Design of biochemical pattern forming systems from minimal motifs

Noise-guided tuning of synthetic protein waves in living cells

Biological systems use protein circuits to organize cellular activities in space and time, but engineering synthetic dynamics is challenging due to stochastic effects of genetic and biochemical variation on circuit behavior. Genetically encoded oscillators (GEOs) built from bacterial MinDE-family ATPase and Activator modules generate fast orthogonal protein waves in eukaryotic cells, providing an experimental model system for genetic and biochemical coordination of synthetic protein dynamics. Here, we use budding yeast to experimentally define and model phase portraits that reveal how the breadth of frequencies and amplitudes available to a GEO are genetically controlled by ATPase and Activator expression levels and noise. GEO amplitude is encoded by ATPase absolute abundance, making it sensitive to extrinsic noise on a population level. In contrast, GEO frequency is remarkably stable because it is controlled by the Activator:ATPase ratio and thus affected primarily by intrinsic noise. These features facilitate noise-guided design of different expression strategies that act as filters on GEO waveform, enabling us to construct clonal populations that oscillate at different frequencies as well as independently tune frequency and amplitude variation within a single population. By characterizing 169 biochemically distinct GEOs, we provide a rich assortment of phase portraits as starting points for application of our waveform engineering approach. Our findings suggest noise-guided design may be a valuable strategy for achieving precision control over dynamic protein circuits.

Controlled Protein-Membrane Interactions Modulate Self-Organization of Min Protein Patterns

Self-organizing protein patterns are crucial for living systems, governing important cellular processes such as polarization and division. While the field of protein self-organization has reached a point where basic pattern-forming mechanisms can be reconstituted in vitro using purified proteins, understanding how cells can dynamically switch and modulate these patterns, especially when transiently needed, remains an interesting frontier. Here, we demonstrate the efficient regulation of self-organizing protein patterns through the modulation of simple biophysical membrane parameters. Our investigation focuses on the impact of membrane affinity changes on Min protein patterns at lipid membranes composed of Escherichia coli lipids or minimal lipid compositions, and we present three major results. First, we observed the emergence of a diverse array of pattern phenotypes, ranging from waves over flower-shaped patterns to snowflake-like structures. Second, we demonstrated the dependency of these patterns on the density of protein-membrane linkers. Finally, we demonstrate that the shape of snowflake-like patterns is fine-tuned by membrane charge. Our results demonstrate the significant influence of membrane linkage as a straightforward biophysical parameter governing protein pattern formation. Our research points towards a simple yet intriguing mechanism by which cells can adeptly tune and switch protein patterns on the mesoscale.

Machine learning-aided design and screening of an emergent protein function in synthetic cells

Recently, utilization of Machine Learning (ML) has led to astonishing progress in computational protein design, bringing into reach the targeted engineering of proteins for industrial and biomedical applications. However, the design of proteins for emergent functions of core relevance to cells, such as the ability to spatiotemporally self-organize and thereby structure the cellular space, is still extremely challenging. While on the generative side conditional generative models and multi-state design are on the rise, for emergent functions there is a lack of tailored screening methods as typically needed in a protein design project, both computational and experimental. Here we describe a proof-of-principle of how such screening, in silico and in vitro, can be achieved for ML-generated variants of a protein that forms intracellular spatiotemporal patterns. For computational screening we use a structure-based divide-and-conquer approach to find the most promising candidates, while for the subsequent in vitro screening we use synthetic cell-mimics as established by Bottom-Up Synthetic Biology. We then show that the best screened candidate can indeed completely substitute the wildtype gene in Escherichia coli. These results raise great hopes for the next level of synthetic biology, where ML-designed synthetic proteins will be used to engineer cellular functions.

