From Math to Biology: A Random Talk

Transcription

1 From Math to Biology: A Random Talk Yang Cao Department of Computer Science

2 About Mathematics Computation and Biology mathematics Biology Simple Abstract Connection Structure Computational Science Accurate Stable Efficient Application Experiments Noisy Data Hypothesis Diverse

3 Mathematician s Biology

4 Mathematician s Biology Gheorghe Craciun and Martin Feinberg, Multiple Equilibria in Complex Chemical Reaction Networks: I. The Injectivity Property, SIAM Journal on Applied Mathematics 65:5, , 2005

5 ODEs and Question Question: Can we find out multiple equilibrium points without solving the corresponding ODEs?

6 c ' = p( c, k) Write the corresponding ODEs c ' = p( c, k) Equilibrium point satisfies 0 = p( c, k) 0 = p( c, k) For a given network structure, can we find a set of positive rate constants k, so that 0 = p( c, k has multiple positive solutions? )

7 Solution from a Math Point of View

8 Solution from a Math Point of View Results on the condition for the system does not have multiple equilibria

9 Solution from a Math Point of View Results on the condition for the system does have multiple equilibria

10 Solution from a Math Point of View

11 Propogation of Errors in Simple Network How does a network topology affect the variance? noise Var( xn 1) Var( xn) x0 + Jonathan C. Mattingly, Associate Professor, Math Department, Duke University x1 x2 x3 x4 Noise? Close System Negative feedback

12 Delay-Induced Stochastic Oscillations Bratsun D, Volfson D, Tsimring LS, et al. Delay-induced stochastic oscillations in gene regulation PNAS,102 (41): OCT Consider a very simple model Ruth Williams, Professor. Mathematics Department, UC, San Diego. which describes protein production and degradation, ruled by an ODE

13 Delayed Gillespie Algorithm

14 the coupling between noise and delay leads to the oscillatory behavior, whereas each element separately does not.

15 Gene Regulation Model

16 Power Spectrum Analysis and Simulation

17 HIV-1 Tat Fluctuations Drive Phenotypic Diversity Stochastic Gene Expression in a Lentiviral Positive-Feedback Loop: HIV-1 Tat Fluctuations Drive Phenotypic Diversity, Leor S. Weinberger, John C. Burnett, Jared E. Toettcher, Adam P. Arkin, and David V. Schaffer, Volume 122, Issue 2, 29 July 2005, Pages Weinberger, a researcher at the University of California in Berkeley, was having breakfast with a colleague. They were contemplating the failure of conventional therapies to get to grips with HIV. The vast majority of scientists don t believe it s possible to eliminate the virus or to develop a protective vaccine, he says. So he got to thinking of an entirely different approach: rather than destroying the virus, try instead to live with it. The result of his research is the design for a genetically modified virus that he hopes will be every bit as pervasive as HIV. Adam Arkin Leor Weinberger

18 HIV-1 Tat Fluctuations Drive Phenotypic Diversity

19 Model with BioChemical Reactions

20 Phenotypic Diversity (A) LG-infected Jurkats 7 days after infection (4% are GFP+). (B) LG Dim and Mid bulk sorts (104 cells sorted) were analyzed 1 month postsorting, and no bifurcation or relaxation kinetics were observed. (C) LGIT-infected Jurkats 7 days after infection (6.5% are GFP+). (D) LGIT Off and Bright bulk sorts 7 days postsorting. Postsort analysis confirmed 98% sorting fidelity (data not shown). (E) GFP expression dynamics of LGIT Dim bulk sorted cells (black outline) together with a computer-simulated Dim sort (solid gray) of Equations 1 13 (see text). Dim sorted cells trifurcated in GFP expression after 7 days, such that 30% of cells remained Dim, 30% switched Off and, most strikingly, 30% turned Bright, as seen more easily in the time course (F). Thirtyfive days later, the remaining Dim population had completely relaxed into only Off and Bright populations. Stochastic simulations (initiated with 50,000 GFP molecules) successfully reproduced the trifurcation and relaxation dynamics. (G) Mid sorts relaxed into the Bright region over 20 days, as seen in the 3D overlay of histograms. Notably, the GFP axis is measured on a log scale, and there is a significant difference in GFP between days 7 and 20 (by a chi-square test), as seen more easily in the time course (H). (I) Conversion of the LGIT Mid sort histograms into a color-map representation (the histogram peak is depicted as pink, and each row is a different day) shown together with a grayscale color-map of an in silico LGIT Mid bulk sort (using Equations 1 13 initiated with 300,000 GFP molecules).

21 Phenotypic Diversity (A) Proportions of phenotypes exhibited by clonal populations generated from FACS sorting of single cells from the LGIT Dim GFP region. Of the 30% of Dim cells that successfully expanded, 73% generated clones with no/off GFP expression, 2% produced Bright clones, and 25% of clones exhibited phenotypic bifurcation (PheB). (B) A representative fluorescence micrograph of an LGIT PheB clone (green = GFP, blue = DAPI nuclear staining). (C) Flow histograms of LGIT PheB clones. (D) Clones sorted from the LGIT Bright region do not exhibit PheB. (E) A representative LGIT Bright clone micrograph. (F) Clones sorted from the LG Dim region do not exhibit PheB. (G) LGIT PheB clones (red) can be fully transactivated by chemical perturbation, including a 17 hr incubation in TNFα (green), TSA (yellow), and exogenous Tat protein (1 µg, lightest blue; 6 µg, darker blue; and 12.5 µg, darkest blue). (H) Infection of cells with the two-reporter control LTRmRFP-IRES-TatGFP shows a strong correlation between the first and second cistrons. (I) Additional FACS subsorting from the Bright, Dim, and Off regions of three LGIT PheB clones. For all clones, Dim sorted cells rapidly relaxed into the Off and Bright regions recapitulating a bifurcating phenotype in the first several days after sorting, whereas Bright and Off sorted cells appeared significantly more stable.

22 Thanks! Questions? Plato is my friend, Aristotle is my friend, but my best friend is truth --- Newton