W e are pleased that our essay on process thinking in contemporary science (Jaeger & Monk, 2015) still elicits reflection and constructive discussion. We thank Horsting & Hartjes (2022) for their thoughtful engagement with this fundamental philosophical issue, which has lost none of its relevance since the publication of our essay. Horsting & Hartjes (2022) take issue with our definition of “substance” and our claim that process-based explanations are more fundamental than substance-based ones. We would like to clarify three specific points in reply. First, there is an important reason we use the original philosophical definition of “substance” as that which is “universally and eternally unchangeable” as it reflects the perennial search for simple universal principles that underlie the confusing richness and transient nature of our world. The classic example of such substances is the atoms of Greek philosophy: in the atomist view, processes are mere epiphenomena, more or less fleeting rearrangements of unchangeable atoms. Our essay criticises the echoes of this notion, which still reverberate in contemporary science. Horsting & Hartjes, in contrast, use a more modern definition of “substance” as “material or matter, with definite chemical composition and distinct properties.” Substance defined in this way is not useful as a contradistinction to process. Indeed, it embodies the approach of representing dynamic processes in terms of interactions of relatively autonomous and stable material components. Importantly, however, such components can only be defined in terms of the processes from which they are abstracted: the concepts of “composition” and “properties” themselves are fundamentally relational (e.g. “mass” and “charge” refer to interaction propensities). Furthermore, such components are never truly immutable and must ultimately themselves be viewed as processes. It was never our aim to contrast such physical matter with process; in fact, we argue that it is process. Instead, we compare two different modes of explaining phenomena in the material world, arguing that a process perspective leads to better understanding. Second, we do not say that explanation in terms of process should ignore components and their interactions. What we are saying concretely is that to understand thunderstorms, diseases, thought processes and the like, it is not enough to identify their component parts. Material components are interchangeable and are not what define these processes. We, therefore, advocate a shift of focus from the necessary step of identifying components (composition) to the dynamic interactions between them (activity). This does not overrule or replace the analysis of systems through decomposition into material parts but complements it by a dynamic recomposing of the activity that arises from the concerted interaction of these parts (see DiFrisco & Jaeger, 2019). It is not either components or activities, but both components and activities, that are understood as a (limited but useful) representation of process. In this sense, we agree with Horsting & Hartjes (2022) that the two aspects of a system deserve attention. Third, this does, however, not mean that “substances and processes deserve equal credit in science” or that “process-based research should not come at the expense of the importance of substance-based research.” We highlight that there is still far too much emphasis on the search for unchangeable fundamental principles (substance). Since scientific funding is a zero-sum game, this does indeed imply that we need less substance-based research and more process-oriented approaches, simply because the latter have been sorely neglected. More fundamentally, substances and processes are not equivalent in terms of their explanatory power. Substances ultimately cannot be defined and cannot be conceived as existing outside the context of a process. In contrast, there can be subject-less processes, whose nature involves material entities, but does not depend on any individual component’s presence (the hurricane is not in the air molecules, but in their collective pattern). In this sense, processes are more fundamental than substances—they convey greater explanatory power, capturing phenomena that substancebased approaches cannot. The same is not true the other way around. While representation in terms of material composition provides a powerful analytical tool for the study of processes, the organised activity of components can only be properly understood in a process-oriented framework. In summary, we agree with Horsting & Hastjes that material aspects matter; to deny this point was never our intention. Instead, our essay addresses the philosophical issue

An organism’s phenotype can be thought of as consisting of a set of discrete traits, able to evolve relatively independently of each other. This implies that the developmental processes generating these traits—the underlying genotype-phenotype map—must also be functionally organised in a modular manner. The genotype-phenotype map lies at the heart of evolutionary systems biology. Recently, it has become popular to define developmental modules in terms of the structure of gene regulatory networks. This approach is inherently limited: gene networks often do not have structural modularity. More generally, the connection between structure and function is quite loose. In this chapter, we discuss an alternative approach based on the concept of dynamical modularity, which overcomes many of the limitations of structural modules. A dynamical module consists of the activities of a set of genes and their interactions that generate a specific dynamic behaviour. These modules can be identified and characterised by phase-space analysis of data-driven models. We showcase the power and the promise of this new approach using several case studies. Dynamical modularity forms an important component of a general theory of the evolution of regulatory systems and the genotype-phenotype map they define.

