of mass action} to describe and predict enzymatic reactions in solution. Precisely, the law of mass action states that the rate of an elementary reaction, i.e., a reaction that proceeds in one step, is proportional to the product of the amounts (or concentrations) of participating molecules \cite{gulberg, lund}. More complex reactions that occur in multiple steps may be written as a series of elementary reactions.
Thus, consider the (elementary) reaction:
\begin{equation}
\mathrm{A}+\mathrm{B}\rightarrow\mathrm{C}
\end{equation}
where substrates $\mathrm{A}$, $\mathrm{B}$ react (irreversibly) to form product $\mathrm{C}$, and define $k$ as the (positive, forward) reaction rate constant. Then, the rate of change of these molecules are written as:
\begin{equation}
\frac{d\mathrm{A}}{dt}=\frac{d\mathrm{B}}{dt}=-\frac{d\mathrm{C}}{dt}=-k\mathrm{A}\mathrm{B}
\end{equation}
Although the law of mass action was developed to describe molecules in solution, it may also be generalized to describe interactions between large numbers of individuals, e.g., wildlife populations in ecosystems. In theory, GMA is a simple and intuitive model that approximates aggregate interactions between individuals in large populations. In other words, the (kinetic) probability of interaction is proportional to the number of individuals. However, in practice, the proportionality constant is not easily obtained. In this case, this means that it is experimentally difficult to measure, or even estimate, the required enzymatic reaction rate constants.
\subsubsection{Flux balance analysis}
In flux balance analysis (FBA), metabolic system dynamics is described in terms of steady-state flux between metabolites under the assumption that particular objectives, e.g., homeostasis, are optimized \cite{kauffman, orth, raman, varma}. This approach combines metabolite amount (concentration), rate of reaction, and stoichiometry to represent metabolic flux. Thus, FBA is a constraint-based approach that seeks to describe the rates at which metabolites (variables) are exchanged between various states (compartments), i.e., flux, in a given system.
Thus, consider the reaction:
\begin{equation}
\mathrm{A}+\mathrm{B}\leftrightarrow\mathrm{C}
\end{equation}
where substrates $\mathrm{A}$, $\mathrm{B}$ react (reversibly) to form product $\mathrm{C}$, and define $v_{forward}$, $v_{reverse}$ as the forward and reverse fluxes. Then, under the steady-state assumption, the rate of change of the balanced fluxes are written as:
\begin{equation}
\frac{d\vec{x}}{dt}=\mathcal{S}\cdot\vec{v}=\vec{0}
\end{equation}
where
\begin{equation}
\vec{x}=\left(
\begin{array}{c}
\mathrm{A} \\
\mathrm{B} \\
\mathrm{C}
\end{array} \right),\qquad
\mathcal{S}=\left[
\begin{array}{cc}
-1 & 1 \\
-1 & 1 \\
1 & -1
\end{array} \right],\qquad
\vec{v}=\left(
\begin{array}{c}
v_{forward} \\
v_{reverse}
\end{array} \right) \nonumber
\end{equation}
$\vec{x}$ is a vector of molecular amounts (or concentrations), and $\mathcal{S}$ is the (balanced) stoichiometric matrix of the system that describes $\vec{v}$, a vector of (forward and reverse) fluxes that are unknown quantities of interest. Furthermore, fluxes may also be specified as internal, as shown in this example, or external to a system. FBA has been reported in models of biochemical processes in a variety of organisms that includes \emph{E. coli} \cite{edwards}, yeast, and plants \cite{grafahrend}. In theory, even for dense networks, fluxes in systems at equilibrium may be balanced to give linear equations. However, in practice, it is difficult to verify that flux is a constant quantity in metabolic systems.
