Oren Shoval

The Incoherent Feedforward Loop Can Provide Fold-Change Detection in Gene Regulation

Many sensory systems (e.g., vision and hearing) show a response that is proportional to the fold-change in the stimulus relative to the background, a feature related to Webers Law. Recent experiments suggest such a fold-change detection feature in signaling systems in cells: a response that depends on the fold-change in the input signal, and not on its absolute level. It is therefore of interest to find molecular mechanisms of gene regulation that can provide such fold-change detection. Here, we demonstrate theoretically that fold-change detection can be generated by one of the most common network motifs in transcription networks, the incoherent feedforward loop (I1-FFL), in which an activator regulates both a gene and a repressor of the gene. The fold-change detection feature of the I1-FFL applies to the entire shape of the response, including its amplitude and duration, and is valid for a wide range of biochemical parameters.

Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space

Managing Trade-Offs Most organisms experience selection on a host of traits to determine their likelihood to succeed evolutionarily. However, specific traits may experience trade-offs in determining an organisms optimal phenotype. Shoval et al. (p. 1157; see the Perspective by Noor and Milo) relate physical traits to the task that they are optimizing using a Pareto curve, a power law probability distribution, to show that a single set of trait values optimizes performance at a given task and that performance decreases as an organisms phenotype moves away from this set of trait values. The results suggest how selection makes the best trade-offs for an arbitrary number of tasks and traits and may explain examples of evolutionary variation. The fitness of an organism can be modeled graphically to determine how phenotypic trade-offs are maximized. Biological systems that perform multiple tasks face a fundamental trade-off: A given phenotype cannot be optimal at all tasks. Here we ask how trade-offs affect the range of phenotypes found in nature. Using the Pareto front concept from economics and engineering, we find that best–trade-off phenotypes are weighted averages of archetypes—phenotypes specialized for single tasks. For two tasks, phenotypes fall on the line connecting the two archetypes, which could explain linear trait correlations, allometric relationships, as well as bacterial gene-expression patterns. For three tasks, phenotypes fall within a triangle in phenotype space, whose vertices are the archetypes, as evident in morphological studies, including on Darwin’s finches. Tasks can be inferred from measured phenotypes based on the behavior of organisms nearest the archetypes.

Fold-change detection and scalar symmetry of sensory input fields

Recent studies suggest that certain cellular sensory systems display fold-change detection (FCD): a response whose entire shape, including amplitude and duration, depends only on fold changes in input and not on absolute levels. Thus, a step change in input from, for example, level 1 to 2 gives precisely the same dynamical output as a step from level 2 to 4, because the steps have the same fold change. We ask what the benefit of FCD is and show that FCD is necessary and sufficient for sensory search to be independent of multiplying the input field by a scalar. Thus, the FCD search pattern depends only on the spatial profile of the input and not on its amplitude. Such scalar symmetry occurs in a wide range of sensory inputs, such as source strength multiplying diffusing/convecting chemical fields sensed in chemotaxis, ambient light multiplying the contrast field in vision, and protein concentrations multiplying the output in cellular signaling systems. Furthermore, we show that FCD entails two features found across sensory systems, exact adaptation and Webers law, but that these two features are not sufficient for FCD. Finally, we present a wide class of mechanisms that have FCD, including certain nonlinear feedback and feed-forward loops. We find that bacterial chemotaxis displays feedback within the present class and hence, is expected to show FCD. This can explain experiments in which chemotaxis searches are insensitive to attractant source levels. This study, thus, suggests a connection between properties of biological sensory systems and scalar symmetry stemming from physical properties of their input fields.

