TBA

---

Session 3 of 4

Fast and accurate predictions for complex physical dynamics are a big challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in the real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. In this study, we propose HyperDeepONet, which uses the expressive power of the hypernetwork to enable learning of a complex operator with smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a special case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet was successfully applied to various operator learning problems using low computational resources compared to other benchmarks.

---

Session 4 of 4 Please be aware of the change of venue.

In this talk, we will discuss the problem of determining the maximum number of edges in an n-vertex k-critical graph. A graph is considered k-critical if its chromatic number is k, but any proper subgraph has a chromatic number less than k. The problem remains open for any integer k ≥ 4. Our presentation will showcase an improvement on previous results achieved by employing a combination of extremal graph theory and structural analysis. We will introduce a key lemma, which may be of independent interest, as it sheds light on the partial structure of dense k-critical graphs. This is joint work with Cong Luo and Jie Ma.

Configurations of axis-parallel boxes in $\mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their perceived simplicity, there are many problems involving their structure that are not well understood. I will talk about a construction that shows that their structure might be more complicated than people conjectured.

We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of biophysical systems and for discovering hidden mechanisms and pathways from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and on generative adversarial networks (GANs). Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the mathematical physics, e.g.., conservation laws, reaction kinetics, etc,. which are used to obtain informative priors or regularize the neural networks. We will demonstrate how we can train BINNs from multifidelity/multimodality data, and we will present several examples of inverse problems, e.g., in systems biology for diabetes and in biomechanics for non-invasive inference of thrombus material properties. We will also discuss how operator regression in the form of DeepOnet can be used to accelerate inference based on historical data and only a few new data, as well its generalization and transfer learning capacity.

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Pivot-minors can be thought of as a dense analogue of graph minors. We shall discuss pivot-minors and two recent results for proper pivot-minor-closed classes of graphs. In particular, that for every graph H, the class of graphs containing no H-pivot-minor is 𝜒-bounded, and also satisfies the (strong) Erdős-Hajnal property.

The well-known two-process model of sleep regulation makes accurate predictions of sleep timing and duration, as well as neurobehavioral performance, for a variety of acute sleep deprivation and nap sleep scenarios, but it fails to predict the effects of chronic sleep restriction on neurobehavioral performance. The two-process model belongs to a broader class of coupled, non-homogeneous, first-order, ordinary differential equations (ODEs), which can capture the effects of chronic sleep restriction. These equations exhibit a bifurcation, which appears to be an essential feature of performance impairment due to sleep loss. The equations implicate a biological system analogous to two connected compartments containing interacting compounds with time-varying concentrations, such as the adenosinergic neuromodulator/receptor system, as a key mechanism for the regulation of neurobehavioral functioning under conditions of sleep loss. The equations account for dynamic interaction with circadian rhythmicity, and also provide a new approach to dynamically tracking the magnitude of sleep inertia upon awakening from restricted sleep. This presentation will describe the development of the ODE system and its experimental calibration and validation, and will discuss some novel predictions.

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

TBA

---

Session 2 of 4

Small regulatory RNA molecules such as microRNA modulate gene expression through inhibiting the translation of messenger RNA (mRNA). Such post-transcriptional regulation has been recently hypothesized to reduce the stochastic variability of gene expression around average levels. Here we quantify noise in stochastic gene expression models with and without such regulation. Our results suggest that silencing mRNA post-transcriptionally will always increase rather than decrease gene expression noise when the silencing of mRNA also increases its degradation as is expected for microRNA interactions with mRNA. In that regime we also find that silencing mRNA generally reduces the fidelity of signal transmission from deterministically varying upstream factors to protein levels. These findings suggest that microRNA binding to mRNA does not generically confer precision to protein expression

Many questions in everyday life as well as in research are causal in nature: How would the climate change if we lower train prices or will my headache go away if I take an aspirin? Inherently, such questions need to specify the causal variables relevant to the question and their interactions. However, existing algorithms for learning causal graphs from data are often not scaling well both with the number of variables or the number of observations. This talk will provide a brief introduction to causal structure learning, recent efforts in using continuous optimization to learn causal graphs at scale and systematic approaches for causal experimental design at scale.