\subsection{Pathway regulation}
In existing approaches to model metabolic systems, control is not usually differentiated from dynamics. Existing approaches describe only metabolic system dynamics, i.e., motion, in terms of parameters that must be specified, either by measurement or estimation. Examples of parameters in such approaches include enzymatic reaction rate constants (mass action) \cite{gulberg, lund}, stoichiometric coefficients and flux rates (flux balance) \cite{varma}, control coefficients and elasticities (metabolic control analysis) \cite{fell, heinrich, kacser}, and kinetic orders (biochemical systems theory) \cite{savageau-a, savageau-b}. Because metabolic system dynamics are described only in terms of such parameters, the problem of modeling metabolic systems can be rephrased into a question of parameter estimation, or system identification in engineering terms.
Importantly, using such approaches, it is implied that certain properties of metabolic system dynamics, in particular robustness, can be fully explained with the associated parameters. However, to do so using -omic data, the number of parameters in recently published models of metabolic systems tends to be significantly larger than the number of variables. This leads to the problem of data over-fitting. Furthermore, despite the numbers, these parameters contain proportionately little information about the metabolic system dynamics, e.g., in terms of parameter sensitivity \cite{alvarez-vasequez,gupta}.
\subsubsection{Differentiating control from dynamics}
Especially in complex systems, control, i.e., how system dynamics is regulated, is essential to achieve system stability. Furthermore, it is necessary in reverse engineering stable metabolic systems that components of control be differentiated from components of dynamics and motion.
To illustrate the point, consider self-righting objects for instance (Figure 1). Precisely, the object geometry, a round bottom, is the basis for dynamics and motion, i.e., rotation about a point of stable equilibrium that represents the point of minimum gravitational potential energy. To ensure self-righting behavior, such objects usually contain ballast so that the center of mass is lower than its geometry suggests (assuming uniform density). This ensures that any tilting about the fixed point raises the center of mass and is followed by a subsequent return to equilibrium.
Thus, for a self-righting object, the ballast provides the necessary control mechanism to regulate the object dynamics and motion due to its geometry. If not for the recent construction of a self-righting object of uniform density, it may be easy to overlook the role of ballast as a control mechanism in such self-righting objects \cite{domokos-a,varkonyi-a}. \footnote{In addition, see \cite{domokos-b,varkonyi-b} for how such geometry is found naturally in the shell of the Indian Star Tortoise.}
\begin{figure}[p]%
\label{fig:self-right}%
\begin{center}
\includegraphics[width=4.5in]{./figures/self-righting_comp.png}%
\caption[Stability of motion in self-righting objects]
{Stability of motion in self-righting objects: \emph{(top)} ballast regulates motion in objects of non-uniform density; \emph{(bottom)} geometry is solely responsible for self-righting in an object of uniform density, the G\"{o}mb\"{o}c. (G\"{o}mb\"{o}c image taken from \textsf{http://www.thestar.com/article/269792})}%
\end{center}
\end{figure}
\subsubsection{Control theory in biology}
Cybernetics is the study of structural complexity in animal and machine that enables communication and control \cite{wiener}, and is closely related to control theory. In biology, this approach was first applied to study organ systems in physiology, e.g., in circulation \cite{guyton,pickering}, immunology \cite{kochel,tauber}, the central \cite{bianchi,cariani} and peripheral \cite{french,kiehn} nervous systems, and even bone remodeling \cite{frost}. Recently, this approach has been applied to study metabolic systems that include biochemical pathways in \emph{E. coli}, human hepatocytes and erythrocytes \cite{behre}, as well as in adult rat cardiomyocytes and human skeletal muscle \cite{guzun}. Nonetheless, while cybernetics emerged as the science of effective organization within systems, control theory was developed to influence and guide the dynamics of complex systems.
