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Learning from the Steersman: A Natural History of Cybernetic Models Jamey D. Young* Department of Chemical and Biomolecular Engineering and Department of Molecular Physiology and Biophysics, Vanderbilt University, PMB 351604, Nashville, Tennessee 37235-1604, United States ABSTRACT: Cybernetic models rely on optimal control heuristics to predict the effects of metabolic regulation on the dynamics of biochemical reaction networks. Over the past 30+ years, this fertile paradigm has brought forth scores of research publications and has witnessed diverse applications ranging from bioprocess optimization and control to the redesign of cellular hosts through rational metabolic engineering. This review traces the historical development of the cybernetic modeling framework, beginning with its philosophical presuppositions and then following its progress from first-generation “lumped” cybernetic models to biochemically detailed second-generation models to recent “hybrid” cybernetic models that combine aspects of both first- and second-generation models. The broad and lasting influence of the cybernetic modeling approach is due in large part to the fact that it has remained true to its foundational principles while adapting over time to meet the changing demands of biotechnology research.
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INTRODUCTION Biological systems are characterized by their ability to grow and reproduce under diverse environmental conditions. Isolates of the archaebacterial genus Pyrodictium, for instance, exhibit optimal growth at temperatures near 105 °C in deep-sea hydrothermal vents, while the bacterium Thiobacillus ferrooxidans can tolerate acidities down to pH 1.0 in mine water runoff streams containing high levels of heavy-metal impurities.1 Even organisms that exist in comparatively mild habitats must cope with frequent cycles of feast and famine along with a constantly changing menu of available nutrients. This astonishing resiliency is owed to a knack for regulating their metabolic machinery in ways that ultimately promote survival. Metabolism, the term used to denote the totality of chemical reactions in an organism, is enabled by a vast collection of enzymes with highly specific catalytic functions. Metabolic regulation is responsible for controlling the amounts and catalytic activities of these enzymes. The rates of metabolic reactions are thus subject to rigorous control aimed at protecting the interests of the organism. Accounting for this metabolic regulation in a rational manner is the main challenge faced by mathematical modelers who aim to predict the outcome of biological experiments based on a quantitative representation of the underlying cellular processes. The traditional kinetic modeling approach involves postulating biochemical and regulatory mechanisms that determine the rate of each enzymatic reaction, which can then be translated into corresponding kinetic expressions. Unfortunately, this requires a detailed knowledge of regulatory interactions and leads to a rapidly expanding set of kinetic parameters. The cybernetic modeling approach, on the other hand, views the complete biological system from an evolutionary perspective. On the basis of a general understanding of the metabolic objectives that an organism must satisfy in order to grow and survive, cybernetic control laws are postulated to reflect putative regulatory programs. These control laws provide a surrogate description of unknown or incomplete regulatory details within the © XXXX American Chemical Society
cybernetic model and facilitate a minimalist representation of complex biochemical networks. Cybernetic control laws are identified by envisaging the cell as an economically efficient system. This idea derives from the observation that certain limited resources are required for the synthesis and activation of enzymes, which in turn catalyze necessary metabolic conversions. Presumably, the cell’s regulatory strategies have evolved in such a way as to ensure that said resources are allocated efficiently in meeting metabolic demands. Cybernetic models therefore depend upon concepts from automatic control theory to determine the regulatory policies that best meet the metabolic objectives of the organism and lead to optimal utilization of metabolic resources. Accounting for regulatory control actions in this manner reduces the complexity of kinetic formulations and leads to simulations that are stable and reliable over a wide range of possible phenotypes. In this contribution, I provide a historical perspective on the development of this theoretically rich and practically useful class of mathematical models, beginning with the foundational underpinnings of the cybernetic modeling approach and then tracing the progress from first-generation “lumped” cybernetic models to biochemically detailed second-generation models to “hybrid” cybernetic models that combine aspects of both first- and second-generation models. I provide a high-level view of the control architecture that defines each major subdivision in the cybernetic modeling taxonomy and emphasize the conceptual progression from one to the next rather than the mathematical details. For a more in-depth and comprehensive review of the cybernetic modeling approach and its applications, the reader is referred to the excellent review by Ramkrishna and Song.2 Special Issue: Doraiswami Ramkrishna Festschrift Received: April 8, 2015 Revised: July 4, 2015 Accepted: July 6, 2015