A programmable reaction-diffusion system for spatiotemporal cell signaling circuit design

Cells self-organize molecules in space and time to generate complex behaviors, but we lack synthetic strategies for engineering spatiotemporal signaling. We present a programmable reaction-diffusion platform for designing protein oscillations, patterns, and circuits in mammalian cells using two bacterial proteins, MinD and MinE (MinDE). MinDE circuits act like "single-cell radios," emitting frequency-barcoded fluorescence signals that can be spectrally isolated and analyzed using digital signal processing tools. We define how to genetically program these signals and connect their spatiotemporal dynamics to cell biology using engineerable protein-protein interactions. This enabled us to construct sensitive reporter circuits that broadcast endogenous cell signaling dynamics on a frequency-barcoded imaging channel and to build control signal circuits that synthetically pattern activities in the cell, such as protein condensate assembly and actin filamentation. Our work establishes a paradigm for visualizing, probing, and engineering cellular activities at length and timescales critical for biological function.

The mechanism of MinD stability modulation by MinE in Min protein dynamics

The patterns formed both in vivo and in vitro by the Min protein system have attracted much interest because of the complexity of their dynamic interactions given the apparent simplicity of the component parts. Despite both the experimental and theoretical attention paid to this system, the details of the biochemical interactions of MinD and MinE, the proteins responsible for the patterning, are still unclear. For example, no model consistent with the known biochemistry has yet accounted for the observed dual role of MinE in the membrane stability of MinD. Until now, a statistical comparison of models to the time course of Min protein concentrations on the membrane has not been carried out. Such an approach is a powerful way to test existing and novel models that are difficult to test using a purely experimental approach. Here, we extract time series from previously published fluorescence microscopy time lapse images of in vitro experiments and fit two previously described and one novel mathematical model to the data. We find that the novel model, which we call the Asymmetric Activation with Bridged Stability Model, fits the time-course data best. It is also consistent with known biochemistry and explains the dual MinE role via MinE-dependent membrane stability that transitions under the influence of rising MinE to membrane instability with positive feedback. Our results reveal a more complex network of interactions between MinD and MinE underlying Min-system dynamics than previously considered.

Multiscale Biofabrication: Integrating Additive Manufacturing with DNA-Programmable Self-Assembly

Structure and hierarchical organization are crucial elements of biological systems and are likely required when engineering synthetic biomaterials with life-like behavior. In this context, additive manufacturing techniques like bioprinting have become increasingly popular. However, 3D bioprinting, as well as other additive manufacturing techniques, show limited resolution, making it difficult to yield structures on the sub-cellular level. To be able to form macroscopic synthetic biological objects with structuring on this level, manufacturing techniques have to be used in conjunction with biomolecular nanotechnology. Here, a short overview of both topics and a survey of recent advances to combine additive manufacturing with microfabrication techniques and bottom-up self-assembly involving DNA, are given.

Bridging scales in a multiscale pattern-forming system

Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

ChemChaste: Simulating spatially inhomogeneous biochemical reaction-diffusion systems for modeling cell-environment feedbacks

BACKGROUND

Spatial organization plays an important role in the function of many biological systems, from cell fate specification in animal development to multistep metabolic conversions in microbial communities. The study of such systems benefits from the use of spatially explicit computational models that combine a discrete description of cells with a continuum description of one or more chemicals diffusing within a surrounding bulk medium. These models allow the in silico testing and refinement of mechanistic hypotheses. However, most existing models of this type do not account for concurrent bulk and intracellular biochemical reactions and their possible coupling.

CONCLUSIONS

Here, we describe ChemChaste, an extension for the open-source C++ computational biology library Chaste. ChemChaste enables the spatial simulation of both multicellular and bulk biochemistry by expanding on Chaste's existing capabilities. In particular, ChemChaste enables (i) simulation of an arbitrary number of spatially diffusing chemicals, (ii) spatially heterogeneous chemical diffusion coefficients, and (iii) inclusion of both bulk and intracellular biochemical reactions and their coupling. ChemChaste also introduces a file-based interface that allows users to define the parameters relating to these functional features without the need to interact directly with Chaste's core C++ code. We describe ChemChaste and demonstrate its functionality using a selection of chemical and biochemical exemplars, with a focus on demonstrating increased ability in modeling bulk chemical reactions and their coupling with intracellular reactions.

AVAILABILITY AND IMPLEMENTATION

ChemChaste version 1.0 is a free, open-source C++ library, available via GitHub at https://github.com/OSS-Lab/ChemChaste under the BSD license, on the Zenodo archive at zendodo doi, as well as on BioTools (biotools:chemchaste) and SciCrunch (RRID:SCR022208) databases.

