Control theory and the life sciences

Control theory is the mathematical study of systems that respond to uncertain and dynamic environments. In engineering, control theory is used to design decision systems in applications like aerospace, electronics, industrial-scale chemical synthesis, and learning-driven technologies like robotics and autonomous vehicles. In the life sciences (biology, medicine, ecology), control theory encompasses a set of emerging techniques for designing experiments, making inferences about, and interfacing with natural or extant control systems. Control systems in the life sciences include biochemical processes in a single cell; immune, nervous, and developmental processes in an organism; clinical decision-making processes and treatment protocols; medical devices like pacemakers and ventilators; and stable ecosystems, arguably including the Earth system.

What connects these quite different systems is a notion I’ll call implementability.

We can give this notion a precise mathematical definition when we want to work out a precise mathematical theorem, but it can be a mistake to be too precise, too fast. Yes, much of what we know about implementability will come from theorems concerning ordinary differential equations, but the deeper questions – mathematically, scientifically, practically – are about the notion itself.

To illustrate what makes a system implementable, let's consider a simple example of a system that is not implementable. Suppose you want to ride a bicycle across a very long, very narrow bridge. Is it possible to cross the bridge without using your handlebars?

There is a theoretical answer to this question, which is: within some angle of error at launch, for a given length and width of bridge, you will successfully cross the bridge, and while this angle will tend smaller as the bridge becomes longer and narrower, a nonzero angle will always exist. But there is a better theoretical answer to this question, which is: no. You cannot cross the bridge. You must in fact use your handlebars.

This is not the old complaint about simplification. Simplification is often illuminating. It is also not a problem of assumptions, or at least, not just a problem of assumptions. An implementable, handlebar-based model might still assume favorable weather conditions and fail if it rains. The problem with the no-handlebars model is the problem with the system it represents: that it is fragile, that it exists only in the mathematical universe, that a slight breeze will immediately render it useless. To say that a no-handlebars bicycle on a long, narrow bridge would be selected against by evolution is too weak a claim; it would be selected against by reality.

The weaker forms of implementability are interesting in their own right. We might find that a system in not implementable under resource constraints, or that there are tradeoffs between two favorable properties but no implementable system exists that achieves both, or that the system is implementable, but seems unlikely to be competitive with other implementable systems. These weaker statements require more interpretation but can still be used for both design and discovery.

Feedback, in space and in life

The key difference between riding a bike with and without using the handlebars is feedback: the continuous process of sensing, deciding, and acting. Feedback is probably the central topic of control theory today, and it certainly was in the mid- to late-twentieth century. At that time, the canonical control problem was spaceflight. The space shuttle, on re-entry into the atmosphere, would reach speeds in the thousands of miles per hour and temperatures in the thousands of degrees. If the shuttle lost orientation, it would quickly be torn apart. Yet it was buffeted by complex turbulences, too unpredictable for human piloting to handle. Instead, onboard computers took in data from electronic sensors and converted the data from these sensors into actions to stabilize the re-entry trajectory. The algorithms that transformed these kinds of sensed data to actions are the subject of the robust control problem. Another way of understanding these algorithms is as functions that take an unknown signal through time with large variation, like turbulence, and transform it into a known signal through time with small variation, like (by design) the re-entry trajectory.

Even in the mid-twentieth century, however, control theorists were aware that this loop of sensing, deciding, and acting was not specific to spaceflight: sensing, deciding, and acting to maintain a trajectory is a description of natural movement as well. Maintaining small variation in some key signal through time, despite large variation in the unknowns, is a description of homeostasis.

There might be many reasons why, despite this early recognition of key similarities, the applications of control theory to the life sciences only started to flourish more recently. Technological development is part of the answer (computing power on the theoretical and quantitative side, new tools for data collection on the life sciences side), but not fully explanatory, in my estimation (that is, I suspect there are still interesting low-technology questions to be asked). Whatever the reasons for the delay, however, it is apparent that control theory as it exists today, as the foundation for a general theory of implementable systems and implementable decision systems, has explanatory power in the life sciences.

Theory for basic science and translation

Let’s take an applied problem, like, “What kinds of interventions would reduce rates of cancer in adults with a smoking history?” This leads naturally to several basic science questions. “Why do some people with a smoking history get cancer while others don’t?” leads to “What kinds of processes make people vulnerable to cancer, and how do they vary across people?” leads to “What kinds of mechanisms protect organisms from cancer, and how do they vary across individuals and species?