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PHILOSOPHICAL PRELIMINARIES The term “cybernetics” was coined in 1948 by the mathematician Norbert Wiener to describe what he envisioned as a unified science of “control and communication theory, whether in the machine or in the animal”.3 He speculated that biological science could draw upon concepts from the then-fledgling field of automatic control theory to better understand feedback processes and other regulatory phenomena that occur in biological systems. At the same time, he felt that insights from biology might inspire improved control systems for human-made devices. The word “cybernetics” comes from the Greek word κυβερνήτης, which means “steersman”. Wiener chose this word to reflect his notion that steering engines on ships were one of the earliest and best-developed forms of feedback control during his time. Above all else, Wiener’s key insight was in recognizing that living organisms are unique from other natural systems because of their dependence on feedback mechanisms, which have evolved over time to enable precise control of physiological functions. As a result, evolutionary principles are central to biological understanding and oftentimes supersede fundamental physical and chemical laws when interpreting the cause of an observed effect. A central theme of biological research, then, should involve the discovery and analysis of fundamental control laws that govern cell behavior. The evolutionary biologist Ernst Mayr would later echo these same ideas in defending the status of biology as an autonomous science. This issue took center stage in the 19th century when vitalists clashed with materialists over whether living organisms possess a “vital fluid” or “life force” that distinguishes biomatter from inert matter. Mayr4 points out that it is unnecessary to postulate a metaphysical “life force” to explain the seemingly goal-directed nature of biological systems because such behavior is easily rationalized from an evolutionary standpoint. In a sense, the genetic code represents a concrete replacement for the abstract notion of “life force” inasmuch as it endows living organisms with a capacity for storing historically acquired information that influences future behavior. Mayr often uses the word “teleonomic” to describe the apparently end-directed processes of biological systems. In contrast to the ancient Greek notion of teleology, teleonomic processes owe their goaldirectedness to the operation of a program, not to any preconception of the goal itself. The program to which Mayr refers is the genetic program, whose directives serve to regulate cellular processes in the interest of survival. Viewed from the foregoing perspective, it becomes apparent that a cybernetic treatment of biological systems is eminently within the realm of rational science. Building on the seminal observations of intellectual giants such as Wiener and Mayr, Ramkrishna5 introduced the concept of cybernetic modeling as a way to codify the teleonomic principles of biologial systems within a quantitative and predictive mathematical framework. Cybernetic models are dynamic models of metabolism comprising a set of ordinary differential equations that describe the time evolution of enzyme levels and metabolite concentrations within the cell and its external environment. The governing equations are derived by balancing the rates of formation, breakdown, accumulation, and depletion of all biochemical components represented in the model. The key challenge, as stated in the Introduction, is to express these rates in a manner that accounts for pertinent regulatory influences, especially those that control the synthesis and activity of metabolic enzymes. Cybernetic models rely on optimal
control heuristics in meeting this challenge and can often provide reliable approximations to the true system response even when the underlying regulatory mechanisms are incompletely understood. In what follows, we outline the historical development of cybernetic modeling concepts as they have been applied to various biological systems. We begin by reviewing the important phenomenon of diauxie, which provided the impetus for much of the pioneering work with cybernetic models.
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DIAUXIE: A PARADIGM OF BIOLOGICAL REGULATION A classic example of biological control is found in the diauxic growth behavior of Escherichia coli and other microbes. Originally discovered by Monod,6 diauxie is a growth pattern that often arises when multiple growth-supporting substrates are simultaneously present in the external environment. Instead of consuming the available nutrients in proportion to their abundance, the bacterial cells actively control substrate uptake by regulating their internal repertoire of enzymes. As shown in Figure 1, culturing E. coli on a mixture of two substitutable
Figure 1. Diauxic growth of E. coli on two substitutable substrates, S1 and S2, which are converted to biomass (B) by the key enzymes E1 and E2, respectively. The substrate concentrations are symbolized by s1 and s2, while the biomass concentration is denoted as c.