Bulk-surface coupling identifies the mechanistic connection between Min-protein patterns in vivo and in vitro

Self-organisation of Min proteins is responsible for the spatial control of cell division in Escherichia coli, and has been studied both in vivo and in vitro. Intriguingly, the protein patterns observed in these settings differ qualitatively and quantitatively. This puzzling dichotomy has not been resolved to date. Using reconstituted proteins in laterally wide microchambers with a well-controlled height, we experimentally show that the Min protein dynamics on the membrane crucially depend on the micro chamber height due to bulk concentration gradients orthogonal to the membrane. A theoretical analysis shows that in vitro patterns at low microchamber height are driven by the same lateral oscillation mode as pole-to-pole oscillations in vivo. At larger microchamber height, additional vertical oscillation modes set in, marking the transition to a qualitatively different in vitro regime. Our work reveals the qualitatively different mechanisms of mass transport that govern Min protein-patterns for different bulk heights and thus shows that Min patterns in cells are governed by a different mechanism than those in vitro.

Increasing MinD's Membrane Affinity Yields Standing Wave Oscillations and Functional Gradients on Flat Membranes

The formation of large-scale patterns through molecular self-organization is a basic principle of life. Accordingly, the engineering of protein patterns and gradients is of prime relevance for synthetic biology. As a paradigm for such pattern formation, the bacterial MinDE protein system is based on self-organization of the ATPase MinD and ATPase-activating protein MinE on lipid membranes. Min patterns can be tightly regulated by tuning physical or biochemical parameters. Among the biochemically engineerable modules, MinD's membrane targeting sequence, despite being a key regulating element, has received little attention. Here we attempt to engineer patterns by modulating the membrane affinity of MinD. Unlike the traveling waves or stationary patterns commonly observed in vitro on flat supported membranes, standing-wave oscillations emerge upon elongating MinD's membrane targeting sequence via rationally guided mutagenesis. These patterns are capable of forming gradients and thereby spatially target co-reconstituted downstream proteins, highlighting their functional potential in designing new life-like systems.

A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks

Cancers are complex dynamical systems. They remain the leading cause of disease-related pediatric mortality in North America. To overcome this burden, we must decipher the state-space attractor dynamics of gene expression patterns and protein oscillations orchestrated by cancer stemness networks. The review provides an overview of dynamical systems theory to steer cancer research in pattern science. While most of our current tools in network medicine rely on statistical correlation methods, causality inference remains primitively developed. As such, a survey of attractor reconstruction methods and machine algorithms for the detection of causal structures applicable in experimentally derived time series cancer datasets is presented. A toolbox of complex systems approaches are discussed for reconstructing the signaling state space of cancer networks, interpreting causal relationships in their time series gene expression patterns, and assisting clinical decision making in computational oncology. As a proof of concept, the applicability of some algorithms are demonstrated on pediatric brain cancer datasets and the requirement of their time series analysis is highlighted.

Probing Biomolecular Interactions by a Pattern-Forming Peptide-Conjugate Sensor

As a key mechanism underpinning many biological processes, protein self-organization has been extensively studied. However, the potential to apply the distinctive, nonlinear biochemical properties of such self-organizing systems to biotechnological problems such as the facile detection and characterization of biomolecular interactions has not yet been explored. Here, we describe an in vitro assay in a 96-well plate format that harnesses the emergent behavior of the Escherichia coli Min system to provide a readout of biomolecular interactions. Crucial for the development of our approach is a minimal MinE-derived peptide that stimulates MinD ATPase activity only when dimerized. We found that this behavior could be induced via any pair of foreign, mutually binding molecular entities fused to the minimal MinE peptide. The resulting MinD ATPase activity and the spatiotemporal nature of the produced protein patterns quantitatively correlate with the affinity of the fused binding partners, thereby enabling a highly sensitive assay for biomolecular interactions. Our assay thus provides a unique means of quantitatively visualizing biomolecular interactions and may prove useful for the assessment of domain interactions within protein libraries and for the facile investigation of potential inhibitors of protein–protein interactions.