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

TBA

---

Session 1 of 4

The standard approach to problem-solving in physics consists of identifying state variables of the system, setting differential equations governing the state evolution, and solving the obtained. The behavior of the system for different values of parameters can be examined only as a fourth step. On the contrary, the modern approach to studying dynamical systems relies on Morphological/Topological analysis which alleviates the necessity for the explicit solution of differential equations. The stability analysis of the parabolic swing will demonstrate the merit of such an approach. It will be shown how to construct a qualitatively correct model of system dynamics that is surprisingly quantitatively correct as well. The sudden (catastrophic) change in the swing’s stability, caused by a slight change in the critical value of system parameters, will be linked to the drastic topological change of the corresponding phase-space portraits. It will be shown that for a system’s parameters close to critical ones, the system’s behavior is identical to a specific simple universal prototype given by catastrophe theory. A short survey of the simplest elementary catastrophes will be given that represents the basis for applying catastrophe theory in other fields of science.

The seasonal outbreaks of influenza infection cause globally respiratory illness, or even death in all age groups. Given early-warning signals preceding the influenza outbreak, timely intervention such as vaccination and isolation management effectively decrease the morbidity. However, it is usually a difficult task to achieve the real-time prediction of influenza outbreak due to its complexity intertwining both biological systems and social systems. By exploring rich dynamical and high-dimensional information, our dynamic network marker/biomarker (DNM/DNB) method opens a new way to identify the tipping point prior to the catastrophic transition into an influenza pandemics. In order to detect the early-warning signals before the influenza outbreak by applying DNM method, the historical information of clinic hospitalization caused by influenza infection between years 2009 and 2016 were extracted and assembled from public records of Tokyo and Hokkaido, Japan. The early-warning signal, with an average of 4-week window lead prior to each seasonal outbreak of influenza, was provided by DNM-based on the hospitalization records, providing an opportunity to apply proactive strategies to prevent or delay the onset of influenza outbreak. Moreover, the study on the dynamical changes of hospitalization in local district networks unveils the influenza transmission dynamics or landscape in network level.

Multi-omics technologies, and in particular those with single-cell and spatial resolution, provide unique opportunities to study the deregulation of intra- and inter-cellular signaling processes in disease. I will present recent methods and applications from our group toward this aim, focusing on computational approaches that combine data with biological knowledge within statistical and machine learning methods. This combination allows us to increase both the statistical power of our analyses and the mechanistic interpretability of the results. These approaches allow us to identify key processes, that can be in turn studied in detailed with dynamic mechanistic models. I will then present how cell-specific logic models, trained with measurements upon perturbations, can provides new biomarkers and treatment opportunities. Finally, I will show how, using novel microfluidics-based technologies, this approach can also be applied directly to biopsies, allowing to build mechanistic models for individual cancer patients, and use these models to prose new therapies.

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

We give a summary on the work of the last months related to Frankl's Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $\mathcal F$), then $\mathcal F$ contains an (abundant) element that belongs to at least half of the sets. Meanwhile, there are many related versions of the conjecture due to Frankl. For example, graph theorists may prefer the equivalent statement that every bipartite graph has a vertex that belongs to no more than half of the maximal independent sets. While even proving the ε-Union-Closed Sets Conjecture was out of reach, Poonen and Cui & Hu conjectured already stronger forms. At the end of last year, progress was made on many of these conjectures. Gilmer proved the ε-Union-Closed Sets Conjecture using an elegant entropy-based method which was sharpened by many others. Despite a sharp approximate form of the union-closed conjecture as stated by Chase and Lovett, a further improvement was possible. In a different direction, Kabela, Polak and Teska constructed union-closed family sets with large sets and few abundant elements. This talk will keep the audience up-to-date and give them insight in the main ideas. People who would like more details, can join the Ecopro-reading session on the 7th of March (10 o'clock, B332) as well. Here we go deeper in the core of the proofs and discuss possible directions for the full resolution.

