Thinking Dynamically About Biological Mechanisms: Networks of Coupled Oscillators

Explaining the complex dynamics exhibited in many biological mechanisms requires extending the recent philosophical treatment of mechanisms that emphasizes sequences of operations. To understand how nonsequentially organized mechanisms will behave, scientists often advance what we call dynamic mechanistic explanations. These begin with a decomposition of the mechanism into component parts and operations, using a variety of laboratory-based strategies. Crucially, the mechanism is then recomposed by means of computational models in which variables or terms in differential equations correspond to properties of its parts and operations. We provide two illustrations drawn from research on circadian rhythms. Once biologists identified some of the components of the molecular mechanism thought to be responsible for circadian rhythms, computational models were used to determine whether the proposed mechanisms could generate sustained oscillations. Modeling has become even more important as researchers have recognized that the oscillations generated in individual neurons are synchronized within networks; we describe models being employed to assess how different possible network architectures could produce the observed synchronized activity. Although the construction of mechanistic accounts to explain phenomena has been the dominant research strategy in biology since the 19 century, philosophy of science has only recently started to catch up by examining the character of mechanistic explanation (Bechtel & Richardson, 1993/2010; Bechtel & Abrahamsen, 2005; Machamer, Darden, & Craver, 2000; Thagard, 2003; Wimsatt, 1976). This new mechanistic philosophy of science has emphasized the same explanatory endeavor that most biologists themselves emphasize: the decomposition of the system responsible for a given phenomenon into component parts and operations. Identifying components – treating the responsible system as a mechanism to be taken apart – is quintessentially reductionistic. In both biology and philosophy of science there has been much less attention to the converse endeavor: the recomposition of the mechanism. The components will not produce the phenomenon unless they are put back together properly; that is, they must be organized so as to produce the phenomenon of interest. Even as staunch a reductionist as E. O. Wilson has urged attention to recomposition: The greatest challenge today, not just in cell biology and ecology but in all of science, is the accurate and complete description of complex systems. Scientists have broken down many kinds of systems. They think they know most of the elements and forces. The next task is to reassemble them, at least in mathematical models that capture the key properties of the entire ensembles (Wilson, 1998, p. 85). Recomposition – reassembly of the components – means figuring out the spatial organization of the parts and the temporal organization of the operations in the system producing the phenomenon targeted for explanation. This is more difficult, and tends to begin later, than decomposition of the system, but is essential for achieving a complete mechanistic explanation. Through most of the history of biology, such work has been guided by rather limited, though tractable, conceptions of organization: spatial adjacency and connectedness (for parts) and stepwise temporal sequencing (for operations). Those developing the new mechanistic philosophy of science have incorporated this preference for the simplicity of sequential thinking. One influential account (Machamer, Darden, & Craver, 2000) portrayed mechanisms as “productive of regular changes from start or set-up conditions to finish or termination conditions.” As they point out, the simplest sequential accounts guided by this conceptualization in science often have been useful, at least as idealizations, but not uncommonly more complex organizational schemes are required. A classic example is the Krebs cycle in oxidative metabolism: after years of struggle to compose an adequate sequential account, closing the sequence into a loop was the key innovation leading to success. Even these accounts, though, lack attention to dynamics in real time. In recent years biologists have increasingly incorporated a concern with dynamics into their mechanistic accounts of various living systems. The simplest possible assumption regarding the dynamics of a mechanism is that it will operate at a constant or steady rate once the start or set-up conditions are met and continue at that rate until the finish or termination conditions are achieved. But many biological mechanisms exhibit oscillations or more complex dynamical behavior, and this can be crucial for orchestrating operations within the mechanism. Such complex dynamical behavior is possible when the organization is nonsequential, at least some of the interactions in the mechanism are nonlinear, and the system is open to energy. When complex dynamics are not present, or are present but not addressed, simple stepwise sequences of operations can be specified on paper or even in the researcher’s head. Achieving an account