Phase Transition to Chaos in Complex Ecosystems with Nonreciprocal Species-Resource Interactions.

Nonreciprocal interactions between microscopic constituents can profoundly shape the large-scale properties of complex systems. Here, we investigate the effects of nonreciprocity in the context of theoretical ecology by analyzing a generalization of MacArthur's consumer-resource model with asymmetric interactions between species and resources. Using a mixture of analytic cavity calculations and numerical simulations, we show that such ecosystems generically undergo a phase transition to chaotic dynamics as the amount of nonreciprocity is increased. We analytically construct the phase diagram for this model and show that the emergence of chaos is controlled by a single quantity: the ratio of surviving species to surviving resources. We also numerically calculate the Lyapunov exponents in the chaotic phase and carefully analyze finite-size effects. Our findings show how nonreciprocal interactions can give rise to complex and unpredictable dynamical behaviors even in the simplest ecological consumer-resource models.

A universal niche geometry governs the response of ecosystems to environmental perturbations.

How ecosystems respond to environmental perturbations is a fundamental question in ecology, made especially challenging due to the strong coupling between species and their environment. Here, we introduce a theoretical framework for calculating the linear response of ecosystems to environmental perturbations in generalized consumer-resource models. Our construction is applicable to a wide class of systems, including models with non-reciprocal interactions, cross-feeding, and non-linear growth/consumption rates. Within our framework, all ecological variables are embedded into four distinct vector spaces and ecological interactions are represented by geometric transformations between these spaces. We show that near a steady state, such geometric transformations directly map environmental perturbations - in resource availability and mortality rates - to shifts in niche structure. We illustrate these ideas in a variety of settings including a minimal model for pH-induced toxicity in bacterial denitrification.

Phase transition to chaos in complex ecosystems with non-reciprocal species-resource interactions.

Non-reciprocal interactions between microscopic constituents can profoundly shape the large-scale properties of complex systems. Here, we investigate the effects of non-reciprocity in the context of theoretical ecology by analyzing a generalization of MacArthur's consumer-resource model with asymmetric interactions between species and resources. Using a mixture of analytic cavity calculations and numerical simulations, we show that such ecosystems generically undergo a phase transition to chaotic dynamics as the amount of non-reciprocity is increased. We analytically construct the phase diagram for this model and show that the emergence of chaos is controlled by a single quantity: the ratio of surviving species to surviving resources. We also numerically calculate the Lyapunov exponents in the chaotic phase and carefully analyze finite-size effects. Our findings show how non-reciprocal interactions can give rise to complex and unpredictable dynamical behaviors even in the simplest ecological consumer-resource models.

Emergent competition shapes top-down versus bottom-up control in multi-trophic ecosystems.

Ecosystems are commonly organized into trophic levels-organisms that occupy the same level in a food chain (e.g., plants, herbivores, carnivores). A fundamental question in theoretical ecology is how the interplay between trophic structure, diversity, and competition shapes the properties of ecosystems. To address this problem, we analyze a generalized Consumer Resource Model with three trophic levels using the zero-temperature cavity method and numerical simulations. We derive the corresponding mean-field cavity equations and show that intra-trophic diversity gives rise to an effective "emergent competition" term between species within a trophic level due to feedbacks mediated by other trophic levels. This emergent competition gives rise to a crossover from a regime of top-down control (populations are limited by predators) to a regime of bottom-up control (populations are limited by primary producers) and is captured by a simple order parameter related to the ratio of surviving species in different trophic levels. We show that our theoretical results agree with empirical observations, suggesting that the theoretical approach outlined here can be used to understand complex ecosystems with multiple trophic levels.

Engineering synthetic phosphorylation signaling networks in human cells.

Protein phosphorylation signaling networks play a central role in how cells sense and respond to their environment. Here, we describe the engineering of artificial phosphorylation networks in which "push-pull" motifs-reversible enzymatic phosphorylation cycles consisting of opposing kinase and phosphatase activities-are assembled from modular protein domain parts and then wired together to create synthetic phosphorylation circuits in human cells. We demonstrate that the composability of our design scheme enables model-guided tuning of circuit function and the ability to make diverse network connections; synthetic phosphorylation circuits can be coupled to upstream cell surface receptors to enable fast-timescale sensing of extracellular ligands, while downstream connections can regulate gene expression. We leverage these capabilities to engineer cell-based cytokine controllers that dynamically sense and suppress activated T cells. Our work introduces a generalizable approach for designing and building phosphorylation signaling circuits that enable user-defined sense-and-respond function for diverse biosensing and therapeutic applications.