In general, control theory deals with the design of particular controllers to influence and guide the dynamics of complex systems. In biology, some examples include the design of optimal control strategies to manage wildlife populations \cite{runge,williams}, or to maintain desirable pest populations in agricultural systems \cite{bor-a,bor-b,bor-c}. Recently, control theory has also been applied at the level of metabolic systems, e.g., to regulate the amount of cells in a bioreactor using robust sliding mode control \cite{fossas}, or to regulate cell function in \emph{Escherichia coli} (E. coli) by constructing a bi-stable protein switch based on the switching properties of $lambda$ phage \cite{hasty}. In such applications for controller design, the key constraint is the difficulty involved in implementing the requisite actuators to effect control. In other words, the control of metabolic systems is limited by the extent to which biochemical mechanisms in these complex systems can be manipulated.
\clearpage
\section{Reverse engineering homeostasis}
Homeostatic pathways resemble engineering systems in that both types of systems are robust to disturbance within limit, i.e., system stability is maintained under small perturbations. Based on such observations, recent opinion express cautious optimism to reverse engineer the complexity and regulation of metabolic pathways by applying control theory in the context of systems biology \cite{csete,tomlin}. In particular for metabolic systems, the robustness of some cellular functions may be described in terms of control theory \cite{lauffenburger,stelling}. For even more practical applications, it is envisioned that a theory of biological robustness may be developed to help understand the robustness of metabolic drug responses so as to improve drug therapy \cite{kitano-a,kitano-b}. Thus, and for various other reasons, reverse engineering robustness of metabolic systems, as a result of homeostasis, has attracted interest in the community not only to understand how homeostatic pathways may have come to be, but also with the potential of influencing and guiding the dynamics of deviated pathways.
\subsection{Homeostasis as control}
Homeostasis is the process of control, i.e., the regulation of pathway dynamics, to maintain a stable condition in metabolic systems. While homeostasis is highly complex, homeostatic mechanisms can be, and have been, simplified in terms of control theory. To illustrate the point, two examples of homeostasis in synaptic signaling and glycolysis are discussed in terms of open- and closed-loop control.
\subsubsection{Open-loop control in synaptic signaling}
Action potentials are propagated by the regulated flow of ions -- calcium (Ca$^{2+}$), sodium (Na$^{+}$) and potassium (K$^{+}$) -- into and out of cells via voltage- and receptor-gated channels on the cell membrane (Figure 2). Across gaps between cells, or synapses, propagation of action potentials is mediated by neurotransmitter molecules, e.g., acetylcholine, dopamine, norepinephrine, and glutamate \cite{holz,pollard,soreq}.
Starting at rest, the post-synaptic membrane assumes a resting potential. An action potential arrives at the pre-synaptic membrane and depolarizes the membrane to open voltage-gated Ca$^{2+}$ channels. Ca$^{2+}$ influx into the cell triggers exocytosis of small synaptic vesicles that contain neurotransmitters. These molecules couple with ion channels, e.g., Na$^{+}$ channels, and other receptors that activate secondary messengers in the post-synaptic membrane. Ligand-receptor binding at the post-synaptic membrane depolarizes the membrane to generate an action potential. As the action potential at the post-synaptic membrane travels away from the synapse, its resting potential is restored.
To regulate synaptic signaling, neurotransmitter receptors are also present on the pre-synaptic membrane that may either inhibit or enhance exocytosis of synaptic vesicles. At the same time, released neurotransmitters may be $(a)$ (re-)taken up by transport proteins on the pre-synaptic terminal membrane, e.g., dopamine, norepinephrine, and glutamate; $(b)$ degraded, e.g., acetylcholine, or $(c)$ taken up by neighboring glial cells, e.g., glutamate. Synaptic vesicles are recycled at the post-synaptic membrane by endocytosis.
Membrane potentials at the pre- and post-synaptic cells can be measured in terms of voltage. In this system, pre-synaptic membrane potential is the input and post-synaptic membrane potential is the output. The post-synaptic membrane potential depends on neurotransmitters, released at the pre-synaptic membrane, which diffuse across the synapse to bind to receptors on the post-synaptic membrane. The post-synaptic membrane potential does not affect the pre-synaptic membrane potential. Thus, synaptic signaling is regulated based on open-loop control, where the output does not inform the input.
