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Geometric Theory of Distribution Shapes for Autoregulatory Gene Circuits
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| Pengarang | : | Ying Sheng, Genghong Lin, Feng Jiao, Chen Jia |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 636-661 |
| Abstrak | : | In this study, we provide a complete mathematical characterization of the phase diagram of distribution shapes in an extension of the two-state telegraph model of stochastic gene expression in the presence of positive or negative autoregulation. Using the techniques of second-order difference equations and nonlinear discrete dynamical systems, we prove that the feedback loop can only produce three shapes of steady-state protein distributions (decaying, bell-shaped, and bimodal), corresponding to three distinct parameter regions in the phase diagram. The boundaries of the three regions are characterized by two continuous curves, which can be constructed geometrically by the contour lines of a series of ratio operators. Based on the geometric structure of the phase diagram, we then provide some simple and verifiable sufficient and/or necessary conditions for the existence of the bimodal parameter region, as well as the conditions for the steady-state distribution to be decaying, bell-shaped, or bimodal. Finally, we also investigate how the phase diagram is affected by the strength of positive or negative feedback. |
Reconstructing the Kinetic Chemotaxis Kernel Using Macroscopic Data: Well-Posedness and Ill-Posedness
| Pengarang | : | Kathrin Hellmuth, Christian Klingenberg, Qin Li, Min Tang |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 613-635 |
| Abstrak | : | Bacterial motion is guided by external stimuli (chemotaxis), and the motion described on the mesoscopic scale is uniquely determined by a parameter ? that models velocity change response from the bacteria. This parameter is termed a chemotaxis kernel. In a practical setting, experimental data was collected to infer this kernel. In this article, a PDE-constrained optimization framework is deployed to perform this reconstruction using velocity-averaged, localized data taken in the interior of the domain. The problem can be well-posed or ill-posed depending on the data preparation and the experimental setup. In particular, we propose one specific design that guarantees numerical reconstructability and local convergence. This design is adapted to the discretization of ? in space and decouples the reconstruction of local values of ? into smaller cell problems, opening up parallelization opportunities. Numerical evidence supports the theoretical findings. |
A Stochastic Version of the Hopfield–Ninio Kinetic Proofreading Model
| Pengarang | : | E. Franco, J. J. L. Velázquez |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 576-612 |
| Abstrak | : | In this paper we study a stochastic version of the Hopfield–Ninio kinetic proofreading model. The model is characterized by means of two parameters, the unbinding time, which depends on the binding energy between a ligand and a receptor, and the number of times ?≥1 that a ligand attaches to a receptor. We prove that, under suitable assumptions on ?, our model has an extreme specificity, i.e., it is capable of discriminating between different ligands, and has a high sensitivity, i.e., the response of the system does not change in a significant manner for ranges of ligands varying within several orders of magnitude. Additional quantities like the amount of energy used by the network or the time required to yield a response will be also computed. We also show that our results are robust, i.e., they do not depend on the specific choice of parameters that we make in this paper. |
Exponential Asymptotics and Higher-Order Stokes Phenomenon in Singularly Perturbed ODEs
| Pengarang | : | Josh Shelton, Samuel Crew, Philippe H. Trinh |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 548-575 |
| Abstrak | : | The higher-order Stokes phenomenon can emerge in the asymptotic analysis of many problems governed by singular perturbations. Indeed, over the last two decades, the phenomenon has appeared in many physical applications, from acoustic and optical wave phenomena and gravity-capillary ripples to models of crystal growth and equatorial Kelvin waves. It emerges in a generic fashion in the exponential asymptotics of higher-order ordinary and partial differential equations. The intention of this work is to highlight its importance, and develop further practical methodologies for the study of higher-order Stokes phenomena, primarily for general nonintegrable problems. Our formal methodology is demonstrated through application to a second-order linear inhomogeneous ODE that exemplifies the simplest example of higher-order Stokes phenomena. In this model problem, the Borel transform can be derived explicitly, and this gives insight into the beyond-all-orders structure. We review and study additional examples, with physically important connections, including higher-order ODEs and eigenvalue problems. |
Probability Flow Approach to the Onsager–Machlup Functional for Jump-Diffusion Processes
| Pengarang | : | Yuanfei Huang, Xiang Zhou, Jinqiao Duan |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 524-547 |
| Abstrak | : | The Onsager–Machlup action functional is an important concept in statistical mechanics and thermodynamics to describe the probability of fluctuations in nonequilibrium systems. It provides a powerful tool for analyzing and predicting the behavior of complex stochastic systems. For the diffusion process, the path integral method and the Girsanov transformation are two main approaches to construct the Onsager–Machlup functional. However, it is a long-standing challenge to apply these two methods to the jump-diffusion process, because the complexity of jump noise presents intrinsic technical barriers to deriving the Onsager–Machlup functional. In this work, we propose a new strategy to solve this problem by utilizing the equivalent probabilistic flow between the pure diffusion process and the jump-diffusion process. For the first time, we rigorously establish the closed-form expression of the Onsager–Machlup functional for jump-diffusion processes with finite jump activity, which includes an important term of the Lévy intensity at the origin. The same probability flow approach is further applied to the Lévy process with infinite jump activity and yields a time-discrete version of the Onsager–Machlup functional. |