Since these are already big enough questions to motivate several years of clinical or basic research, we might consider stopping there. We could follow a well-worn path: (1) identify some Molecule X that is correlated with higher rates of post-smoking lung cancer, (2) verify in appropriate experimental systems that inhibiting Molecule X decreases rates of post-smoking lung cancer phenotypes and supplementing Molecule X increases rates of post-smoking lung cancer phenotypes, and (3) design a series of safety and efficacy clinical studies, culminating in a randomized control trial, to treat individuals at risk for post-smoking lung cancer with Molecule X Inhibitors and measure their rates of lung cancer development.

What if no Molecule X can be found? What if there is a Molecule X correlated with higher rates of post-smoking lung cancer, but inhibiting Molecule X has no apparent effect on post-smoking lung cancer phenotypes in experiments? What if inhibiting Molecule X reduces rates of post-smoking lung cancer phenotypes in experiments, but in randomized control trials, there is no apparent effect on real-world outcomes?

These what-ifs describe the fate of the overwhelming majority of basic-to-applied translational efforts. Of course, efforts that don’t yield widely adopted therapies can be successful and informative in other ways, and it would be unreasonable to expect that every promising preclinical idea achieved its maximal clinical potential. Improving translational yield is an important goal, but I want to make an even more elementary point. The typical translational path is reasonable for certain kinds of systems: those in which molecules match one-to-one with phenotypes and are homologous across species, in which anything we haven’t measured can be aggregated into benign noise that we can mostly overcome with statistical tests. Crucially, we already know that other kinds of systems can exist.

Suppose we were studying the activity of a car’s brakes, but we didn’t know anything about drivers. We would notice that cars whose brakes were constantly applied tended not to leave the lot. But if we tried to fix this problem by removing the brakes from those cars, then for entirely different reasons, those cars wouldn’t leave the lot either.

When we encounter some of the expected difficulties translating the basic to the applied, we can take steps to make sure we aren’t studying cars without studying drivers. We necessarily start to ask questions like, “What kinds of systems could we be dealing with?” and “What kinds of multicellular systems develop cancer, and what kinds of systems don’t?” and “What kinds of multicellular systems can exist?” For systems with only two or three interacting parts, we might be able to talk through these questions informally at a whiteboard. For larger systems, with more interactions and more feedback, these questions are at the boundary of expert intuition, and at the boundary of what we have rigorous and broadly applicable theory or computational tools to describe.

Characterization of systems that can or can’t exist is essential for ruling out hypotheses; it is also, less obviously, an accelerating step in experimental design and engineering design. The boundary between the feasible system and the infeasible system is the boundary between one kind of system behavior and another, the boundary where experiments are most interesting. The successful characterization helps you target new tools at the lab bench to modulate your system more dramatically, reveal new behaviors, respond to new contexts. The same tools that allow you to design new experiments at the bench allow you to design new interventions. In this way, when the clinician asks, “What works?” and the experimentalist asks, “How does it work?” and the theorist asks, “What might work?” – they are all asking the same question within a well-structured framework.

The problem of complex system failure

Often when we say a system is complex, we are not really naming a property of the system so much as the absence of its complement, simplicity. We might also, implicitly, be naming any of several properties familiar to the control theorist: robustness, instability, autocatalysis, self-replication, spatial localization, decentralization, multiple scales, multiple agents, multiple feedback loops, lossy communication, mutation, evolution, learning, adaptation, and so on. Complexity may be more than the absence of simplicity, but a universal notion of complexity would include many properties and exceptions, while the absence of simplicity can result from adding any single property.

So rather than defining complexity, we can develop a working definition of simplicity. Let's say simplicity is a property of a system whereby, if we perturb some components or behaviors X, we know exactly or at least directionally what will happen to some components or behaviors Y. An interesting (and nontrivial) consequence of this definition is that a system can be simple in some descriptions and complex in others. Typing a letter onto paper using a typewriter and typing a letter into a web editor using a laptop are both simple in the map from keystroke to letter, but the laptop is complex in the map from laptop bits encoding a keystroke to server bits encoding a letter.

Defining complexity in opposition to simplicity saves us the trouble of asking whether any property is a property of all complex systems, or even if it is a universal property of the complex system we’re looking at right now. We just want to know whether the particular X to Y map we’re looking at is simple or not. If not, we want to know what properties are complex-ifying that map, and whether we have the mathematical tools to make it simple.