Phase transition to chaos in complex ecosystems with non-reciprocal species-resource interactions.

Non-reciprocal interactions between microscopic constituents can profoundly shape the large-scale properties of complex systems. Here, we investigate the effects of non-reciprocity in the context of theoretical ecology by analyzing a generalization of MacArthur's consumer-resource model with asymmetric interactions between species and resources. Using a mixture of analytic cavity calculations and numerical simulations, we show that such ecosystems generically undergo a phase transition to chaotic dynamics as the amount of non-reciprocity is increased. We analytically construct the phase diagram for this model and show that the emergence of chaos is controlled by a single quantity: the ratio of surviving species to surviving resources. We also numerically calculate the Lyapunov exponents in the chaotic phase and carefully analyze finite-size effects. Our findings show how non-reciprocal interactions can give rise to complex and unpredictable dynamical behaviors even in the simplest ecological consumer-resource models.

Ultra-high throughput mapping of genetic design space.

Massively parallel genetic screens have been used to map sequence-to-function relationships for a variety of genetic elements. However, because these approaches only interrogate short sequences, it remains challenging to perform high throughput (HT) assays on constructs containing combinations of sequence elements arranged across multi-kb length scales. Overcoming this barrier could accelerate synthetic biology; by screening diverse gene circuit designs, "composition-to-function" mappings could be created that reveal genetic part composability rules and enable rapid identification of behavior-optimized variants. Here, we introduce CLASSIC, a generalizable genetic screening platform that combines long- and short-read next-generation sequencing (NGS) modalities to quantitatively assess pooled libraries of DNA constructs of arbitrary length. We show that CLASSIC can measure expression profiles of >10 5 drug-inducible gene circuit designs (ranging from 6-9 kb) in a single experiment in human cells. Using statistical inference and machine learning (ML) approaches, we demonstrate that data obtained with CLASSIC enables predictive modeling of an entire circuit design landscape, offering critical insight into underlying design principles. Our work shows that by expanding the throughput and understanding gained with each design-build-test-learn (DBTL) cycle, CLASSIC dramatically augments the pace and scale of synthetic biology and establishes an experimental basis for data-driven design of complex genetic systems.

Emergent competition shapes the ecological properties of multi-trophic ecosystemss.

Ecosystems are commonly organized into trophic levels - organisms that occupy the same level in a food chain (e.g., plants, herbivores, carnivores). A fundamental question in theoretical ecology is how the interplay between trophic structure, diversity, and competition shapes the properties of ecosystems. To address this problem, we analyze a generalized Consumer Resource Model with three trophic levels using the zero-temperature cavity method and numerical simulations. We find that intra-trophic diversity gives rise to "emergent competition" between species within a trophic level due to feedbacks mediated by other trophic levels. This emergent competition gives rise to a crossover from a regime of top-down control (populations are limited by predators) to a regime of bottom-up control (populations are limited by primary producers) and is captured by a simple order parameter related to the ratio of surviving species in different trophic levels. We show that our theoretical results agree with empirical observations, suggesting that the theoretical approach outlined here can be used to understand complex ecosystems with multiple trophic levels.

Bias-variance decomposition of overparameterized regression with random linear features.

In classical statistics, the bias-variance trade-off describes how varying a model's complexity (e.g., number of fit parameters) affects its ability to make accurate predictions. According to this trade-off, optimal performance is achieved when a model is expressive enough to capture trends in the data, yet not so complex that it overfits idiosyncratic features of the training data. Recently, it has become clear that this classic understanding of the bias variance must be fundamentally revisited in light of the incredible predictive performance of overparameterized models-models that avoid overfitting even when the number of fit parameters is large enough to perfectly fit the training data. Here, we present results for one of the simplest examples of an overparameterized model: regression with random linear features (i.e., a two-layer neural network with a linear activation function). Using the zero-temperature cavity method, we derive analytic expressions for the training error, test error, bias, and variance. We show that the linear random features model exhibits three phase transitions: two different transitions to an interpolation regime where the training error is zero, along with an additional transition between regimes with large bias and minimal bias. Using random matrix theory, we show how each transition arises due to small nonzero eigenvalues in the Hessian matrix. Finally, we compare and contrast the phase diagram of the random linear features model to the random nonlinear features model and ordinary regression, highlighting the additional phase transitions that result from the use of linear basis functions.

