# Chih-Yun's Page

Name: Chih-Yun Tseng ch [dot] tseng [at] rutgers [dot] edu OR ctseng98 [at] terpmail [dot] umd [dot] edu University of Maryland, College Park Validating Invariant Sets from DSGRN

DSGRN is a software package that takes as input gene regulatory networks and outputs a database the describes the global dynamics over all of parameter space. The description of the dynamics is given in terms of order theory and algebraic topology (Conley index) and is valid for large classes of nonlinearities. Many biologists use Hill function nonlinearities to describe the dynamics of gene regulation.

The goal of this project would be to start with the coarse description of dynamics given by DSGRN and then rigorously identify specific dynamics.

## Research Log

### Week 1

This week I read a paper about DSGRN [1], two chapters a about rigorous numerics [2][3], and two lecture notes about numerical analysis in higher dimensions [4][5]. I also connected with my peers and came up with a potential goal with my mentors.

Besides readiing these sources, I also attended the HTML workshop which I thought was very helpful.

### Week 2

This week I had my first formal group meeting with my PI and the entire research group. I learned about how DSGRN works and what my role is in the group. I also had a presentation about our project on Monday and I listened to what others will be working on throughout the summer.

For the rest of the week, I have been learning multivariate Newton method and computing radii polynomials. In the process of learning radii polynomials, I found out that there are some issues with computing symbolically verses numerically in MatLab. I read a few notes about radii polynomials and is still in the progress of understanding how to set bounds for the Newton Method.

### Week 3

This week, I attended the data science boot camp that is instructed by Professor Matthew Stone. The course was really interesting and I have learned a lot about python in a short period of time.

In addition, I have also been dealing with finding the radii polynomial for a simple toggle-switch model and have been trying to come up with a matlab code to compute radii polynomials automatically.

### Week 4

This week, I have been working on turning hill functions into piecewise linear functions on MatLab for any given parameters. This program will be useful for me because I would like to examine the dynamics behind the systems of equations. In addtion, I have been able to plot the linear functions and solve for their intersections in order to retrieve a crude approximation of the equilibrium so that I can plug them back to the Newton method.

### Week 5

This week, I have finished combining my radii polynomial function, detect intersection function, and Newton method all together for 2D toggle-switch. I also adjusted how I computed my alpha's by abandoning the crude and inefficient method of taking 100 iterations and averaging them all out. I am currently working on merging the linearization of hill functions and try to produce similar results with what I had for simple hill functions (ie hill functions without interactions).

### Week 6

This week, I have been working on self-interaction functions and trying to employ similar methods that I did last week to produce linear approximations and corresponding graphs. Some issues that I encountered were how to divide each 'block' and solve for the intersections. In addition, I tried to explicitly solve for the linear systems and the self-interaction hill function but made a couple mistakes. Overall, I was able to fixed the problems and am working on 'filter systems' that allows me to not solve for piecewise intersections explicitly.

### Week 7

This week, I have completed a very rough approximation for solving the piecewise linear self-interaction function. The approach is just utilizing the intermediate value theorem (but there are still a few errors left). I wanted to start using IntLab to prove the absense of equilibria but in the meanwhile I'm trying to prove the number of equilibria for a simple toggle-switch model by hand.

### Week 8

This week, I have resolved the small errors with the intermediate value theorem. Now, I have implemented some code for excluding other equilibria in the complement region with INTLAB.

### Week 9

This week, I have been focusing on my paper and the final presentation.