Viscoelastic fluids can be difficult to model due to the wide range of different physical behaviors that polymer melts can exhibit. One such feature is the viscous elastic boundary layer. We address the particular problem of a viscoelastic shear-dependent fluid flowing past a corner and investigate how the properties of the boundary layer change for a White-Metzner fluid. The boundary layer equations are derived and the upstream layer is matched with the far-field flow. It was found that if the fluid is sufficiently shear thinning then the viscoelastic boundary layer formulation fails due to the inertial forces becoming dominant. The depth of the boundary layer is controlled by the shear-thinning parameters. These effects are not a feature of other shear-thinning models, such as the Phan-Thien-Tanner model. This study provides insight in the different effects of some commonly used viscoelastic models in corner flows in the upstream boundary layer, the downstream boundary layer is not addressed.

Transfer learning considers a learning process where a new task is solved by transferring relevant knowledge from known solutions to related tasks. While this has been studied experimentally, there lacks a foundational description of the transfer learning problem that exposes what related tasks are, and how they can be exploited. In this work, we present a definition for relatedness between tasks and identify foliations as a mathematical framework to represent such relationships.

Studying the dynamics of networks rather than the individual components is essential for our understanding of complex regulatory phenomena. Only by adopting process philosophy as the appropriate conceptual framework can the true potential of systems biology be realized.

In this paper, we illustrate how dynamical systems theory can provide a unifying conceptual framework for evolution of biological regulatory systems. Our argument is that the genotype-phenotype map can be characterized by the phase portrait of the underlying regulatory process. The features of this portrait - such as attractors with associated basins and their bifurcations - define the regulatory and evolutionary potential of a system. We show how the geometric analysis of phase space connects Waddington's epigenetic landscape to recent computational approaches for the study of robustness and evolvability in network evolution. We discuss how the geometry of phase space determines the probability of possible phenotypic transitions. Finally, we demonstrate how the active, self-organizing role of the environment in phenotypic evolution can be understood in terms of dynamical systems concepts. This approach yields mechanistic explanations that go beyond insights based on the simulation of evolving regulatory networks alone. Its predictions can now be tested by studying specific, experimentally tractable regulatory systems using the tools of modern systems biology. A systematic exploration of such systems will enable us to understand better the nature and origin of the phenotypic variability, which provides the substrate for evolution by natural selection.