Global Dynamics of Three-Dimensional Lotka–Volterra Competition Models with Seasonal Succession: I. Classification of Dynamics
| Pengarang | : | Lei Niu, Yi Wang, Xizhuang Xie |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 499-523 |
| Abstrak | : | The current series of two papers focus on a three-dimensional Lotka–Volterra competition model of differential equations with seasonal succession, which exhibits that populations experience an external periodically forced environment. In the first part of the series, we first use a novel technique to construct an index formula for the associated Poincaré map, by which we thoroughly classify the dynamics of the model into 33 classes via the equivalence relation relative to boundary dynamics. More precisely, we show that in classes 1–18, there is no positive fixed point and that every orbit tends to a certain boundary fixed point, while for classes 19–33, there exists at least one (but not necessarily unique) positive fixed point, that is, a positive harmonic time-periodic solution of the model. Among them, the dynamics is trivial in classes 19–25 and 33 provided that the positive fixed point is unique. We emphasize that, unlike the corresponding two-dimensional system, a major significant difference and difficulty for the analysis of the global dynamics for the three-dimensional system is that it may not possess the uniqueness of the positive fixed point. In the forthcoming second part of the series, we shall address the issues of (non)uniqueness of the positive fixed points for the associated Poincaré map. |
A Bimaterial Plane with a Nano-Sized Flexible Line Inclusion on the Interface
| Pengarang | : | Anna Y. Zemlyanova |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 477-498 |
| Abstrak | : | In this paper, a semianalytic solution of a problem for a flexible line inclusion on the interface in a bimaterial plane is given. The line inclusion resists stretching and bending and is mathematically modeled using the Steigmann–Ogden surface elasticity model. The bimaterial plane consists of two linearly elastic isotropic semiplanes with different material parameters. The problem is solved by utilizing Fourier transforms of the unknown functions in each of the semiplanes. It is shown that in the most general case, the problem can be reduced to a system of two singular integral equations on the segment occupied by the inclusion. The existence and uniqueness of the solution of this system are studied by reduction to the system of Fredholm equations of the second kind. The behavior of the solutions at the ends of the inclusion is investigated. The numerical solution is obtained by using Chebyshev polynomial series approximations of the unknown functions. Parametric studies and comparisons with the known results are given. |
The Velocity Field and Birkhoff–Rott Integral for Nondecaying, Nonperiodic Vortex Sheets
| Pengarang | : | David M. Ambrose |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 456-476 |
| Abstrak | : | The Birkhoff–Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, nondecaying, nonperiodic cases are also of interest, such as the interaction of periodic wave trains with noncommensurate periods (i.e., spatially quasiperiodic solutions), or nonperiodic disturbances to periodic wave trains. We therefore develop a more general single formula for the Birkhoff–Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff–Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of nondecaying, nonperiodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, we expect the methodology will apply equally well to three-dimensional fluids. |
Calculation of Expected Reward Rate in a Two-Alternative Decision Process
| Pengarang | : | Lara Dolecek |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 433-466 |
| Abstrak | : | The drift-diffusion model (DDM) has long been recognized as a useful representation of the algorithmic process of decision-making. Specifically, the process of selecting between two options can be considered as the accumulation of evidence subject to noise until one of two accumulation thresholds or boundaries is reached, with past experience setting a bias or drift toward one of the boundaries. In this work, we provide a mathematical analysis to compare two notions of expected reward rate associated with the DDM: the standard formulation based on a ratio of averages, \(\operatorname{E}[R]/\operatorname{E}[T]\), and an alternative based on direct averaging of rate, \(\operatorname{E}[R/T]\), where \(R\) denotes reward size and \(T\) denotes decision time. Both theoretical and empirical results suggest that \(\operatorname{E}[R/T]\) may in fact better describe behavior in many cases. This analysis leads to a new formula for the expected frequency of decisions for arbitrary DDM parameters. It also provides insights about what criteria should be used to select DDM parameters in various settings, which may be helpful both for the design of systems that gather evidence in order to select between available options and for analyzing experimental data related to decision-making tasks. |
Ellipsoidal Embeddings of Graphs
| Pengarang | : | Michaël Fanuel, Antoine Aspeel, Michael T. Schaub, Jean-Charles Delvenne |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 2) |
| Halaman | : | 413-432 |
| Abstrak | : | Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis tools are based on the algebraic representation of graphs via matrices such as the graph Laplacian, or on associated graph embeddings. Such embeddings associate to each node a set of coordinates in a vector space, a representation that can then be employed for learning tasks such as the classification or alignment of the nodes of the graph. As the geometric picture provided by embedding methods enables the use of a multitude of methods developed for vector space data, embeddings have thus gained interest from a theoretical as well as a practical perspective. Inspired by trace optimization problems, often encountered in the analysis of graph-based data, here we present a method to derive ellipsoidal embeddings of the nodes of a graph, in which each node is assigned a set of coordinates in a hyperellipsoid. Our method may be seen as an alternative to popular spectral embedding techniques, with which it shares certain similarities we discuss. To illustrate the utility of the embedding we conduct a case study in which we analyze synthetic and real world networks with modular structure, and compare the results obtained with known methods in the literature. |