of complex dynamics such as oscillations, though, requires the tools of computational (mathematical) modeling. A typical dynamical model consists of a set of differential equations that indicate how the values of certain variables change over time relative to parameters (constants for which appropriate values must be estimated) and other variables. Of particular interest here are those dynamical models for which key variables, and terms in the equations 1 One of the first philosophical characterizations of a mechanism was due to Bechtel and Richardson (1993/2010): “A machine is a composite of interrelated parts, each performing its own functions, that are combined in such a way that each contributes to producing a behavior of the system. A mechanistic explanation identifies these parts and their organization, showing how the behavior of the machine is a consequence of the parts and their organization.” This account did not impose sequential organization and indeed one of the objectives of the analysis was to trace the process by which scientists moved from early accounts which focused on only one component (we called such accounts simple or direct localizations) or a sequence of operations performed by different parts (complex localizations) to accounts that identified cycles and other complex arrangements of parts that we termed integrated systems. But we did not address the complex temporal behavior of such systems in that book. correspond to selected properties of the parts and operations of the target mechanism. The tools of dynamical systems theory can elucidate such models, and often are explicitly called upon. Bechtel and Abrahamsen (2010, 2011) have characterized the endeavor of integrating mechanistic decomposition of a system into its parts and operations and computational modeling of its real-time dynamics as dynamic mechanistic explanation. We illustrate the project of dynamic mechanistic explanation by discussing how computational models have been used to explore the dynamics of the mechanisms responsible for circadian rhythms. Research aimed at decomposition has, over the past two decades, elucidated many of the parts and operations in the molecular mechanisms that are responsible for circadian oscillations in organisms ranging from bacteria to mammals. Recomposition has kept pace, because these molecular mechanisms utilize an already well-known cyclic organizational scheme (gene expressed as a protein with negative feedback inhibiting further expression). This scheme enables, but does not guarantee, oscillatory behavior. Assessing the dynamics of networks requires that computational modeling be carried out alongside mechanistic modeling. In particular, the finding in mammals that individual neurons vary significantly in periodicity when dispersed in culture has made it important to address the question of how biological oscillators such as neurons synchronize their behavior and how this is affected by the their organization. 1. From Reactive to Endogenously Active Mechanisms The initial metaphor giving rise to mechanistic accounts of living systems is that they bear some resemblance to machines designed by humans. Many machines are set up to begin working when appropriate conditions are met, carry out a fixed sequence of steps, and then stop. Such machines are reactive: they react to the onset of their start or set-up conditions and continue until they reach the finish or termination condition. Scientists using the machine metaphor have therefore tended to conceptualize biological systems in this way. Thus, gene expression is characterized as beginning when a specialized molecule “unzips” the helical structure of the relevant segment of DNA and continuing through transcription and then translation until synthesis of the relevant protein has been achieved. Glycolysis is described as a sequence of reactions that begins when glucose is present and terminates in the production of pyruvic acid. Visual processing in the brain is conceived as beginning with the presentation of a visual stimulus and continuing through numerous steps until an object is recognized and localized in space. We will argue that this reactive conception is inadequate to account for the behavior of numerous biological mechanisms. The first step away from a reactive conception of mechanisms is to advance beyond purely sequential organization. The first non-sequential design principle to be discovered was negative feedback. Tellingly, two thousand years intervened between its first introduction by Ktesbios in his design for a water clock and the cyberneticists’ recognition in the 1940s that negative feedback is a general organizational principle for living and social systems (see Mayr, 1970). Humans, even scientists, find it difficult or inconvenient to stretch their thinking beyond sequential organization. Even the cyberneticists, moreover, treated negative feedback primarily as a way of maintaining a system in a steady state—a means of achieving what Claude Bernard (1865) referred to as the constancy of the organism’s “internal environment.” Walter Cannon (1929) coined the term homeostasis for this phenomenon and explored ways in which