Periodic Reporting for period 1 - STOPATT (Stochastic pattern formation in biochemical systems)
Período documentado: 2020-04-01 hasta 2022-03-31
What is the problem/issue being addressed?
Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Many sorts of motile cells undergo chemotaxis. For example, bacteria and many amoeboid cells can move in the direction of a
food source. In our bodies, immune cells like macrophages and neutrophils can move towards invading cells. Other cells, connected with the immune response and wound healing, are attracted to areas of inflammation by chemical signals.
This macroscopic system of equations is usually derived from microscopic behavior by studying the limit behavior. From the microscopic perspective, one interprets the movements of the cells as a result of microscopic irregular movement of the single members of the population of cells. Taking the limit and passing from the microscopic to the macroscopic equation, one is neglecting the fluctuations. Another source of randomness is due to the environment. Here, biologists distinguish between internal (or intrinsic) noise caused by the irregular movement of the cells and external noise caused by a random environment. An appropriate mathematical approach to establishing more realistic models is the incorporation of stochastic processes. To model the randomness, one adds a random forcing term to the system.
Adding a stochastic driving term has a highly non–trivial impact on the behavior of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behavior, which is most helpful in understanding the real processes and is also often more realistic. E.g. if the system starts to switch between two quasi-stable states, the pattern may change or starts to wander around. Bifurcations are smeared out, and the importance of tipping points may change.
Although the project is of theoretical nature, the findings may be used to develop a more realistic model in biology and medicine and to explain phenomena that appear but can not be explained by purely deterministic systems. Improving models in biology or medicine have a natural impact on society. Understanding, e.g. the self-organization of cells, means, e.g. understanding biological processes like wound healing or cancer in a better way. This can lead to the design of new strategies against cancer.
The aim of the project was first to investigate the mathematical existence, uniqueness, and dynamical behavior of these processes. Then, to investigate its long-time dynamical behavior, and, finally, the modelisation of these processes.
Why is it important for society?
Model organisms are species that are extensively studied to understand biological phenomena that provide insight into the functions of other organisms. Researching these organisms explores the basic mathematics, biology, and chemistry of life. As a next question, we ask ourselves, why choose to study model organisms instead of the organism of interest itself? Research of biological systems, development, or genetics on humans is complicated and poses ethical challenges in many situations. In biological systems, it is very difficult to separate the effects of a gene from the effects of the environment. However, with models, variables of interest can be engineered and confounding variables can be controlled accordingly.
What are the overall objectives?
The overall objective of the project was first to investigate the mathematical solvability theory (existence, uniqueness), and then the dynamical behavior of these processes. As a next step, we investigated its long-time dynamical behavior, and, finally, the modelization of these processes. This mathematical analysis establishes important discoveries in the world of mathematical biological models in presence of random perturbations.
Chemotaxis is defined as the oriented movement of cells (or an organism) in response to a chemical gradient. Many sorts of motile cells undergo chemotaxis. For example, bacteria and many amoeboid cells can move in the direction of a
food source. In our bodies, immune cells like macrophages and neutrophils can move towards invading cells. Other cells, connected with the immune response and wound healing, are attracted to areas of inflammation by chemical signals.
This macroscopic system of equations is usually derived from microscopic behavior by studying the limit behavior. From the microscopic perspective, one interprets the movements of the cells as a result of microscopic irregular movement of the single members of the population of cells. Taking the limit and passing from the microscopic to the macroscopic equation, one is neglecting the fluctuations. Another source of randomness is due to the environment. Here, biologists distinguish between internal (or intrinsic) noise caused by the irregular movement of the cells and external noise caused by a random environment. An appropriate mathematical approach to establishing more realistic models is the incorporation of stochastic processes. To model the randomness, one adds a random forcing term to the system.
Adding a stochastic driving term has a highly non–trivial impact on the behavior of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behavior, which is most helpful in understanding the real processes and is also often more realistic. E.g. if the system starts to switch between two quasi-stable states, the pattern may change or starts to wander around. Bifurcations are smeared out, and the importance of tipping points may change.
Although the project is of theoretical nature, the findings may be used to develop a more realistic model in biology and medicine and to explain phenomena that appear but can not be explained by purely deterministic systems. Improving models in biology or medicine have a natural impact on society. Understanding, e.g. the self-organization of cells, means, e.g. understanding biological processes like wound healing or cancer in a better way. This can lead to the design of new strategies against cancer.
The aim of the project was first to investigate the mathematical existence, uniqueness, and dynamical behavior of these processes. Then, to investigate its long-time dynamical behavior, and, finally, the modelisation of these processes.
Why is it important for society?
Model organisms are species that are extensively studied to understand biological phenomena that provide insight into the functions of other organisms. Researching these organisms explores the basic mathematics, biology, and chemistry of life. As a next question, we ask ourselves, why choose to study model organisms instead of the organism of interest itself? Research of biological systems, development, or genetics on humans is complicated and poses ethical challenges in many situations. In biological systems, it is very difficult to separate the effects of a gene from the effects of the environment. However, with models, variables of interest can be engineered and confounding variables can be controlled accordingly.
What are the overall objectives?