substrates produces two distinct exponential growth phases separated by an intermediate lag phase in which growth is stagnant. The first growth phase involves exclusive consumption of the preferred substrate (S1), which often confers a higher nutritional benefit to the cells, as evidenced by a higher growth rate. After the preferred substrate has been completely exhausted, the cells undergo a lag period during which they switch to synthesizing the enzymes needed to metabolize the less preferred substrate (S2). The second growth phase promptly commences once these enzymes have been amassed to sufficient levels. From a cybernetic viewpoint, the diauxie phenomenon is the outward manifestation of a biological control circuit that has evolved a capacity for optimal decision-making, which in this case serves to maximize the instantaneous growth rate of the culture.
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FIRST-GENERATION CYBERNETIC MODELS The cybernetic modeling approach was first introduced by Ramkrishna and co-workers in the mid-1980s.5,7−10 Early investigations were primarily concerned with describing macroscopic regulatory features of microbial systems, including the emergence of diauxic growth patterns in multisubstrate bacterial B
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preferred or when switching should take place. These are outputs of the model that are determined by the optimization procedure. Although Dhurjati et al.’s model was successful in reproducing the experimental observations of diauxie, it had some conceptual and practical disadvantages that warranted further development of the cybernetic modeling groundwork. First of all, since the model takes a long-term perspective of cellular optimization, it must determine the open-loop policy of substrate utilization over a finite period of time based on some prescribed environmental conditions. As a result, one must assume that either (i) the cell possesses foreknowledge of future environmental perturbations that allows it to reckon a policy that is optimal over the entire simulation interval or (ii) the cell lacks an ability to anticipate future events and therefore responds in a suboptimal manner to any imposed disturbances. Both positions have their associated difficulties. The first is obviously untenable from a logical standpoint, while the second suffers from the inability to appropriately model perturbed batch situations or other scenarios in which the environment is manipulated by outside forces. Moreover, the concepts of initial and terminal times are not well-defined for a continuous culture, and the frequency with which the control policy should be revised is open to interpretation. A second drawback of Dhurjati et al.’s model is that it results in a discontinuous control policy that cannot readily simulate simultaneous uptake of substrates, as often occurs in chemostat cultures at low dilution rates. One final objection is that the model requires intense numerical computations to simulate even a very simple growth curve, which amounts to a case of “the tail wagging the dog” in more complex situations. In this respect, we should remain mindful that computing the optimal control is merely a means to an end, not the end itself. Model of Kompala et al. The work of Kompala et al.7,9 set out to rectify many of the shortcomings of Dhurjati et al.’s approach. In Kompala et al.’s model of diauxic growth, regulation of both enzyme synthesis and enzyme activity was included by introducing cybernetic control variables u and v, respectively. As in Dhurjati et al.’s model, the u vector mimics transcriptional and translational control inputs that influence the rates of enzyme synthesis. The v vector subsequently determines the relative activities of these enzymes. Incorporating activity control enables the model to describe important regulatory features such as feedback inhibition of enzymes that were not addressed by Dhurjati et al. Similar to the foregoing treatment, a cellular objective that represents the overriding goal of metabolism is postulated. All regulation is assumed to center around achieving this objective in an optimal manner. However, Kompala et al. based their interpretation of optimality on a short-term perspective whereby the control policy is continuously updated at each instant solely on the basis of an inspection of the current state vector. This amounts to a closed-loop control policy, which avoids many of the logical pitfalls associated with controls that are implemented over a finite time horizon. The system equations for the model of Kompala et al. can be represented abstractly as
cultures and the shift between growth and maintenance metabolism that often occurs under starvation conditions. The models were based on an unstructured or “lumped” representation of the organism’s biochemical pathways, meaning that overall metabolic conversions were not resolved into individual reaction steps. Such intracellular details were deemed unnecessary in describing the relevant macroscopic dynamics. The metabolic state of the system was accordingly condensed to include only the concentrations of a few (mostly extracellular) biochemical species, a handful of key enzymes, and the unstructured biomass component. We can encapsulate this information in the state vector ⎡y⎤ x = ⎢e⎥ ⎢ ⎥ ⎣c⎦
(1)
where y is the vector of biochemical species concentrations, e is the vector of key enzyme levels, and c is the aggregate biomass concentration. By formulating and solving the governing balance equations that describe how x changes over time, early cybernetic modelers were able to analyze the dynamic behavior of biological systems in much the same way as engineers have traditionally studied industrial chemical processes. Model of Dhurjati et al. Much of the initial debate in the cybernetic modeling literature centered around elucidating the proper scope of cybernetic control laws. Dhurjati et al.8 investigated control policies based on a long-term perspective whereby the average cell mass productivity was maximized over a finite time interval. They developed a model of diauxic growth by defining a control input vector u such that each element ui represents the fractional amount of cellular resources dedicated to synthesizing the ith key enzyme, which in turn catalyzes uptake of the ith extracellular substrate. According to this definition, the elements of u must obey the summation constraint ∑iui = 1. The performance index for the system was chosen to minimize the amount of time needed to convert all of the available carbon sources into biomass, denoted as T. The resulting optimal control problem can be stated as min T s.t.