\subsubsection{Closed-loop control in glycolysis}
Glycolysis is the primary pathway that breaks down glucose to synthesize adenosine triphosphate (ATP), the energy currency of the cell. The enzyme phosphofructokinase (PFK) is a key regulator of this pathway (Figure 3). In particular, PFK catalyzes the first committed step in glycolysis, i.e., the irreversible conversion of fructose-6-phosphate (F6P) into fructose-1,6,-bisphosphate (FBP) \cite{berg,paricharttanakul}.
At high levels of ATP, i.e., high energy levels in the cell, ATP allosterically inhibits the enzyme PFK by lowering the binding affinity of PFK for its substrate F6P at the catalytic site. Where energy (in the form of ATP) is spent, i.e., at low energy levels in the cell, ATP is converted to adenosine monophosphate (AMP). AMP reverses the inhibition of ATP on the enzyme PFK. In other words, AMP enhances the activity of the enzyme PFK to increase glycolysis, so as to increase the production of ATP.
Amounts of intracellular F6P and ATP/AMP can be measured. In this system, F6P is the input and ATP/AMP are the output. The production of ATP/AMP depends on glycolysis, where PFK catalyzes the first committed step of the pathway. Critically, PFK activity is moderated by ATP/AMP. Thus, glycolysis is regulated based on closed-loop control, where the output informs the input.
\begin{figure}[p]%
\begin{center}
\includegraphics[width=5in]{./figures/neuron.png}%
\caption[Open-loop control in synaptic signaling]
{Synaptic transmission: an example of homeostasis in biochemical processes using open-loop control \-- [1] Ca$^{2+}$ enters to trigger [2] exocytosis and release neurotransmitters (NT) that [3] bind to ion channels, e.g., Na$^{+}$, and other receptors in the post-synaptic terminal, [4] synaptic vesicles are recycled by endocytosis; released neurotransmitters may be [a] (re-)taken up by transport proteins, [b] degraded, [c] taken up by neighboring glial cells. (Image modified from source: Holz and Fisher \cite{holz} \copyright 1999 American Society for Neurochemistry, National Center for Biotechnology Information (NCBI) Bookshelf)}%
\end{center}
\label{fig:neuron}%
\end{figure}
% insert figure of synaptic signaling
\begin{figure}[p]%
\begin{center}
\includegraphics[width=5in]{./figures/pfkinase.png}%
\caption[Closed-loop control in cellular energy metabolism]
{Structure of phosphofructokinase (PFK): this allosteric enzyme is a tetramer of four identical subunits and is the key regulator of glycolysis. PFK catalyzes the conversion of fructose-6-phosphate (F6P) to fructose-1,6,-bisphosphate (FBP). Adenosine triphosphate (ATP) inhibits, while adenosine monophosphate (AMP) enhances, PFK enzymatic activity by binding at the allosteric sites. (Image source: Berg \emph{et al} \cite{berg} \copyright 2002 W. H. Freeman and Company, National Center for Biotechnology Information (NCBI) Bookshelf)}%
\end{center}
\label{fig:pfkinase}%
\end{figure}
\subsubsection{A theory of stability}
To aid the study of robustness in metabolic systems, a theory of stability is undoubtedly helpful to further reduce the complexity of homeostasis in terms of underlying principles. In particular, the Lyapunov theory of stability \cite{lasalle, lyapunov} is especially useful because it provides: (a) an analytical definition of stability, and (b) a method of controller design that ensures system stability and consequent robustness. In this dissertation, this method of controller design, also known as the direct method of Lyapunov, is used to develop and apply a comparator model to infer potential feedback in a highly regulated metabolic pathway.