Julian Lewis, developmental biologist, theoretician, writer, polyglot, friend, and inspiration to many, has died aged 67. Known for his gentle personality and rare combination of expertise in physics, mathematics, and biology, Julian was widely acknowledged as a brilliant and original scientist who made many important contributions to the field of developmental biology. In particular, it was through his analysis of the Notch signaling pathway—in a whole range of tissues—that Julian made his mark, driving this research field in new directions using both careful theoretical reasoning and elegant experimental design. Julian began his career in the physical sciences, reading physics at Balliol College as an undergraduate and staying on in Oxford to study for a DPhil. After a postdoctoral year at the Institute for Physical Problems in Moscow, he began work on the chick embryo in the laboratory of Lewis Wolpert at the Middlesex Hospital Medical School, London. He worked as a lecturer in the Department of Anatomy at King’s College London for 8 years before he joined the Imperial Cancer Research Fund’s Developmental Biology Unit in Oxford as a group leader in 1986. Ten years later, the lab moved to the London Research Institute at Lincoln’s Inn Fields in London, where Julian worked until his retirement in 2012. He continued with research and writing in an emeritus capacity, with his latest paper appearing in April 2014. Those who were fortunate enough to have passed through Julian’s lab recall a supportive and delightfully quirky mentor who was a mine of information and expertise. Julian was interested in everything and read extremely widely. Conversations ranged from the family to French grammar. His physics background meant that he understood the optics of a compound or confocal microscope inside out, and he loved to help in setting everything up to get the very best out of any equipment in the lab. But he wasn’t only concerned with scientific contributions: Julian was always interested to hear how the families of lab members were doing and loved nothing better than to hold and admire a new baby. Julian’s research interests were broad, spanning several different animal model systems and a plethora of different tissues. Nevertheless, a common thread ran through much of his work: he would distill a theoretical model from biological observations and then set out to test it rigorously. Graduate students and post docs were kept busy with his demands for quantitative data: how long, how many, how much, how far, how fast? This approach is illustrated well by his work on the Notch pathway—an enduring interest spanning more than 20 years of his research career. Julian’s interest in Notch stemmed partly from his fascination with the vertebrate inner ear. He admired the beauty of the regularly spaced pattern of hair cells in the sensory patches of the ear—each hair cell surrounded and insulated by supporting cells—and surmised that this arrangement must be the result of a lateral inhibition mechanism, like that operating in the fly nervous system (Lewis, 1991Lewis J. Rules for the production of sensory cells.in: Bock G.R. Whelan J. Regeneration of Vertebrate Sensory Receptor Cells. Volume 160. John Wiley and Sons, Chichester1991Google Scholar). Julian created and explored a simple yet influential mathematical model of lateral inhibition (Collier et al., 1996Collier J.R. Monk N.A.M. Maini P.K. Lewis J.H. Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling.J. Theor. Biol. 1996; 183: 429-446Crossref PubMed Scopus (329) Google Scholar) and, at the same time, set out to demonstrate that Notch signaling did indeed mediate lateral inhibition in the ear. It was at this time that the first large-scale zebrafish mutagenesis screens were uncovering genes with key roles in vertebrate embryonic development. Julian was impatient to know the results of the screens; he would predict the phenotypes that should be revealed, and, when the classes of mutants were announced, he was intrigued by those that had not appeared in the collection. One mutant in particular caught his eye: mind bomb, so called for its huge excess of neurons in the developing brain, which developed at the expense of other cell types, such as the neural crest. Julian had been expecting just such a mutant to turn up, and he speculated that the mind bomb mutant ear should contain supernumerary sensory hair cells at the expense of supporting cells. Catherine Haddon performed the analysis, and the rest of the lab waited for the result with a real sense of suspense. The ear phenotype was exactly as Julian had predicted: where there should have been just a pair of sensory hair cells, a group of 30 or so were differentiating all together in the sensory patches, and supporting cells were missing (Haddon et al., 1998Haddon C. Jiang Y.-J. Smithers L. Lewis J. Delta-Notch signalling and the patterning of sensory cell differentiation in the zebrafish ear: evidence from the mind bomb mutant.Development. 1998; 125: 4637-4644PubMed Google Scholar). The task now was to identify the gene disrupted in mind bomb. Julian was entirely wedded to the idea that the mutation must abrogate Notch signaling, and post doc Yun-Jin Jiang set out on a candidate approach, sequencing the zebrafish homolog of every known Notch pathway component in the mutant—a painstaking and lengthy exercise at that time. As gene after gene was ticked off the list, with no mutation in sight, one wondered whether Julian’s logic was flawed. Nevertheless, he was proved right in the end: mind bomb turned out to code for a ubiquitin ligase with a critical role in the Notch pathway (Itoh et al., 2003Itoh M. Kim C.H. Palardy G. Oda T. Jiang Y.