Some diseases are simple, or at least conceal their complexity behind simple maps. Localized bacterial infections, for example: we know that the bacteria cause the disease, we know that clearing the bacteria will resolve the disease, and we know that once the bacteria are cleared the disease will not come back. Simple diseases often respond to simple small-molecule treatments, like antibiotics.

Other diseases are complex: varied and cryptic in mechanism, presentation, and progression. They may not result from a single infectious agent, mutation, or injury. They are often hard to treat and hard to reverse. Examples of complex diseases include cancer, sepsis, acute respiratory distress syndrome, epilepsy, vascular diseases, autoimmune diseases, fibrotic diseases, neurodegenerative diseases, and psychiatric disorders.

At first glance, this list is too sprawling to have much in common. Yet there are commonalities. The repeated involvement of nervous, immune, and developmental processes is no coincidence. Complex diseases, rather than resulting from an external cause, are dysregulations of the complex systems that keep us healthy. And complexity in health means something narrower than complexity in disease: it means coordination and control. This matters because it means we can narrow our consideration to implementable systems and their inherent failure modes.

Narrowing the scope of the problem expands our scope of treatment approaches beyond what pathways we target (though this is still a key question!) to include when, where, and in what combinations we target those pathways. In order to be successful, these new approaches should lead to specific treatments that target specific pathways. Fortunately, there are commonalities in specifics as well. I think it is reasonable to describe the complex diseases I’ve listed above in three overlapping categories: immune dysfunction, tissue-homeostatic disorders, and multicellular coordination disorders. These are provisional categories – the boundaries between immune and tissue-homeostatic function are porous, and there are roles for immune and tissue-homeostatic function in epilepsy and psychiatric disorders. It seems likely to me that over time, we will have better names for these things.

Projects past and future, briefly

This essay is a work in progress, a sort of living document meant to clarify for myself and anyone who might be interested what control theory means, as it exists today, in my own work and in the work of the vibrant community that the topic still attracts. For now, this essay is non-technical, though I may incorporate more technical elements in later revisions. A basic technical introduction to implementability in control theory can be found here.

The ideas in this essay have developed through experiences extending and applying control theory in the life sciences. Because of the way published research articles work – the experimental results reviewed and read by experimentalists, the theoretical results reviewed and read by theorists, math models inspired by theorems published separately from theorems that don’t perfectly describe them – many of the key ideas have lived in the interstices. My hope is always that the published research stands on its own. But if you’ve read to this point, you may be wondering how to approach that published work with these ideas in mind. If the preceding essay has sketched a map of a whole field, this is a guidebook for my little corner of it.

Most people, when they think of control theory, think of specific technical problems involving ordinary differential equations, and that is a reasonable starting point for many of the major theorems in the field. But I think many working control theorists are aware that there is more to life than differential equations – that what we call control is a refraction of some other thing with other refractions that people call computation, information processing, optimization, learning, and so on. I persist in calling this field control theory, an unsatisfying name, because I work on some of those specific technical problems involving ordinary differential equations, and because any other name is equally unsatisfying, and because I do not enjoy naming things. However, I think that colloquially, decision-making is the closest phrase to the actual unrefracted thing, so I will use that term here, for this draft.

My perspective on this is stereoscopic, at the exact meeting point of (1) the life sciences encompass many special instances of a general class of implementable decision systems, that might or might not have anything to do with phospholipids and nucleic acids; and (2) the life sciences, in all of their particular details that have everything to do with phospholipids and nucleic acids, can be studied and harnessed for good in the world.

This lends itself to a hierarchy of increasingly specific questions. What decision systems can be implemented? What decision systems can be implemented in groups (of species, of people, of cells, of molecules)? How do groups of cells implement decision-making for specific functions like tissue homeostasis? What are the natural failure modes of groups of cells implement decision-making for specific functions like tissue homeostasis? What mathematical tools can we use to describe groups of cells implementing decision-making for specific functions like tissue homeostasis? What molecular or technological tools can we use to guide those cells towards healthy function? What mix of math, science, and technology best advances global health at scale? What mix of math, science, and technology best advances pediatric immunology (prevention and treatment of infections, autoimmunity, allergies, asthma, cancer)?