Cancer: A turbulence problem

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Application of nonlinear dynamics to cancer ecosystems.

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Chemical turbulence and strange attractor models in tumor growth, invasion and pattern formation are investigated.

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Computational algorithms for detecting such structures are proposed.

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Complex systems applications to cancer dynamics.

Cancers are complex, adaptive ecosystems. They remain the leading cause of disease-related death among children in North America. As we approach computational oncology and Deep Learning Healthcare, our mathematical models of cancer dynamics must be revised. Recent findings support the perspective that cancer-microenvironment interactions may consist of chaotic gene expressions and turbulent protein flows during pattern formation. As such, cancer pattern formation, protein-folding and metastatic invasion are discussed herein as processes driven by chemical turbulence within the framework of complex systems theory. To conclude, cancer stem cells are presented as strange attractors of the Waddington landscape.

Turing Patterning in Stratified Domains

Reaction-diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal-mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction-diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction-diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

Conformational equilibrium of MinE regulates the allowable concentration ranges of a protein wave for cell division

The Min system for determining the cell division position at the center in bacteria has a unique character that uses a protein wave (Min wave) that emerges from its components (MinD and MinE). The Min wave emerges under the coupling of chemical reactions and molecular diffusions of MinDE and appears when the concentrations of MinD and MinE are similar. However, the nanoscale mechanism to determine their concentration ranges has remained elusive. In this study, by using artificial cells as a mimic of cells, we showed that the dominant MinE conformations determined the allowable concentration ranges for the emergence of the Min wave. Furthermore, the deletion of the membrane-binding region of MinE indicated that the region was essential for limiting the concentration ranges to be narrower. These findings illustrate a parameter tuning mechanism underlying complex molecular systems at the nanoscale for spatiotemporal regulation in living cells and show a possibility that the regulation of the equilibrium among molecular conformations can work as a switch for cell division.

C-terminal eYFP fusion impairs Escherichia coli MinE function

The Escherichia coli Min system plays an important role in the proper placement of the septum ring at mid-cell during cell division. MinE forms a pole-to-pole spatial oscillator with the membrane-bound ATPase MinD, resulting in MinD concentration being the lowest at mid-cell. MinC, the direct inhibitor of the septum initiator protein FtsZ, forms a complex with MinD at the membrane, mirroring its polar gradients. Therefore, MinC-mediated FtsZ inhibition occurs away from mid-cell. Min oscillations are often studied in living cells by time-lapse microscopy using fluorescently labelled Min proteins. Here, we show that, despite permitting oscillations to occur in a range of protein concentrations, the enhanced yellow fluorescent protein (eYFP) C-terminally fused to MinE impairs its function. Combining in vivo, in vitro and in silico approaches, we demonstrate that eYFP compromises the ability of MinE to displace MinC from MinD, to stimulate MinD ATPase activity and to directly bind to the membrane. Moreover, we reveal that MinE-eYFP is prone to aggregation. In silico analyses predict that other fluorescent proteins are also likely to compromise several functionalities of MinE, suggesting that the results presented here are not specific to eYFP.

Pattern localization to a domain edge

The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can "sense" the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.

Design of biochemical pattern forming systems from minimal motifs

Although molecular self-organization and pattern formation are key features of life, only very few pattern-forming biochemical systems have been identified that can be reconstituted and studied in vitro under defined conditions. A systematic understanding of the underlying mechanisms is often hampered by multiple interactions, conformational flexibility and other complex features of the pattern forming proteins. Because of its compositional simplicity of only two proteins and a membrane, the MinDE system from Escherichia coli has in the past years been invaluable for deciphering the mechanisms of spatiotemporal self-organization in cells. Here, we explored the potential of reducing the complexity of this system even further, by identifying key functional motifs in the effector MinE that could be used to design pattern formation from scratch. In a combined approach of experiment and quantitative modeling, we show that starting from a minimal MinE-MinD interaction motif, pattern formation can be obtained by adding either dimerization or membrane-binding motifs. Moreover, we show that the pathways underlying pattern formation are recruitment-driven cytosolic cycling of MinE and recombination of membrane-bound MinE, and that these differ in their in vivo phenomenology.