-J. Maust D. Yeo S.Y. Lorick K. Wright G.J. Ariza-McNaughton L. et al.Mind bomb is a ubiquitin ligase that is essential for efficient activation of Notch signaling by Delta.Dev. Cell. 2003; 4: 67-82Abstract Full Text Full Text PDF PubMed Scopus (632) Google Scholar). The mind bomb hunt exemplifies how Julian clung to his ideas tenaciously. He would not hesitate to fire off an email to authors asking for more information if he doubted the results in a paper and was reluctant to accept anything—sometimes including the opinions of his collaborators—that went against his own convictions. This could result in surprisingly fierce exchanges from someone who was usually so mild mannered. Even so, unexpected experimental findings often resulted in his delighted surprise. His papers are full of disarming expressions such as “This is not what we found” and “Our expectations were confounded” and “Curiously, however,…”. Julian also studied Notch signaling in the central nervous system, gut, vasculature, and somites. In the latter, components of the Notch pathway were found to be oscillating as part of a molecular clock. The data, however, were confusing; although it was clear that disruptions to the Notch pathway perturbed somite formation, it was not obvious what role Notch was playing. Julian cut through the molecular complexity by proposing a strikingly simple hypothesis: that during somitogenesis, Notch signaling couples neighboring cells to keep their oscillations in synchrony (Jiang et al., 2000Jiang Y.-J. Aerne B.L. Smithers L. Haddon C. Ish-Horowicz D. Lewis J. Notch signalling and the synchronization of the somite segmentation clock.Nature. 2000; 408: 475-479Crossref PubMed Scopus (402) Google Scholar). Such synchronization is well known (and very well studied) in physical systems, but the boldness of Julian’s idea was in giving Notch such a generic role in a biological system. He went on to propose a simple mechanism for the clock itself, speculating that oscillations were driven by a delayed transcriptional feedback (Lewis, 2003Lewis J. Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator.Curr. Biol. 2003; 13: 1398-1408Abstract Full Text Full Text PDF PubMed Scopus (552) Google Scholar). Guided by Julian’s modeling, the group of Ryoichiro Kageyama soon provided corroboration of the clock model by breaking the mechanism itself, while a beautiful set of experiments from the group of Andy Oates lent convincing support to the synchronization proposal. Less-tenacious researchers would have left it there, but Julian wanted to go further, aiming to measure the basic parameters of his proposed mechanisms. This was a daunting prospect, and to achieve his goal he had to develop creative and original approaches. An elegant paper followed, in which his group determined the production delays and molecular lifetimes of the core clock components, demonstrating that they fit perfectly with the requirements of his models (Giudicelli et al., 2007Giudicelli F. Özbudak E.M. Wright G.J. Lewis J. Setting the tempo in development: an investigation of the zebrafish somite clock mechanism.PLoS Biol. 2007; 5: e150Crossref PubMed Scopus (127) Google Scholar). He even went so far as to measure the elongation rate of RNA polymerase II in zebrafish embryos (Hanisch et al., 2013Hanisch A. Holder M.V. Choorapoikayil S. Gajewski M. Özbudak E.M. Lewis J. The elongation rate of RNA polymerase II in zebrafish and its significance in the somite segmentation clock.Development. 2013; 140: 444-453Crossref PubMed Scopus (38) Google Scholar); it was unexpectedly high, overwriting the oft-quoted textbook values. This showed that other delays, such as those resulting from splicing, must be important for clock function. In his latest paper, Julian and colleagues extended these techniques to provide a direct test of the idea that Notch signaling acts to synchronize the somite oscillator. Here, they restored clock function to a zebrafish deltaC mutant by giving repeated pulses of deltaC expression from a heat-shock-driven transgene (Soza-Ried et al., 2014Soza-Ried C. Öztürk E. Ish-Horowicz D. Lewis J. Pulses of Notch activation synchronise oscillating somite cells and entrain the zebrafish segmentation clock.Development. 2014; 141: 1780-1788Crossref PubMed Scopus (36) Google Scholar). Somitogenesis was rescued, with somite size corresponding to the time interval between heat shock pulses: small somites for rapid pulsing and larger ones for longer time intervals. There is a beautiful and powerful symmetry in these most recent papers. It is widely appreciated that during somitogenesis the embryo uses a cellular temporal oscillation to establish a multicellular spatial pattern. Julian’s ingenuity was to invert this logic, inferring quantitative details of the underlying cellular processes from the resulting spatial pattern. This is a clever example of reverse engineering and illustrates Julian’s creative thinking at its best. Although Julian himself was comfortable juggling complex theoretical ideas with experimental results, he always took the trouble to explain what he meant very carefully. He was fond of analogies, which kept popping up in his papers: sailors at sea, children swinging in a playground, the parts of a physical clock, and a magnetic tape recorder were