Correlation of plastic events with local structure in jammed packings across spatial dimensions.

In frictionless jammed packings, existing evidence suggests a picture in which localized physics dominates in low spatial dimensions, d = 2, 3, but quickly loses relevance as d rises, replaced by spatially extended mean-field behavior. For example, quasilocalized low-energy vibrational modes and low-coordination particles associated with deviation from mean-field behavior (rattlers and bucklers) all vanish rapidly with increasing d. These results suggest that localized rearrangements, which are associated with low-energy vibrational modes, correlated with local structure, and dominant in low dimensions, should give way in higher d to extended rearrangements uncorrelated with local structure. Here, we use machine learning to analyze simulations of jammed packings under athermal, quasistatic shear, identifying a local structural variable, softness, that correlates with rearrangements in dimensions d = 2 to d = 5. We find that softness­and even just the local coordination number Z­is essentially equally predictive of rearrangements in all d studied. This result provides direct evidence that local structure plays an important role in higher d, suggesting a modified picture for the dimensional cross-over to mean-field theory.

Memorizing without overfitting: Bias, variance, and interpolation in overparameterized models.

The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using "over-parameterized models" where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in over-parameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of over-parameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold, the test error of the two-layer neural network decreases due to a monotonic decrease in both the bias and variance in contrast with the classical bias-variance trade-off. We also show that in contrast with classical intuition, over-parameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in over-parameterized models and relate our results to random matrix theory.

Hidden Topological Structure of Flow Network Functionality.

The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function: networks with different local architectures can perform the same functions. Here, we answer the question of how changes at the level of individual edges collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of nearly uniform pressure. Although these sectors are not apparent in the underlying network structure, they correlate strongly with the tuned function. The connectivity of these sectors, rather than that of individual nodes, provides a quantitative relationship between structure and function in flow networks.

Limits of multifunctionality in tunable networks.

Nature is rife with networks that are functionally optimized to propagate inputs to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single "target" link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of constraint-satisfaction problems.

Designing allostery-inspired response in mechanical networks.

Recent advances in designing metamaterials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally efficient approach, we extend this idea to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any two target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ∼1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic 2D and 3D systems that exhibit these responses. This work is inspired by the long-range coupled conformational changes that constitute allosteric function in proteins. The fact that allostery is a common means for regulation in biological molecules suggests that it is a relatively easy property to develop through evolution. In analogy, our results show that long-range coupled mechanical responses are similarly easy to achieve in disordered networks.

Mechanical signaling coordinates the embryonic heartbeat.

In the beating heart, cardiac myocytes (CMs) contract in a coordinated fashion, generating contractile wave fronts that propagate through the heart with each beat. Coordinating this wave front requires fast and robust signaling mechanisms between CMs. The primary signaling mechanism has long been identified as electrical: gap junctions conduct ions between CMs, triggering membrane depolarization, intracellular calcium release, and actomyosin contraction. In contrast, we propose here that, in the early embryonic heart tube, the signaling mechanism coordinating beats is mechanical rather than electrical. We present a simple biophysical model in which CMs are mechanically excitable inclusions embedded within the extracellular matrix (ECM), modeled as an elastic-fluid biphasic material. Our model predicts strong stiffness dependence in both the heartbeat velocity and strain in isolated hearts, as well as the strain for a hydrogel-cultured CM, in quantitative agreement with recent experiments. We challenge our model with experiments disrupting electrical conduction by perfusing intact adult and embryonic hearts with a gap junction blocker, ß-glycyrrhetinic acid (BGA). We find this treatment causes rapid failure in adult hearts but not embryonic hearts-consistent with our hypothesis. Last, our model predicts a minimum matrix stiffness necessary to propagate a mechanically coordinated wave front. The predicted value is in accord with our stiffness measurements at the onset of beating, suggesting that mechanical signaling may initiate the very first heartbeats.