For the time-invariant case, the Lyapunov theory of stability is as follows:
\begin{lyap}
Let $\vec{x} = \vec{0}$ be an equlibrium point for $\dot{\vec{x}}(t) = f(\vec{x}(t))$, where $\vec{x}(0) = \vec{0}$, $x \in \Re^n$, $f$ Lipschitz continuous in $\mathcal{D} \subset \Re^n$, and $\mathcal{D}$ contains the origin. Suppose that $V(\vec{x}(t)) \in \mathbf{C}^1$ positive definite in $\mathcal{D}$ such that $\dot{V}(\vec{x}(t)) \leq 0$, $\forall \vec{x}(t) \in \mathcal{D}$, then the equilibrium point is stable. If $\dot{V}(\vec{x}(t)) < 0$, $\forall \vec{x}(t) \in \mathcal{D} \backslash \vec{0}$, then the equilibrium point is asymptotically stable.
\end{lyap}
A major outcome of the Lyapunov theory of stability is the provision of \emph{sufficient} conditions to determine the stability of the origin, e.g., (0, 0) in 2-dimensional space, of complex systems. By specifying a candidate function, i.e., the candidate stability function (or \emph{Lyapunov} function), that satisfies these conditions, a controller may be designed that ensures system stability without the need to solve accompanying differential equations that describe the system dynamics. This is known as the Lyapunov direct method, or the second method of Lyapunov. For specific cases, the definition of candidate Lyapunov functions is left to the user.
The Lyapunov direct method is commonly used to design stabilizing controllers in dynamic systems. To do this, the control objective can be posed as a problem of stability of motion, where the origin represents desired (stable) steady-state dynamics. Then, a controller or steering vector $\vec{u}(t)$ may be specified with respect to the plant, so as to guide the system dynamics towards the origin. The key is to determine the dynamics of the controller based on the choice of an appropriate candidate Lyapunov function. In these cases, where controllers need not be unique, the choice of one controller over another may also be informed by other practical considerations, e.g., the cost of implementation.
In terms of regulation in metabolic systems, homeostasis may be interpreted as a steering vector in a complex system. At the same time, because the Lyapunov theory of stability provides only \emph{sufficient} but not \emph{necessary} conditions, the uniqueness of potential mechanisms that may be responsible for homeostasis in metabolic pathways requires additional experimental evidence to verify. When pathway redundancy in metabolic systems is also considered, it is likely that multiple modes of control, i.e., different enzymatic reactions, may exist to serve the same purpose.
\subsection{Classical and modern control approaches}
Classical control approaches deal mainly with single-input/single-output (SISO) systems in the frequency domain. The transfer function, i.e., function of the (scalar) system output in terms of the input, is computed from observations of frequency responses. Then based on the transfer function, system stability and performance is analyzed in terms of the roots of the characteristic equation, or 'poles' and 'zeros', using graphical methods. Feedback as a result of classical control is specified in terms of the system output, i.e., \emph{closing the loop} based on open-loop behavior. Common modes of feedback in classical control are described in terms of proportional, integral, and derivative (PID) control.
In general, current models of homeostasis in metabolic systems are developed in terms of classical control and deal only with relatively simple cell behaviors that are well-studied in the literature. This is because the key issue in applying control theory to model homeostasis in metabolic systems is the difficulty in observing pathway dynamics with sufficient temporal resolution, i.e., to measure metabolite amounts quickly enough. Classical control approaches, developed to work with SISO systems, involve less variables and parameters that require less data for abstraction.
In addition, because homeostatic mechanisms that may be responsible for control in metabolic systems are difficult to verify experimentally, current studies on homeostasis in metabolic systems are also focused on cases where extensive knowledge of the relevant biology is available \emph{a priori}. Furthermore, in many cases, public literature is the only feasible resource to support any reasonable speculation on potential homeostatic mechanisms.
For example, bacterial chemotaxis in \emph{Escherichia coli} is an instance of relatively simple homeostatic behavior that may be described in terms of classical control. Exact adaptation in bacterial chemotaxis is observed to be robust and not affected by changes in protein levels \cite{alon,barkai}. Furthermore, such behavior may be described as the result of integral control that ensures convergence to steady state without error \cite{yi}. Using a classical control approach, details of the relevant transfer function can also be illustrated using dynamic input/output measurements, which provides insight into possible mechanisms that enable the robust behavior \cite{shimizu}.