all used to illustrate his scientific ideas. From 1979 until his death, Julian was often occupied with “The Book”—Molecular Biology of the Cell—which he coauthored with several others, all highly influential leaders in their fields (Alberts et al., 2008Alberts B. Johnson A. Lewis J. Raff M. Roberts K. Walter P. Molecular Biology of the Cell. Garland Science, New York2008Google Scholar). First published in 1983, MBoC has been widely accepted as the text for students of molecular and cellular biology for over 30 years; now in its fifth edition, it has been translated into many different languages. According to Martin Raff, Julian rescued MBoC from a near nonstart when he joined the team, bearing a chapter on developmental biology that he had coauthored with Cheryll Tickle: “It was a gem—beautifully written and covering the subject with clarity, originality, and flair—better than anything we could have hoped for. It lifted our spirits and enabled us to keep going.” Through the book, Julian and the other authors have inspired innumerable undergraduate students, and his crystal-clear prose will reach students for many years to come. The book meetings occupied a lot of time, as Julian pointed out in the preface to the first edition, in his own inimitable way: “This is a large book, and it has been a long time in gestation—three times longer than an elephant, five times longer than a whale.” Marine mammals perhaps provide a clue to the distractions in getting the book written, as the meetings were often held on the Californian coast. Julian wrote: “Skived off today at lunchtime to see the elephant seals, whose way of life is frankly decadent. Nine months diving up and down (to 2,000–5,000 ft!) gorging themselves and getting fat (5,000 lbs), then three months lying on a beach doing nothing—not even talking to one another.” Other projects sometimes got rather behind, but it was impossible to be cross with someone with such a delicious sense of humor. In contrast to the fluency of his written prose, Julian had a manner of speaking that could appear hesitant, even unsure, at times. But this was deceptive; far from being uncertain, he was merely taking his time to construct the best and clearest way of getting his argument across. He had the uncanny ability to appear to be asleep throughout a seminar and then ask a highly pertinent question at the end. Nicolas Daudet recalls: “Julian’s ‘power naps’ were legendary and a topic of amusement in the lab, not least for him: he arrived one day for his journal club with a paper demonstrating the benefits of short daytime naps on cognitive performance.” Journal editors have appreciated his thoughtful and constructive comments as a manuscript reviewer; authors, too, often unwittingly, will have benefitted from his intellectual insights. The British Society for Developmental Biology awarded Julian the Waddington Medal in 2003 for his lifetime contributions to the community. He was elected an EMBO member in 2005 and a Fellow of the Royal Society in 2012. Thankfully, his inaugural presentation was recorded (https://royalsociety.org/people/fellowship/2012/julian-lewis), and we can listen again to his gentle voice and clear reasoning. Julian was funded for much of his independent career by a cancer charity—initially the Imperial Cancer Research Fund, which later merged with the Cancer Research Campaign to become Cancer Research UK in 2002. The ICRF and CRUK were insightful in providing funding to scientists working on problems of basic developmental biology, recognizing that such research could lead to valuable and relevant insights into disease mechanism. Julian was skillful in explaining his research in simple terms to visitors to the lab, including charity supporters, providing a persuasive argument for why the secrets of cancer and future cures could be revealed through study of the somitogenesis clock. And, of course, he was absolutely right: disrupted Notch signaling is now widely appreciated to underlie many cancers. Julian’s work stands out as a powerful illustration of the value and impact of basic science on clinical understanding. Julian was diagnosed with prostate cancer in 2004. He bore his illness with great stoicism and positivity, combined with a genuine curiosity and interest about the biological mechanisms underlying his disease. Still working at CRUK, he was enrolled in clinical trials for new treatments and would give talks to both scientists and patient groups about his condition. “This is me,” he would say, pointing to one of the dots on a graph of the trial results, without either self-aggrandizement or self-pity. Leading by example, he became a most powerful advocate and ambassador for the charity that had funded his own research for so long. He also helped and supported others with cancer, including Paul Martin, his former PhD student. Julian died at home, surrounded by his family. He leaves his wife Sherry and three daughters, Emma, Sarah, and Rebecca. In addition to being a wonderful and loving father to his own children, he was also a father figure to many who passed through the lab. Julian’s death was summed up poignantly by Paul: “Feels like my dad has gone.” He will be much missed. We thank several former Lewis lab members—Nicolas Daudet, Suresh Jesuthasan, Yun-Jin Jiang, Paul Martin, and Sylvie Schneider-Maunoury—together with Sherry Granum, David Ish-Horowicz, and Martin Raff for sharing their reminiscences and recollections of Julian for the preparation of this piece.