dx = f(x , u) dt
∑ ui = 1 i
∑ si(T ) = ε ≈ 0 i
(2)
where si is the concentration of the ith substrate and ε is a tolerance chosen suitably close to zero. The authors solved eq 2 numerically by guessing an initial u trajectory, integrating the state equation forward in time, using the state solution to integrate the adjoint equation backward in time, and finally updating the value of u on the basis of a steepest-descent rule. Iterations were continued in this manner until no further improvement was obtained in the performance index. Because the Hamiltonian function for this problem is linear in the control, the optimal solution turned out to be what is called a “bang-bang” policy. The preferred substrate S1 is exclusively utilized (u1 = 1) during the first growth stage, followed by a discontinuous switch to S2 at the computed optimal switching time. It should be noted that there is no need to “tell” the model which substrate is
dx = f(x , u , v) dt
(3)
The authors relied upon heuristic arguments to arrive at the following policies for computing the elements of u and v: ui = C
p
i N ∑i =S1 pi
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pi max i ∈ {1,2,..., NS} pi
of growth maximization as governed by the Matching and Proportional Laws. Extensions of Kompala et al.’s Model. Several authors attempted to extend the results of Kompala et al. to describe more complicated growth patterns that occur in fed-batch and continuous culture situations. Turner and Ramkrishna12 modified the enzyme synthesis rate expression to include both inducible and constitutive contributions. It was also necessary to include maintenance processes in Kompala et al.’s model to correctly describe the reduced biomass yields that are often observed under low-growth conditions. Turner et al.13,14 incorporated maintenance effects into their model of K. oxytoca growth and applied an objective function that served to maximize substrate uptake. The resulting competition between growth and maintenance processes tended to upregulate maintenance whenever the growth rate was depressed. The model provided accurate predictions of fed-batch growth on both single and mixed substrates. Furthermore, it correctly described the transient response evoked by the addition of a substrate pulse during the middle of the fed batch. The model of Baloo and Ramkrishna15,16 was similar to that of Turner et al., but it included some additional features that allowed it to more accurately describe continuous culture experiments, especially during the transient periods following a step change in dilution rate or a switch in feed composition. These improvements were mainly a result of adding an explicit balance on the translational resource, i.e., the cellular ribosome content, which is present at high levels when substrates are plentiful and gradually decays when substrates are scarce. Narang et al.17 later performed batch growth experiments on binary substrate mixtures containing at least one organic acid. In contrast to experiments on pure sugar mixtures, these substrates were often utilized simultaneously. Even in mixtures that elicited diauxie, they found the order of substrate preference sometimes to depended upon the preculturing history. Dynamic analysis of Kompala et al.’s cybernetic model showed that it cannot describe these alternative patterns of substrate uptake because it is predisposed to exhibit diauxie under batch conditions.18 However, Ramakrishna et al.19 subsequently showed how Kompala et al.’s model could be refined to admit solutions characteristic of both sequential and simultaneous uptake patterns. These changes involved introducing a structured biomass component to provide a minimalist description of intracellular growth processes. As such, Ramakrishna et al.’s model demarcates the dividing line between totally lumped firstgeneration cybernetic models and structured second-generation models.