A second example deals with the regulation of heat shock response, also in \emph{Escherichia coli}, which may be simplified as a feedback model for analysis using a number of potential feedback designs to study the costs and benefits of mounting the response with specific homeostatic mechanisms \cite{el-samad}. These examples illustrate classical control approaches to model relatively simple homeostatic behavior that involves analyzing single loops in a closed system.
Other examples of classical control models of homeostasis involve analyzing multiple loops in closed systems, e.g., tryptophan regulation and energy metabolism. The tryptophan system in \emph{Escherichia coli} may be considered as three processes in series, i.e., transcription, translation, and synthesis, that involve multiple feedback loops \cite{bhartiya,venkatesh}. Energy metabolism in terms of glucose and ATP is also complex behavior that may be described in terms of closed-loop regulation by various subsystems \cite{cloutier}. Moving on from single-variable analysis, a recent study of the adaptive response in membrane channels is based on using two variables, instead of one, to describe the dynamics of a 3-state model of membrane channel kinetics \cite{friedlander}.
Thus, given single-variable data, classical control approaches are adequate to analyze and model simplified SISO models of metabolic systems. However, for metabolic systems, it is still difficult to interpret system dynamics and feedback in the frequency domain. Where high-throughput -omic technology is available such that there is sufficient data to support multi-input/multi-output (MIMO) models of metabolic systems, modern control approaches may prove to be more suitable to reverse engineer homeostasis in metabolic systems.
Given multi-variable data, modern control approaches were developed to handle MIMO systems, i.e., in terms of the dynamics of multiple internal states of a system (or \emph{state-space}). To do this, the state-space is represented in terms of the rate of change of said states in the time-domain. Then, system stability may be analyzed in terms of stability theorems, usually based on optimizing specific stability functions (as in the case of the Lyapunov theory of stability). As a result of this approach, system feedback can be specified in terms of the state-space, which may prove useful to identify key controllable states. Common modes of feedback in modern control include adaptive, optimal, and robust control. A detailed comparison between classical and modern control approaches is shown in Table 1.
Thus, depending on the research objective, a modern control approach may prove to be more suitable than a classical control approach to reverse engineer homeostasis in metabolic systems using high-throughput -omic data, particularly in terms of:
\begin{itemize}
\item \textbf{handling many variables simultaneously} \-- modern control approaches handle vector systems using a state-space description but classical control approaches handle scalar systems,
\item \textbf{representing time domain dynamics} \-- time rate of change of pathway dynamics is intuitive to biologists but transfer function 'poles' and 'zeros' in the frequency domain do not correspond directly to biological variables,
\item \textbf{interpreting system feedback} \-- state-space feedback subject only to considerations for system stability in modern control are open to interpretation but PID feedback effectively constrains how pathway metabolites may interact in a homeostatic system.
\end{itemize}
\clearpage
\begin{table}[p]
\caption{Comparison of classical vs. modern control approaches}
\begin{center}
\begin{tabular}{lp{5.8cm}p{5.8cm}}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
& Classical control & Modern control\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Data & Single variable & High-throughput (omic)\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Variables & Input/output (scalar) & Internal states (vector)\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Systems & Single-input/single-output & Multi-input/multi-output\\
\noalign{\smallskip}
& (SISO) & (MIMO)\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Representation & Frequency domain & Time domain\\
\noalign{\smallskip}
& Transfer functions & State-space models\\
\noalign{\smallskip}
& Frequency signals, e.g., impulse/step, are not intuitive and usually difficult to implement/manipulate in biology & Dynamics based on first principles, e.g., biochemical kinetics, are intuitive and appeal to current understanding\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Stability & Roots of the characteristic equation, i.e., 'poles' and 'zeros' of the transfer function & Lyapunov stability theory, i.e., generalized concept of energy\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Methods & Graphical analysis in the complex plane, e.g., root-locus & Numerical/computational analysis, e.g., matrix algebra\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\noalign{\smallskip}
Feedback & Proportional, integral, derivative (PID) control, in terms of system output & Adaptive, robust, optimal control, in terms of internal states of a system\\
\noalign{\smallskip}
\noalign{\smallskip}
\hline
\end{tabular}
\label{table0}
\end{center}
\begin{flushleft}
\end{flushleft}
\end{table}
\clearpage
\section{Thesis statement}
In this present age of -omic biotechnology, modern control approaches are useful to model complex behaviors in biological systems, such as homeostasis in metabolic pathways, using high-throughput data. Although they can be useful in other ways, classical control approaches undermine the vast volumes, and wealth, of -omic data that are increasingly commonplace in biology. Thus, using -omic data, a modern control approach is suitable to reverse engineer homeostasis in metabolic systems.