My research has mostly lived at the level of decision-making by groups of cells, i.e. multicellular coordination and control, with applications in neuroscience, immunology, and developmental-regenerative biology. The theorem-based papers on robust, decentralized, and layered control are looking at the characteristics of such systems in general, which is helpful for both design and for building intuition (Sarma and Doyle, 2019; Matni and Sarma, 2020; Sarma and Doyle, 2022). In a robust control system, the unknowns are adversarial; in a decentralized control system, at least some of the unknowns are other, cooperative decision-makers in a networked system; and in a layered control system, cooperating decision-makers are specialized so that they deal with different, separable problems, or so that some decision-makers primarily supervise while others primarily solve the problem.

One theme that has emerged from this inquiry is that certain communication flows are necessary between individual decision-makers in coordinated tasks, and if the tasks and decision-makers are suitably defined, the necessary communication flows can be defined exactly. We see an explosion of complexity related to communication, even to achieve simple tasks, in a decentralized system. A related modeling paper looked specifically at communication flows in the nervous system using this framework (Li*, Sarma*, et al., 2023). Many of the ideas in that paper would apply to other multicellular systems as well.

I’ve also worked on using control-based design theorems as tools themselves to define a class of mathematical models and analyze them as a whole to rule in or rule out hypotheses. So far, I haven’t written about this as its own approach (and I would like to), but we used it in a pre-print modeling complex immune-virus dynamics that would be difficult or impossible to model with typical differential equation approaches (Sarma et al., 2021).

I participated in some applied work on brain-computer interfaces using a mix of control-theoretic techniques and control-theoretic ideas – most importantly, adjusting machine learning models to account for the fact that the unmodeled dynamics in neural data were not noise or adversarial perturbations, but the dynamics of an extant natural control system. Although we didn’t explicitly use decentralized control techniques, conceptualizing the problem as a decentralized control problem led to considerable increases in system robustness and performance (Gilja*, Pandarinath*, et al., 2015; Jarosiewicz, Sarma, et al., 2015; Milekovic*, Sarma*, et al., 2018). There is probably more to say about decentralization and layering as a framework for both conceptual and computational optimization of medical devices and clinical decision-making.

More recently, I’ve worked on some wet lab science studying population-level mechanisms governing the decision of a cell to divide, looking at moon jellies and how cell proliferation in these organisms relates to regeneration and cancer (some related experimental results are presented in Abrams*, Tan*, Li*, et al., 2021). For some ongoing experiments control theory has had a generative conceptual role in framing questions and has also generated mathematical models with testable predictions. I’ll say more about the experimental results in the future, but for this essay, I’ll stick to a few broad observations.

A key idea in developing the mathematical analysis behind these experiments was that a successful homeostatic system necessarily has negative spaces: perturbations that result in no observable physiological change precisely because homeostasis is working effectively. We know from our foundational understanding of implementable systems that all homeostatic systems have failure modes. In robust control theory, we design a decision-maker to be robust to the actions of an adversarial decision-maker. In order to know how the designed decision-maker works, we have to be able to model and simulate the adversary. While the goal of the designed decision-maker is to transform large variations in the unknowns into small variations in the knowns, the goal of the adversary is to transform small variations in the unknowns into large variations in the knowns. This adversarial transformation, which we know, is identical to the behavior of optimal experimental design: you change one thing, and you get a big effect. Conceptually, control theory can be a tool for optimal experimental design as well as engineering design – something I hope to explore further in theorem-based papers.

In addition to the theoretical aspects of experimental design, the particulars of the living system are interesting in their own right. What we are studying in the moon jelly is tissue homeostasis: the mechanisms that maintain a tissue’s size, integrity, and function. From an evolutionary perspective, tissue homeostasis is ancient and a precondition to multicellular life. In vertebrates, immune cells play a key role in regulating tissue homeostasis; arguably, all tissue homeostatic functions are immune functions and all immune system functions are tissue homeostatic functions. If we can learn general principles of tissue homeostasis in the moon jelly, which a theory-driven approach makes possible, these principles could directly inform our understanding of tissue homeostasis in people.

So, somehow, a clear-ish picture emerges. We observe clinical challenges that look too complex for us to model and understand with the math we have. We expand what we know about implementable systems. We use the new mathematical tools to describe general principles in simple systems, to test these principles experimentally, to bound the explosion of complexity and variation that we expect across evolution. We use the new tools to design more robust and higher-performing interventions and test these interventions in optimized and adaptive clinical trials. Sometimes the experiments might guide us back to math. Other times the clinical trials might guide us back to experiments. All of this is possible, all of it proven out in pieces, but there are many surprises yet between the pieces and the whole.