(5)
where pi is the “return on investment” for the ith substrate and NS is the total number of substitutable carbon substrates in the extracellular medium. The return on investment for a substrate reflects the contribution it makes toward increasing the system objective function. The policy for computing u was termed the Matching Law, since it implies that the fractional amount of resources allocated toward synthesizing a particular key enzyme should match the fractional return derived from the reaction it catalyzes. The Matching Law actually finds its roots in the work of Richard Herrnstein, a behavioral psychologist who used the same type of response-matching relationship to describe empirical observations of choice behavior in pigeons and other animals.11 The policy for computing v was dubbed the Proportional Law, reflecting the notion that each reaction should be activated in proportion to its return on investment. The values of both ui and vi can vary between 0 and 1, with 1 representing the fully induced or activated state. The structure of Kompala et al.’s model is summarized in the block diagram shown in Figure 2.
Figure 2. Block diagram representing the cybernetic model of Kompala et al.9 The system boundary is represented by the dashed line, and everything inside this boundary is considered fully observable to the organism’s regulatory machinery. At each instant, the in silico organism examines its current state x in relation to its metabolic objective and computes the returns on investment contained in the vector p. The Matching and Proportional Laws are then applied to compute the control variables u and v, which tend to upregulate those reactions that give the highest returns on investment. The control inputs as well as the current system state are used to compute the vector f, which describes how the state vector is changing over time and can be integrated to give the value of x at the next time instant.
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Kompala et al. employed an objective of growth maximization to simulate diauxie on a number of substrate mixtures involving the sugars glucose, xylose, arabinose, lactose, and fructose. The return on investment for the ith reaction was set to pi = ri, where ri is the growth rate that would be attained if only the ith substrate were consumed. Hence, the model predicts that the enzyme yielding the highest growth rate will be synthesized and activated to the fullest extent. The significant result was that Kompala et al.’s model accurately portrayed the observed mixed-substrate growth behavior of Klebsiella oxytoca solely on the basis of experimental results obtained on single substrates. The model correctly predicted the order in which the substrates would be consumed and the overall growth dynamics of the culture using kinetic parameters determined from single-substrate experiments, without the need for inhibition parameters or other ad hoc devices to capture the substrate interactions. This impressive predictive capability flows directly out of the imposed objective
SECOND-GENERATION CYBERNETIC MODELS As shown by the work of Kompala et al.,7,9 Turner et al.,13,14 and Baloo and Ramkrishna,15,16 first-generation cybernetic models are powerful tools for describing microbial growth dynamics, especially in mixed-substrate environments. Such models rely heavily on process lumping and pathway abstraction to simplify the underlying reaction network. This approach greatly reduces the complexity and dimension of the system to be modeled, thereby facilitating model identification and producing models that are amenable to analysis and interpretation. As a result, the models are readily applied to investigate the optimal design, operation, and control of bioprocesses.20 However, they lack any description of intracellular dynamics aside from those of the highly lumped enzyme systems. Because of their meager internal structure and limited biological detail, first-generation models D
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Industrial & Engineering Chemistry Research cannot address relevant questions in the areas of metabolic engineering and systems biology, which aim to characterize the complex interactions of intracellular components and determine strategies to systematically alter these components to improve and extend cellular capabilities. Starting in the mid-1990s, interest began to shift toward the development of structured cybernetic models aimed specifically at applications in metabolic engineering and systems biology. These models constitute a second generation of cybernetic models. Straight and Ramkrishna21 laid the groundwork for this class of models by extending the Matching and Proportional Laws to describe resource competitions within general metabolic pathways, which could involve either substitutable or complementary processes. Furthermore, they proposed that metabolic networks of arbitrary complexity could be decomposed into smaller functional units, each having its own local objective function. They identified four structural elements that represent the basic building blocks of cellular metabolism: linear segments, convergent branch points, divergent branch points, and cycles (Figure 3). Varner and Ramkrishna22−24 later expanded upon the
Figure 4. Block diagram representing the modular cybernetic modeling approach of Varner and Ramkrishna.22 The local returns on investment for all reactions included in the kth cybernetic unit are represented by the vector pk. The Matching and Proportional Laws are then applied to determine local control variables uk and vk based on the elements of pk. Global control variables U and V are introduced to handle system-wide management decisions, such as switching entire pathways “on” or “off” according to the needs of the organism. The global variables are also required to handle conflict resolution, i.e., the assignment of “complete” control variables u and v to reactions that participate in multiple cybernetic units. (Because the reaction sets constituting different cybernetic units are not required to be disjoint, special treatment is required to define u and v whenever the cybernetic units overlap.).