The overall objective of the project was first to investigate the mathematical solvability theory (existence, uniqueness), and then the dynamical behavior of these processes. As a next step, we investigated its long-time dynamical behavior, and, finally, the modelization of these processes. This mathematical analysis establishes important discoveries in the world of mathematical biological models in presence of random perturbations.
WP1. Existence and Uniqueness Results and Ergodic Properties:
(I) Existence of a local solution to the two-dimensional stochastic Keller Segel Model:
(2) Strong solution to a stochastic chemotaxis system with porous medium diffusion:
(3) Strong solution to a stochastic proliferation chemotaxis system:
(4) Uniqueness of the stochastic Keller - Segel model in one dimension:
The current paper consists of two results about the stochastic PKS system in 1d. First, we establish some additional regularity results on the solutions. The additional regularity is, e.g. important for its numerical modeling.
Then, as a second result, we obtain the pathwise uniqueness of the solution to the stochastic PKS system in 1d. Finally, we conclude the paper with the existence of a strong solution for this system in 1d.
WP2. Long Term Asymptotics and the Dynamical Behaviour of the System:
(1) Landau-Lifshitz-Gilbert equations: Controllability by Low Modes Forcing for deterministic version and Support Theorems for Stochastic version:
WP3. Modeling, Simulation, and Reconstructing of the Stochastic System:
(1) The Wong -Zakai approximation for Landau - Lifshitz -Gilbert equation driven by geometric rough paths:
The goal is to prove a convergence rate for Wong-Zakai approximations of the stochastic PKS system driven by a finite-dimensional Brownian motion. As a first step, we study the piecewise linear approximation (Wong-Zakai approximation).
Due to COVID, we had a problem concerning the dissemination. In particular, several activities like Forschungsnacht were canceled, and workshops were canceled or went online. We could not invite other researchers, nor could we not visit them. We will present in the future our results, but the presentation will not be within the timeline of this project.
In addition, it often takes from seven to eight months to one or two years to submit a paper to be accepted, we have at the moment only one paper, and the rest are preprints. However, we expect some papers to be accepted soon.
(I) Existence of a local solution to the two-dimensional stochastic Keller Segel Model:
(2) Strong solution to a stochastic chemotaxis system with porous medium diffusion:
(3) Strong solution to a stochastic proliferation chemotaxis system:
(4) Uniqueness of the stochastic Keller - Segel model in one dimension:
The current paper consists of two results about the stochastic PKS system in 1d. First, we establish some additional regularity results on the solutions. The additional regularity is, e.g. important for its numerical modeling.
Then, as a second result, we obtain the pathwise uniqueness of the solution to the stochastic PKS system in 1d. Finally, we conclude the paper with the existence of a strong solution for this system in 1d.
WP2. Long Term Asymptotics and the Dynamical Behaviour of the System:
(1) Landau-Lifshitz-Gilbert equations: Controllability by Low Modes Forcing for deterministic version and Support Theorems for Stochastic version:
WP3. Modeling, Simulation, and Reconstructing of the Stochastic System:
(1) The Wong -Zakai approximation for Landau - Lifshitz -Gilbert equation driven by geometric rough paths:
The goal is to prove a convergence rate for Wong-Zakai approximations of the stochastic PKS system driven by a finite-dimensional Brownian motion. As a first step, we study the piecewise linear approximation (Wong-Zakai approximation).
Due to COVID, we had a problem concerning the dissemination. In particular, several activities like Forschungsnacht were canceled, and workshops were canceled or went online. We could not invite other researchers, nor could we not visit them. We will present in the future our results, but the presentation will not be within the timeline of this project.
In addition, it often takes from seven to eight months to one or two years to submit a paper to be accepted, we have at the moment only one paper, and the rest are preprints. However, we expect some papers to be accepted soon.
We have investigated the existence, uniqueness, and regularity of the Keller-Segel process with proliferation. We analyzed the long-time behavior, i.e. the existence of an invariant measure. Here, nearly no work was known. Also, due to our first work, other mathematical biological groups got also interested in the topic. The modelling is still an ongoing project.
Potential Impact: As in Cryptology or data science which were established around 1950 and 1960, and are largely used now, or even the Turing test by Alan Turing, it will take 10 to 15 years to have a big socio-economical impact. In bio-enigeering, control mechanism using chemotaxis are already used, here modelling under random perturbation may have an impact. Including randomness in the calculation would lead to robust algorithm to control e.g. bio-chemical batch reactors. Here, in text books are often still models are used which even do not use the spatial temporal models. Here, one can eventually have some economic developing new strategies in controlling and maintaining bio-chemical reactors in industry.
Potential Impact: As in Cryptology or data science which were established around 1950 and 1960, and are largely used now, or even the Turing test by Alan Turing, it will take 10 to 15 years to have a big socio-economical impact. In bio-enigeering, control mechanism using chemotaxis are already used, here modelling under random perturbation may have an impact. Including randomness in the calculation would lead to robust algorithm to control e.g. bio-chemical batch reactors. Here, in text books are often still models are used which even do not use the spatial temporal models. Here, one can eventually have some economic developing new strategies in controlling and maintaining bio-chemical reactors in industry.