Beyond D’Arcy Thompson: Future challenges for quantitative biology

Abstract

The centennial of “On Growth and Form” is a good opportunity to reflect on the progress in the quantitative study of living systems and where we would like to see it heading. The era of the physical sciences being a mere vehicle for tool building for biological investigations is over. The approaches taken nowadays are analogous to those that physical scientists have taken within their respective fields for centuries, only that now they ask them about biological phenomena and function. Here I give a brief reflection on where we are and where we should head next, both from the perspective of the research endeavor as a whole, but also with respect to teaching the next generation of scientists joining the field.

D’Arcy Thompson’s highly cited book, “On Growth and Form” [1], admirably encapsulates the use of physical and geometric principles in biology, in a manner that was well ahead of its time. Thompson’s work was widely admired by biologists, physical scientists, and anthropologists, and thereby highly influential across these fields. His elegant descriptions of the scale and shape of living structures inspired numerous influential thinkers, including biologist Conrad Waddington, mathematician Alan Turing, and anthropologist Claude Lévi-Strauss. Its impact reached well beyond the scientific community, as a fine work of literature; including an active influence on modern architecture [2].

“On Growth and Form” has played a pioneering role in the emergence of a subfield of developmental biology: the intersection of morphogenesis with the physical sciences and mathematics. Nowadays, this interdisciplinary path has become well established, and its core goal is a mathematical formulation of the physical mechanisms underlying morphogenesis. Strikingly, many current themes in the field are foreshadowed in Thompson’s work: the pursuit of general theories of biology based on physical constraints [3,4,5] and the importance of emergent structures in the organization of biological systems [6,7,8]. The centennial of “On Growth and Form” provides an occasion to reflect on how the study of morphogenesis has progressed and where we would like to see it head. In doing so, it is helpful to not only recognize the founding contributions of Thompson’s quantitative take on morphogenesis to the field, but also its shortcomings, such as the lack of motivating experiments and testable predictions, of which Thompson was well aware, as well as a rejection of natural selection, bordering on the ideas of vitalism [9,10]. The latter, of course, is difficult to reconcile with the central role that evolution plays in modern morphogenesis. As such the development of the field has differed from Thompson’s early vision and these differences will play a prominent role in the future.

Since the publication of Thompson’s book, the field has developed dramatically. We have fully integrated the notion that qualitatively phenomena, similar to those highlighted by Thompson, have correspondingly deep explanations that go beyond a descriptive nature. As such, from a modern perspective, adhering to role models who have lead the transition from qualitative to quantitative in the life sciences may be more productive. The pioneering works of Galvani and Volta on animal electricity that led to the invention of the first battery [11], or the tight collaboration between experiment and mathematical modeling at the core of the triumph of Hodgkin and Huxley’s understanding of action potentials in the squid giant axon [12], are perhaps a better guide and inspiration to the aspiring young scientists. More broadly, most people motivated by problems at the interface between physics and biology have reveled in Schrödinger’s little book “What is Life?”[13]; and some of the excitement in the early part of the twentieth century about the subject is brilliantly caught by the essays by Bohr [14] and Delbrück [15,16]. Furthermore, some of the classical physicists such as Ohm [17], Helmholtz [18,19], and Rayleigh [20] have put their mark at the advent of this field.

Today, the use of physical principles and mathematics in studies of morphogenesis – and generally in the life sciences – is widespread and generally accepted as a powerful and important tool. It is therefore timely to reflect on where we are and where we should head next, also with respect to teaching the next generation of scientists joining the field. The 21^st century marks an era in which a plethora of quantitative data is available. The availability of increasingly precise data opens the door to dissect intricate problems, where qualitative description has merely limited prospects to lead to further progress. Quantitative analyses may now test theoretical predictions and confronted with falsifiable mathematical models. This scenario represents a stark contrast to the past when experimental data collected by biologists was not easily converted into numbers and theories weren’t necessarily falsifiable. With the advent of sequencing and quantitative microscopy, however, data is inherently represented as numbers, and there is a sense that biological complexity can be tackled if the right mathematical framework can be found [21].

The landscape of the field is constantly changing: not only that the availability of quantitative data is increased, but there is also an increased ability of what scientists can do with these numbers. Biochemistry has been, to a certain degree, quantitative for a long time; but even if the outcome of an experiment was represented by a number, there has not been a sense that this number should serve as a test for a mathematical model or a larger physical principle. This fundamental change is also altering the kinds of questions that can be asked, by offering a theory or a model that can drive these questions. The era of the physical sciences being a mere vehicle for tool building for biological investigations is over. The questions we ask nowadays are questions physical scientists have asked within their respective fields for centuries, only that now they ask them about biological phenomena and function [6].

Some examples of systems where a physical science approach has born fruit: cellular identities arise through the interaction of many genes within regulatory networks [22, 23, 24], computation in the nervous system originates from interactions among many neurons [25, 26], schools of fish or flocks of birds or swarms of insects grow out of social interactions among hundreds or thousands of individuals [27, 28, 29], and even the structures of individual protein molecules emerge from interactions among hundreds of amino acids [30, 31]; it should be possible to characterize all these collective phenomena in a common language, e.g. that of statistical physics [4]. Single cells adapt by adjusting the expression levels of genes, and networks of genes determine cellular fates; individual neurons adapt to their inputs, and networks of neurons learn; organisms evolve to match their environments, and the immune system provides an example of evolution in a microcosm; in each case the dynamics implements an algorithm that drives toward improved performance, and in many cases that performance is determined by the physics of the problem that the system has been selected to solve [32–35].