Figure 3. Topological structures commonly found in metabolic networks: (a) linear segment, (b) convergent branch point, (c) divergent branch point, and (d) cyclic process.
work of Straight and Ramkrishna to develop the modular cybernetic modeling approach summarized in the block diagram presented in Figure 4. As shown in Figure 5, the reaction network is first decomposed into a collection of elementary cybernetic units, which are subsets of reactions presumed to compete for a common pool of cellular resources. Each cybernetic unit is assigned a local metabolic objective, and a return on investment is computed for each constituent reaction in reference to this local objective. The Matching and Proportional Laws are then applied to determine local and global cybernetic control variables for each reaction. This modular cybernetic modeling approach was subsequently applied to investigate the effects of gene overexpression on amino acid production in Corynebacterium lactofermentum25 and gene knockouts in E. coli central carbon metabolism.26 Further work has shown that structured cybernetic models are capable of describing many types of sophisticated nonlinear behavior, including experimentally observed steady-state multiplicity in continuous hybridoma27−29 and E. coli30 cultures. Gadkar and Doyle31 have also investigated methods for applying structured cybernetic models to online model-predictive control of bioreactors. As structured cybernetic models were expanded to describe increasingly larger and more complicated metabolic networks, a number of significant obstacles emerged. First, a systematic algorithm for decomposing metabolic networks into elementary cybernetic units was lacking. Second, global cybernetic variables were introduced on an ad hoc basis, which led to inefficient coordination of metabolic pathways. Third, a mechanism was
Figure 5. Hierarchical decomposition of a metabolic pathway into elementary cybernetic units.
needed to accommodate more general (local and global) metabolic objective functions, producing a concurrent need to revisit the optimality criteria of the Matching and Proportional Laws. The latter concern was rigorously addressed by Young and Ramkrishna,32 who showed how the cybernetic control laws could be obtained from the solution to a well-posed optimal control problem. These results were later used by Young et al.33 to replace the loosely defined concept of an elementary cybernetic unit with the more precisely defined concept of an elementary flux mode (EFM).34 Young et al.33 introduced a cybernetic modeling approach that treated each EFM as a standalone metabolic pathway and defined global cybernetic variables for each EFM based on its “composite” flux. The composite flux was defined as the harmonic mean of all individual reaction rates that constitute the EFM. Local cybernetic variables E
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parameters, etc.) with those of biochemically detailed secondgeneration models (e.g., the ability to predict the effects of targeted genetic manipulations). These hybrid cybernetic models (HCMs) apply kinetic expressions and cybernetic control variables to predict the external fluxes but then apply steadystate flux balancing to estimate the internal fluxes. As a result, the local cybernetic variables are eliminated from the model in favor of the internal steady-state assumption. This approach was first introduced by Kim et al.,35 who assigned a kinetic rate law for each EFM in the metabolic network and defined global cybernetic variables to maximize the total substrate uptake rate. It was shown that the intracellular fluxes predicted by an E. coli HCM were largely identical to those predicted by the cybernetic model of Young et al.33 when the two models were trained on the same experimental data. However, both approaches become progressively more challenging as the metabolic network grows, since the number of EFMs increases exponentially with network size.36 To address this issue, Song and Ramkrishna37 developed a rational procedure for reducing the number of EFMs to a minimal subset through the application of “metabolic yield analysis” (MYA). Using this procedure, Song et al.38 successfully applied an HCM to predict the growth of recombinant yeast strains on glucose−xylose mixtures. HCMs were subsequently developed to simulate several industrial bioprocesses, including polyhydroxybutyrate (PHB) production by Ralstonia eutropha,39 lactic acid production by Pediococcus pentosaceus grown on lignocellulose-derived mixed sugars,40 and bioethanol production by mixed yeast cultures.41 It is noteworthy that Franz et al.39 extended the HCM framework to account for intracellular metabolites with slow dynamics (i.e., PHB) that are not subject to the quasi-steady-state approximation, illustrating how HCMs can be applied to unbalanced growth situations.