the autonomic nervous system served to maintain homeostasis. A common, but misleading, way of characterizing homeostasis is that it involves maintaining a state in which the organism is in equilibrium with its environment. This is misleading in that the most fundamental type of equilibrium with an environment – thermodynamic equilibrium – entails the death of the organism. As highly organized systems, organisms are far from thermodynamic equilibrium with their environments, and to perpetuate their identity as organisms they must maintain the nonequilibrium relation. This in turn necessitates that they recruit matter and energy from their environment and deploy these resources to build and repair themselves. Energy and matter can be procured either from non-biological sources (sunlight in the case of photosynthesis) or from foodstuffs provided by other organisms and then utilized in energy demanding activities such as synthesizing and repairing the organism’s own body, locomotion, and transporting substances through physiological systems. Recognizing that maintaining a relation of non-equilibrium with the environment rather than one of equilibrium has fostered an alternative perspective in which organisms are understood as autonomous systems. Ruiz-Mirazo and Moreno characterize an autonomous system as: a far-from-equilibrium system that constitutes and maintains itself establishing an organizational identity of its own, a functionally integrated (homeostatic and active) unit based on a set of endergonic-exergonic couplings between internal self-constructing processes, as well as with other processes of interaction with its environment (RuizMirazo, Peretó, & Moreno, 2004). Since the processes that would serve to bring an organism back into thermodynamic equilibrium with its environment are relentless, every organism needs to be endogenously active – regularly performing the activities that keep it in the appropriate non-equilibrium relation with its environment. The result is that mechanisms in living systems, rather than needing to have their actions initiated, tend to be designed so that by default they are performing operations. One of the simplest forms of dynamical behavior that exhibits sustained activity is oscillation: the values of variables characterizing the behavior of the system repeatedly rise and fall rather than remain constant. Delayed negative feedback in a system driven by an energy source can readily produce such oscillations. For example, adding a thermostat to a heating system turns the furnace off whenever the target temperature is achieved and back on when it drops below the target. Given the inevitable time delay between the thermostat and furnace, room temperature oscillates rather than being maintained at exactly the target value: the temperature gradually rises while the furnace is on and then declines while the furnace is off. Often, as in this example, oscillations are viewed as nuisances and the goal for engineers is to minimize or dampen them as much as possible. When data collected from biological systems reveal oscillations, the oscillations are frequently regarded by investigators as noise to be removed by statistical techniques; what remains is regarded as the signal to be analyzed. But such oscillations are a resource that can be used by the system in generating important biological phenomena. It is then a mistake to treat them as noise; oscillations that are exploited by the system are, rather, an important part of the signal. Circadian rhythms are one important class of biological oscillations; by definition, they include any biological rhythm for which the period (one rise and fall) is approximately 24 hours. These endogenously maintained oscillations affect a wide range of physiological and behavioral processes in virtually all life forms on our planet in which they have been sought. They have been identified and investigated, for example, in cyanobacteria (Synechococcus elongatus), fungi (Neurospora), plants (Mimosa, Arabidopsis), and animals (Drosophila, Mus). The rhythms, in processes ranging from gene expression to sleep, are shown to be endogenous by removing all cues to time of day (Zeitgebers) and determining that the rhythms nonetheless are maintained with only a small discrepancy from 24 hours. Beyond endogenous maintenance of rhythms, two further features that are considered fundamental are the ability to be entrained by Zeitgebers (e.g., light or temperature) and temperature compensation (so that circadian oscillators, unlike most biological processes, do not speed up as temperature rises). In industrialized human society these rhythms tend to be either ignored or regarded as a nuisance; for example, they make life more difficult for shift workers and travelers who cross multiple time zones. Even biological researchers tend not to take into account regular, welldescribed variations with time of day in a host of basic physiological and behavioral processes. For example, body temperature oscillates over a range of 1°C. daily and reaction times are measurably faster in the afternoon, but time of day at which a measurement was made usually is ignored and these periodic variations