In the rest of this dissertation, from an engineering and a modern control perspective, I describe the development and application of a comparator to reverse engineer homeostasis in a highly regulated metabolic pathway using mass spectrometry lipidomic time-series data. The metabolic pathway in question is the C16:0 sphingolipid \emph{de novo} biosynthesis pathway, studied in a human embryonic kidney (HEK) cell line, where the effect of single-gene overexpression on homeostasis in sphingolipid \emph{de novo} biosynthesis is of interest. Precisely, in these single-gene overexpressed (or treated) cells, the gene that codes for serine palmitoyl-transferase (SPT), an enzyme in the \emph{de novo} biosynthesis pathway, is overexpressed. The treated cells are assumed to approach the same steady-state dynamics as the wild type over time. Consequently, the outcome of the comparator is to: (a) model the differences between the treated and wild type cells, and (b) predict feedback in treated cells as a result of homeostasis.
In Chapter 2, I review the relevant sphingolipid biology and biomedical significance of sphingolipid \emph{de novo} biosynthesis; and present experimental materials and methods as well as the development and application of the comparator model in terms of concept, assumptions, mathematical representation, and numerical implementation. In Chapter 3, I report the results of the comparator model in terms of \emph{in silico} simulation; verify and interpret these results from a biological perspective; and demonstrate the generality of the comparator model with respect to additional data (on C26:0 sphingolipids). In Chapter 4, I discuss the biomedical applications and limitations of the proposed comparator model, and suggest what will be needed in the experimental data for the proposed model to be applied more fruitfully in the fields of biology and biomedical engineering.
From a broader perspective, it is also the goal of this thesis research to contribute to "the use of existing techniques in well-developed areas of control theory to analyze problems of interest to biologists" \cite{sontag}. I strive to achieve this objective by proposing and developing the \emph{first} use and application of a comparator model as a viable, albeit presently crude, tool for analysis in traditional case/control studies in biology.
%\subsection{Specific aims}
%In this case study, I use model-reference adaptive control (MRAC) to study metabolic pathway regulation. In particular, the objective is to extract insight into \emph{de novo} sphingolipid biosynthesis regulation in \emph{serine palmitoyltransferase} (SPT) over-expressing human embryonic kidney (HEK) cells. The specific aims are to:
%\begin{itemize}
% \item compare the efficacy of MRAC, to simulate metabolic pathway dynamics, to a standard method of biochemical systems modeling using mass action alone, and
% \item demonstrate the use of MRAC to uncover novel and quantitative insight into \emph{de novo} sphingolipid biosynthesis.
%\end{itemize}
%Results show that the proposed MRAC approach:
%\begin{itemize}
% \item compares well with using mass action alone to simulate pathway dynamics, where overall goodness-of-fit with experimental data between the two approaches differs by only 5\% in terms of root-sum-square, and
% \item suggests possible regulation dynamics, where continuous-time feedback from particular sphingolipids and underlying interactions between several molecules may account for experimentally observed pathway dynamics.
%\end{itemize}