The morphological method – based on the topology and singularity theory and originally developed for the analysis of the scattering experiments – was extended to be applicable for the analysis of biological data. The usefulness of the topological viewpoint was demonstrated by quantification of the changes of collagen fiber straightness in the human colon mucosa (healthy mucosa, colorectal cancer, and uninvolved mucosa far from cancer). This has been done by modeling the distribution of collagen segment angles by the polymorphic beta-distribution. Its shapes were classified according to the number and type of critical points. We found that biologically relevant shapes could be classified as shapes without any preferable orientation (i.e. shapes without local extrema), transitional forms (i.e. forms with one broad local maximum), and highly oriented forms (i.e. forms with two minima at both ends and one very narrow maximum between them). Thus, changes in the fiber organization were linked to the metamorphoses of the beta-distribution forms. The obtained classification was used to define a new, shape-aware/based, measure of the collagen straightness, which revealed a slight, and moderate increase of the straightness in mucosa samples taken 20 cm and 10 cm away from the tumor. The largest increase of collagen straightness was found in samples of cancer tissue. Samples of the healthy individuals have a uniform distribution of beta-distribution forms. We found that this distribution has the maximal information entropy. At 20 cm and 10 cm away from cancer, the transition forms redistribute into unoriented and highly oriented forms. Closer to cancer the number of unoriented forms decreases rapidly leaving only highly oriented forms present in the samples of the cancer tissue, whose distribution has minimal information entropy. The polarization of the distribution was followed by a significant increase in the number of quasi-symmetrical forms in samples 20 cm away from cancer which decreases closer to cancer. This work shows that the evolution of the distribution of the beta-distribution forms – an abstract construction of the mind – follows the familiar laws of statistical mechanics. Additionally, the polarization of the beta-distribution forms together with the described change in the number of quasi-symmetrical forms, clearly visible in the parametric space of the beta-distribution and very difficult to notice in the observable space, can be a useful indicator of the early stages in the development of colorectal cancer.

Cooperation means that one individual pays a cost for another to receive a benefit. Cooperation can be at variance with natural selection. Why should you help competitors? Yet cooperation is abundant in nature and is important component of evolutionary innovation. Cooperation can be seen as the master architect of evolution and as the third fundamental principle of evolution beside mutation and selection. I will present five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, spatial selection, group selection and kin selection. Global cooperation and the cooperation with future generations is necessary to ensure the survival of our species. Further reading: Nowak MA (2006). Evolutionary Dynamics. Harvard University Press Nowak MA & Highfield R (2011) SuperCooperators. Simon & Schuster. Hauser OP, Rand DG, Peysakhovich A & Nowak MA (2014). Cooperating with the future. Nature 511: 220-223 Hilbe C, Šimsa Š, Chatterjee K & Nowak MA (2018). Evolution of cooperation in stochastic games. Nature 559: 246-249 Hauser OP, Hilbe C, Chatterjee K & Nowak MA (2019). Social dilemmas among unequals. Nature 572: 524-527

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

A key goal of synthetic biology is to establish functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other’s biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal of the planar disjoint paths problem is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. This problem has been studied extensively due to its numerous applications such as VLSI layout and circuit routing. However, this problem is NP-complete even for grid graphs. This motivates the study of this problem from the viewpoint of parameterized algorithms. In this talk, I will present a $2^{O(k^2)}n$-time algorithm for the planar disjoint paths problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020]. This is joint work with Kyungjin Cho and Seunghyeok Oh.

1. The “temporal information code” of insulin action: a bottom-up approach One of the essential elements of signaling networks is to encode information from a wide variety of inputs into a limited set of molecules. We have proposed a “temporal information code” that regulates a variety of physiological functions by encoding input information in temporal patterns of molecular activity, and based on this concept, we are analyzing biological homeostasis by insulin signaling. Taking blood insulin as an example, we will explain how the temporal information of blood insulin is selectively decoded by downstream networks. 2. Transomics of insulin action: a top-down approach In order to obtain a complete picture of insulin action, we performed transomics measurements integrating metabolomics and transcriptomics, and found that metabolism is regulated by allosteric regulation in the liver of normal mice and by compensatory gene expression in the liver of obese mice. (Top-down approach). I will talk about approach the principle of homeostasis of living organisms by temporal patterns, using the analysis of systems biology of insulin action using two different approaches.

---

ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cellular dynamics and emerging biological function are governed by patterns of gene expression arising from networks of interacting genes. Inferring these interactions from data is a notoriously difficult inverse problem that is central to systems biology. The majority of existing network inference methods work at the population level and construct a static representations of gene regulatory networks; they do not naturally allow for inference of differential regulation across a heterogeneous cell population. Building upon recent dynamical inference methods that model single cell dynamics using Markov processes, we propose locaTE, an information-theoretic approach which employs a localised transfer entropy to infer cell-specific, causal gene regulatory networks. LocaTE uses high-resolution estimates of dynamics and geometry of the cellular gene expression manifold to inform inference of regulatory interactions. We find that this approach is generally superior to using static inference methods, often by a significant margin. We demonstrate that factor analysis can give detailed insights into the inferred cell-specific GRNs. In application to two experimental datasets, we recover key transcription factors and regulatory interactions that drive mouse primitive endoderm formation and pancreatic development. For both simulated and experimental data, locaTE provides a powerful, efficient and scalable network inference method that allows us to distil cell-specific networks from single cell data.