therefore treated as part of the error variance (noise) in data analysis. Yet many of these rhythms play crucial roles. One is to coordinate essential behaviors with daily and annual changes in the environment (e.g., fruit flies must eclose from their pupae in the morning when temperatures are lower and humidity is higher, and many rodent species move about mainly at night while their predators sleep). Certain activities require several hours of preparation, necessitating some means of anticipating the time at which they must be performed. Another essential role is to segregate in time activities that are incompatible with each other (e.g., nitrogen fixation and photosynthesis in cyanobacteria, since the critical enzyme for nitrogen fixation is destroyed by the oxygen generated by photosynthesis). In this section we have argued that in living organisms researchers confront active, not reactive, systems that maintain themselves in a non-equilibrium condition. These systems exhibit nonsequential organization, such as negative and positive feedback loops, and non-linear interactions between components, all of which contribute to oscillatory behavior and other complex dynamics. Accordingly, the focus on mechanistic explanation in philosophy of biology must be extended to include a conception of dynamical mechanistic explanation. In inquiring into this type of explanation, we will take advantage of the fact that it has been well-developed in biological research on circadian rhythms. . 2. Complex Dynamics: Oscillatory Mechanisms Although we will focus on circadian rhythms, oscillatory processes with both shorter and longer periods are ubiquitous in the biological world. Oscillations with periods of less than 20 hours are termed ultradian, and include oscillations in basic metabolic processes (e.g., glycolytic oscillations), cardiac rhythms, neuronal spiking, the cell division cycle, and sleep cycle. Oscillations with periods longer than 28 hours are termed infradian and include menstrual cycles and hibernation cycles. Whether and how oscillations of different periods either share a common mechanism or affect each other is a topic of significant ongoing investigation. Around 1960 a variety of biological research teams began to focus on oscillatory behavior, including but not limited to circadian rhythms. Notably, while working with yeast extracts Amal Ghosh uncovered an ultradian rhythm in glycolysis: levels of a key product (NADH) oscillated with a period of approximately one minute (Chance, Estabrook, & Ghosh, 1964). Brian Goodwin (1965) developed a computational model to explore the possibility that negative feedback played a key role in producing such oscillations. He linked it to a generalized version of Jacob and Monod’s lac-operon, a mechanistic model of gene regulation in which a repressor molecule provides negative feedback by binding to a specialized segment of DNA. On their account, this suppresses expression of the adjacent gene, which codes for an enzyme needed to process lactose in E. Coli. The repressor molecule contributes to regulation in that it is most active when lactose is absent. The computational model Goodwin proposed to explore the dynamics of such a mechanism, shown in Figure 1, has come to be known as the Goodwin oscillator. It consists of three differential equations that relate the values of three variables, X, Y, and Z. Each variable represents the concentration of one type of molecule. The first equation captures transcription of a gene into mRNA (X) depending on the effect of a repressor (Z); the more repressor molecules are bound to the gene, the less mRNA is transcribed. The second and third equations capture the dynamics of the translation of this mRNA into a protein (Y) and then of the protein catalyzing a reaction that turns a precursor into the repressor (Z) of the initial gene. Each equation also contains a term specifying the gradual degradation of the mRNA, protein, or repressor. Conducting simulations on an analog computer, Goodwin attempted to identify the conditions under which oscillations would occur. What turn out to be the critical variable was n (also known as the Hill coefficient), which specifies the minimum number of interacting repressor molecules needed to inhibit expression of the gene. Goodwin concluded from his simulations that sustained oscillations could arise with n equal to 2 or 3, but subsequent simulations by Griffith (1968) determined that oscillations occurred only with n > 9. Such a value was deemed biologically unrealistic. However, when nonlinearities were introduced elsewhere (e.g., in the degradation term subtracted in each equation), it was possible to obtain oscillations with more realistic values of n. Figure 1. The Goodwin oscillator. The curved arrows specify that a gene, when not repressed, is translated into mRNA, which is transcribed into an enzyme in the ribosome, which then catalyzes the formation of the repressor protein. The repressor then slows down the process that created it. The straight arrows indicate that concentrations of mRNA (X), the enzyme (Y) and the repressor (Z) gradually decline as the molecules Z k Y k dt dZ Y k X k dt dY X k Z k