Developmental bias in the evolution of phalanges

Significance It has long been proposed that rules stemming from the mechanisms used during development can constrain the range of evolvable variations in a given form, but few empirical examples are known. We have focused on developmental processes determining proportions of phalanx size along individual digits (fingers/toes) of vertebrates. We find that phalangeal variation seen in nature is indeed constrained by an ancestral developmental program, limiting morphologies to a continuum from nearly equal-sized phalanges to a large-to-small gradient of relative sizes. Nonetheless, later innovations in distal regulation expanded variational possibilities for groups that needed greater grasping ability. These data provide a better understanding of how properties of developmental systems work in combination with natural selection to guide evolution of skeletal proportions. Evolutionary theory has long argued that the entrenched rules of development constrain the range of variations in a given form, but few empirical examples are known. Here we provide evidence for a very deeply conserved skeletal module constraining the morphology of the phalanges within a digit. We measured the sizes of phalanges within populations of two bird species and found that successive phalanges within a digit exhibit predictable relative proportions, whether those phalanges are nearly equal in size or exhibit a more striking gradient in size from large to small. Experimental perturbations during early stages of digit formation demonstrate that the sizes of the phalanges within a digit are regulated as a system rather than individually. However, the sizes of the phalanges are independent of the metatarsals. Temporal studies indicate that the relative sizes of the phalanges are established at the time of initial cell condensation. Measurements of phalanges across species from six major taxonomic lineages showed that the same predictable range of variants is conserved across vast taxonomic diversity and evolutionary time, starting with the very origins of tetrapods. Although in general phalangeal variations fall within a range of nearly equal-sized phalanges to those following a steep large-to-small gradient, a novel derived condition of excessive elongation of the distal-most phalanges has evolved convergently in multiple lineages, for example under selection for grasping rather than walking or swimming. Even in the context of this exception, phalangeal variations observed in nature are a small subset of potential morphospace.

The geometry of the Pareto front in biological phenotype space

When organisms perform a single task, selection leads to phenotypes that maximize performance at that task. When organisms need to perform multiple tasks, a trade-off arises because no phenotype can optimize all tasks. Recent work addressed this question, and assumed that the performance at each task decays with distance in trait space from the best phenotype at that task. Under this assumption, the best-fitness solutions (termed the Pareto front) lie on simple low-dimensional shapes in trait space: line segments, triangles and other polygons. The vertices of these polygons are specialists at a single task. Here, we generalize this finding, by considering performance functions of general form, not necessarily functions that decay monotonically with distance from their peak. We find that, except for performance functions with highly eccentric contours, simple shapes in phenotype space are still found, but with mildly curving edges instead of straight ones. In a wide range of systems, complex data on multiple quantitative traits, which might be expected to fill a high-dimensional phenotype space, is predicted instead to collapse onto low-dimensional shapes; phenotypes near the vertices of these shapes are predicted to be specialists, and can thus suggest which tasks may be at play.

Comparing Apples and Oranges: Fold-Change Detection of Multiple Simultaneous Inputs

Sensory systems often detect multiple types of inputs. For example, a receptor in a cell-signaling system often binds multiple kinds of ligands, and sensory neurons can respond to different types of stimuli. How do sensory systems compare these different kinds of signals? Here, we consider this question in a class of sensory systems – including bacterial chemotaxis- which have a property known as fold-change detection: their output dynamics, including amplitude and response time, depends only on the relative changes in signal, rather than absolute changes, over a range of several decades of signal. We analyze how fold-change detection systems respond to multiple signals, using mathematical models. Suppose that a step of fold F1 is made in input 1, together with a step of F2 in input 2. What total response does the system provide? We show that when both input signals impact the same receptor with equal number of binding sites, the integrated response is multiplicative: the response dynamics depend only on the product of the two fold changes, F1F2. When the inputs bind the same receptor with different number of sites n1 and n2, the dynamics depend on a product of power laws, . Thus, two input signals which vary over time in an inverse way can lead to no response. When the two inputs affect two different receptors, other types of integration may be found and generally the system is not constrained to respond according to the product of the fold-change of each signal. These predictions can be readily tested experimentally, by providing cells with two simultaneously varying input signals. The present study suggests how cells can compare apples and oranges, namely by comparing each to its own background level, and then multiplying these two fold-changes.

Input symmetry invariance, and applications to biological systems

This paper studies invariance with respect to symmetries in sensory fields, a particular case of which, scale-invariance, has recently been found in certain eukaryotic as well as bacterial cell signaling systems. We describe a necessary and sufficient characterization of symmetry invariance in terms of equivariant transformations, show how this characterization helps find all possible symmetries in standard models of biological adaptation, and discuss symmetry-invariant searches

Response to Comment on "Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space"

Edelaar raises concerns about the way we tested our theory. Our mathematical theorem predicts that despite the high dimensionality of trait space, trade-offs between tasks leads to phenotypes in low-dimensional regions in trait space, such as lines and triangles. We address Edelaars questions with statistical tests that eliminate pseudoreplication concerns, finding that our predictions remain convincingly supported.