were also defined for each EFM in order to maximize the composite flux, thus attracting resources to any “‘lagging’” reactions. Combining cybernetic control laws with elementary mode analysis thus provided a systematic approach for the formulation of cybernetic models. The control circuit that underlies this latter class of cybernetic models is summarized in the block diagram shown in Figure 6. The framework relies upon the simultaneous
Figure 6. Block diagram representing the EFM-based cybernetic modeling approach of Young et al.33 The local layer of control serves to optimize the composite flux Jk through each EFM. The global layer of control acts to supervise the overall operation of metabolism and to select the EFMs that should be implemented on the basis of their contribution to the global objective function. Combining the local and global cybernetic variables results in complete cybernetic variables that embody the entire hierarchy of control decisions. It should be noted that although EFMs are defined on the basis of a steady-state analysis of the underlying reaction network, the resulting cybernetic model is fully dynamic and draws upon elementary mode analysis only to decompose the metabolic network into functional pathways.
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LUMPED HYBRID CYBERNETIC MODELS
Song and Ramkrishna42,43 further expanded on the HCM concept by introducing a class of “lumped hybrid” cybernetic models (L-HCMs). These models predict the intracellular flux distribution in a hierarchical manner: substrate uptake fluxes are first distributed among lumped pathways and then subdivided among the individual EFMs in each lumped pathway. Both splits are dictated by cybernetic control variables. The partitioning of the uptake flux at the first split is dynamically regulated according to environmental conditions, while the subsequent split is based purely on the stoichiometry of the EFMs. This formulation enables the model to account for the complete set of EFMs in arbitrarily large metabolic networks despite containing only a small number of kinetic parameters, which can be identified using minimal data. L-HCMs have proven capable of predicting complex dynamic responses of E. coli and yeast cultures to not only environmental changes42,43 but also genetic perturbations.44 A recenly published L-HCM of the Shewanella oneidensis MR-1 strain in aerobic batch culture was shown to accurately reproduce complex dynamic metabolic shifts, including the sequential use of substrate (i.e., lactate) and byproducts (i.e., pyruvate and acetate). Flux distributions in S. oneidensis predicted by the LHCM compared favorably to 13C metabolic flux analysis results reported in the literature.
application of local controls that maximize the composite flux through each EFM and global controls that modulate the activities of these modes in order to optimize the global objective function of the cell. This global level of control is an essential part of the modeling approach, since it serves to coordinate the activities of all of the EFMs within the network. On the basis of the hypothesis that cellular resources are optimally allocated, only EFMs that provide high returns on investment are utilized by the cell, and the enzymes that catalyze these EFMs are regulated to give maximum productivity. Because of the stabilizing influence of the local control variables, the resulting cybernetic models tend to be more robust in comparison with other dynamic models when simulating the network response to imposed perturbations. This property was illustrated using a cybernetic model of anaerobic E. coli central carbon metabolism.33 The E. coli model successfully described the metabolic shift that occurs upon deletion of the pta-ackA operon that is responsible for fermentative acetate production. The model also furnished predictions that were consistent with experimental results obtained from additional knockout strains as well as strains expressing heterologous genes.