With this context in mind, what are the future challenges for the field? After a long period during which the contribution of the physical sciences was crucial in providing the tools for discovering the building blocks of biology, while at the same time mathematics entered as a language to describe interactions of these components through phenomenological modeling, the 21^st century might be the time when the physical sciences should reach for an even bolder approach. Both at the level of theory and experimentation, the physical scientist should aspire to a level of precision and rigor in the life sciences that have been so successfully used for understanding inert matter: aiming for unifying principles that explain many biological phenomena at once and providing measurements that allow distinctions between competing quantitative theories.

For example, Berg and Purcell have worked out a limit analogous to shot noise in counting photons, at which bacteria cease to sense concentration gradients due to the random nature of molecular motion [36]. Under most conditions, both photons and molecules arrive at random at their respective sensor and thus there must be shot noise in counting such arrival events. It has been shown that bacteria are indeed able to operate reliably at this fundamental limit, but only if the cell compares independent concentration measurements over time [37]. Later it was shown that this same limit is also encountered at the level of gene regulation in the arrival of transcription factors molecules arriving at regulatory DNA binding sites to activate transcription [38,39]. In this case it was even possible to separate the components of spatial vs. temporal averaging necessary to overcome the observed noise levels [40].

The history of physics makes apparent that progress is facilitated by a lively interplay between theory and experiment. On the one hand, theories predict new places to look for insight and thereby generate new experimental challenges, and on the other hand, theories in turn are challenged by new experiments that point to places that are not encompassed by the initial theory. The most prominent recent examples for this interplay is the discovery of the Higgs particle in fundamental high-energy physics [41], or the discovery of gravitational waves and their relationship to Einstein’s general relativity theory [42]. In both cases theories predicted the existence of some new fundamental state of matter, and in both cases the experimental validation that came many years after the prediction brought with it new puzzles that require our thinking to look past the current theories.

Central to both of these examples was that the theories had produced testable quantitative predictions for which experiments could be designed that could challenge the prediction. In fact, for these kinds of interplays between theory and experiment, both the predictions and the measurements produce numbers that can be unequivocally compared to each other in a way that error bars and the origin of these error bars matter. Such interplays, which hold the highest promise for uncovering fundamental laws in the physical sciences, is still largely missing in our investigation of the mysteries of living matter. But therein lies the hope for the future of quantitative biology, in going beyond mathematical descriptions of complex biological phenomena towards a principle driven understanding that employs the formulation of actual theories that can be tested with physics-style measurements [4].

Traditionally, our thinking about biological problems has been centered on a particular model organism these problems could be addressed in, which naturally created boundaries for both tool development and the kinds of questions that could be asked. Some of these boundaries have been overcome by the recent dramatic developments in biology itself, facilitating the use of a common set of molecular tools across a broad swath of organisms. For example, we commonly use the same techniques to analyze genetic circuits in bacteria that we use to understand the differentiation of various types of neurons in the human brain [43,44].

But also the conceptual questions that are asked now increasingly so cut across the natural organization of biological hierarchies. For example the problem of functional reliability in the presence of molecular noise is faced by almost every biological process and can therefore be addressed in a multitude of different contexts. Individual cells must sense their physical environment and determine their own internal state from signals that are encoded, e.g. in graded concentrations of messenger molecules: how do cells tune the signal-to-noise ratio and achieve efficient coding of information in all these cases? The above-mentioned Berg and Purcell limit has been used to understand sensing in bacterial chemotaxis but it has also been successfully applied to gene regulation and cell fate decisions.

It remains to be seen whether these questions, which manifestly have more of a physical science flavor to them, will in fact lead to a more principled and unifying understanding of living matter. But starting to ask these new questions is making an impact on both biology and on the physical sciences, which nurtures the hope that a new branch of science comprising the interplay between theory and experiment might emerge that centers around the beautiful and intricate phenomena of modern biology.

As much as D’Arcy Thompson’s spirit at the foundations of mathematical biology was avant-garde in the previous century, leading to the emergence of the interplay between biology and the physical sciences, there is a renewed sense of excitement at this interface at the beginning of the 21^st century. Many biologists believe that we are at the brink of another revolution that turns biology into a quantitative science, much like the physical sciences themselves. In parallel, the physical science community has embraced the complexity of the striking phenomena of life as a chance to broaden the boundaries of their respective fields rather than just an opportunity for application of well-established tools and approaches to biology. At the end of this process the distinction between life and physical sciences will vanish, and hopefully an understanding of the living world will emerge that matches the profound understanding that the physical sciences have established for the inanimate world.

In order to achieve these goals we need to rethink our approach to teaching aspiring young scientists about the phenomena of life. Biology alone can no longer be taught as a non-quantitative science. The early curriculum of a beginning biologist needs to have a heavy mathematical and physical science component, and that component needs to accompany the students along every step of their life science education. At the same time, aspiring physical scientists need to be brought very early to the beautiful phenomena of the functional behavior of living systems. But these systems provide obvious entry points into our teaching of physics, chemistry, engineering, and even computer science, since living systems can be analyzed at these various levels of description. There is therefore a real opportunity to make a profound impact on the field at the earliest levels of education such that in due time the boundary between the physical sciences and biology will indeed erode.