A central unresolved problem of evolutionary biology concerns the way in which evolution at the genotypic level relates to the evolution of phenotypes. This genotype-phenotype map involves developmental and physiological processes, which are complex and not well understood. These processes co-determine the rate and direction of adaptive change by shaping the distribution of phenotypic variability on which selection can act. In this study, we argue-expanding on earlier ideas by Goodwin, Oster, and Alberch-that an explicit treatment of this map in terms of dynamical systems theory can provide an integrated understanding of evolution and development. We describe a conceptual framework, which demonstrates how development determines the probability of possible phenotypic transitions-and hence the evolvability of a biological system. We use a simple conceptual model to illustrate how the regulatory dynamics of the genotype-phenotype map can be passed on from generation to generation, and how heredity itself can be treated as a dynamic process. Our model yields explanations for punctuated evolutionary dynamics, the difference between micro- and macroevolution, and for the role of the environment in major phenotypic transitions. We propose a quantitative research program in evolutionary developmental systems biology-combining experimental methods with mathematical modeling-which aims at elaborating our conceptual framework by applying it to a wide range of evolving developmental systems. This requires a large and sustained effort, which we believe is justified by the significant potential benefits of an extended evolutionary theory that uses dynamic molecular genetic data to reintegrate development and evolution.

Understanding the relationship of the size and shape of an organism to the size, shape, and number of its constituent cells is a basic problem in biology; however, numerous studies indicate that the relationship is complex and often nonintuitive. To investigate this problem, we used a system for the inducible expression of genes involved in the G1/S transition of the plant cell cycle and analyzed the outcome on leaf shape. By combining a careful developmental staging with a quantitative analysis of the temporal and spatial response of cell division pattern and leaf shape to these manipulations, we found that changes in cell division frequency occurred much later than the observed changes in leaf shape. These data indicate that altered cell division frequency cannot be causally involved in the observed change of shape. Rather, a shift to a smaller cell size as a result of the genetic manipulations performed correlated with the formation of a smoother leaf perimeter, i.e. appeared to be the primary cellular driver influencing form. These data are discussed in the context of the relationship of cell division, growth, and leaf size and shape.

There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated, or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. In particular, the influence of gene expression on a large class of Turing bifurcation models, namely those with pure kinetics such as the Gierer–Meinhardt system, is unexplored. Our investigations demonstrate that the behaviour of the Gierer–Meinhardt model profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression. Features such as concentration blow up, morphogen oscillations and radical sensitivities to the duration of gene expression are observed and, at best, severely restrict the possible parameter spaces for feasible biological behaviour. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics. Finally, this study also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens.

Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles, and feathers. Here, we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case, polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion.

Positional specification by morphogen gradients is traditionally viewed as a two-step process. A gradient is formed and then interpreted, providing a spatial metric independent of the target tissue, similar to the concept of space in classical mechanics. However, the formation and interpretation of gradients are coupled, dynamic processes. We introduce a conceptual framework for positional specification in which cellular activity feeds back on positional information encoded by gradients, analogous to the feedback between mass-energy distribution and the geometry of space-time in Einstein's general theory of relativity. We discuss how such general relativistic positional information (GRPI) can guide systems-level approaches to pattern formation.

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.