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HYBRID CYBERNETIC MODELS The most recent development in the cybernetic modeling literature has been the emergence of so-called “hybrid” models that attempt to combine the advantages of first-generation “lumped” models (e.g., computational efficiency, few kinetic
CYBERNETIC MODELS VERSUS DETAILED KINETIC MODELS There have been repeated efforts to construct detailed kinetic models of cellular metabolism in a variety of species. A few F
DOI: 10.1021/acs.iecr.5b01315 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
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Industrial & Engineering Chemistry Research notable examples are found in the work of Domach et al.45 and Chassagnole et al.46 to model E. coli metabolism and that of Rizzi et al.47 to model yeast. Unlike cybernetic models, these models attempt to account for regulation mechanistically by expanding the set of state variables to include known regulatory components and augmenting the kinetic rate expressions to include known regulatory interactions. Detailed kinetic models have been effective in describing many observable biological phenomena, such as spontaneous oscillations in the glycolytic metabolism of yeast.48 However, they have not been widely used for phenotypic predictions because they often fail to extrapolate outside of the narrow range of state/parameter space from which the training data were obtained. Difficulty in identifying suitable rate expressions, estimating kinetic parameters, and systematically accounting for regulation has limited the scope of detailed kinetic models to relatively small- to medium-sized networks of well-studied biochemical reactions. Therefore, most investigations have been restricted to model strains of bacteria and yeast for which the important kinetic and regulatory features are already thoroughly understood. While the burden of parameter estimation and model identification still remains a major challenge for cybernetic modeling, the presence of the u and v cybernetic variables causes cyberentic models to be less sensitive to incomplete or inaccurate kinetic formulations. Cybernetic models cannot provide the mechanistic insights afforded by detailed kinetic models, but this shortcoming is balanced by their improved robustness and ability to make extrapolated predictions.33 While it is not difficult to formulate a detailed kinetic model that describes a single experimental condition, the complexity and nonlinearity of the kinetic formulation often cause the models to break down when applied to other conditions. For instance, the energy-carrying molecule ATP participates in literally hundreds of metabolic conversions, and the ATP/ADP ratio must be held in a narrow range to ensure proper functioning of these processes. The ATP pool is turned over approximately once every second, so the ATP dynamics are extremely rapid in comparison with the macroscopic processes of growth and substrate consumption. Without a cybernetic mechanism to ensure stability, a kinetic model will most likely fail in response to a perturbation that it has not been expressly built to handle. On the other hand, cybernetic control laws promote robustness so that models can be deployed in a predictive mode to discover unforeseen physiological outcomes that result from genetic or environmental manipulations.
yields and productivities that result from genetic recombination. This is in contrast to stoichiometric modeling approaches that require the substrate uptake rates to be specified before the remaining fluxes can be predicted. It is also interesting to note that many of the fundamental concepts underlying the early cybernetic models (e.g., the application of optimality heuristics to biological models5 and decomposition of metabolic networks into elementary units21) actually predated the application of similar concepts to stoichimetric models (e.g., flux balance analysis49,50 and elementary mode analysis34,51). Likewise, concepts such as the EFM reduction technique of Song and Ramkrishna37 may prove broadly useful for understanding metabolic network function, independent of its contribution to cybernetic modeling. As a result, the impressive legacy of cybernetic modeling, which all began from a very simple idea regarding the optimality of microbial growth, will forever influence our thinking on how biological regulation can (and perhaps should) be modeledfrom the perspective of the steersman.
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AUTHOR INFORMATION
Corresponding Author
*Phone: 615-343-4253. Fax: 615-343-7951. E-mail: j.d.young@ vanderbilt.edu. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (Award DE-SC008118). REFERENCES
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CONCLUSION Now more than 30 years since its initial conception, cybernetic modeling has survived multiple reincarnations and has expanded to fill the entire biotechnology landscape. The framework has evolved from early “lumped” cybernetic models, meant to describe macroscopic patterns of microbial growth and substrate utilization in bioprocesses, to later “second-generation” cybernetic models aimed at metabolic engineering and systems biology applications. Finally, “hybrid” cybernetic models have recently emerged that combine many of the desirable properties of their forebears while addressing key challenges of model identification and computational complexity that plagued earlier efforts. As a result, the current cybernetic modeling framework enables a much richer description of cellular behaviors than previously envisaged, in terms of both macroscopic phenotypes and the underlying molecular dynamics. One key aspect of these models is that because they involve a kinetic description of reaction rates, they have the potential to predict changes in both G
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