Construction of an Effective Landscape for Multistate Genetic Switches

Mingyang Lu, José Onuchic, and Eshel Ben-Jacob

Phys. Rev. Lett. 113, 078102 – Published 13 August 2014

Abstract

Multistate genetic switches play a crucial role during embryonic development and tumorigenesis. An archetypical example is the three-way switch regulating epithelial-hybrid-mesenchymal transitions. We devise a special WKB-based approach to investigate white Gaussian and shot noise effects on three-way switches, and construct an effective landscape in good quantitative agreement with stochastic simulations. This approach allows efficient analytical or numerical calculation of the landscape contours, the optimal path, and the state relative stability for general multicomponent multistate switches.

Received 13 February 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.078102

Authors & Affiliations

Mingyang Lu¹, José Onuchic^1,*, and Eshel Ben-Jacob^1,2,†

¹Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005-1827, USA
²School of Physics and Astronomy and The Sagol School of Neuroscience, Tel-Aviv University, Tel-Aviv 69978, Israel

^*jonuchic@rice.edu
^†eshelbj@gmail.com

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Vol. 113, Iss. 7 — 15 August 2014

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Images

Figure 1
Deterministic dynamics of multistate genetic switches. The plots show the nullclines (navy and brown lines), stable steady states (solid green circles), saddle points (hollow green circles), and deterministic flow field (black arrows). (a) Bistable toggle switch and (b) tristable self-activating toggle switch (parameters in the Supplemental Material [31]).
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Figure 2
Construction of effective landscape by the WKB-based approach. Characteristic curves (solid lines, colors representing the $W$ values) are generated by the method of characteristics, starting from the initial contours around each stable fixed point. Some white spaces are not covered by the curves because of under sampling. (a) Two-way switch with white Gaussian noise and (b) three-way switch with shot noise.
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Figure 3
Illustration of the CE method for computing landscape contours and optimal paths. (a) The interpolation method to construct contour lines. Starting from points on contour 1, characteristic curves are generated until they reach contour 2. The contour line between two adjacent contour points $A$ and $B$ can be interpolated by the quadratic Bezier curves constructed from the points $A$ , $B$ , and $C$ , where the point $C$ is the intersection of the two contour slope lines through $A$ and $B$ . For the next iteration, the initial conditions are selected from points on contour 2. (b) The optimal path for the two-way switch. The plot shows the nullclines (navy and brown dotted lines) and the optimal path connecting the stable point and the saddle point. Our numerical methods generate two sets of line segments of the characteristic curves, shown in blue and red solid lines (which almost overlap, see inset).
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Figure 4
Comparison of the landscape contours with stochastic simulations for the three-way switch. Left panels: Gaussian noise; right panels: shot noise. Panels (a) and (b) show the CE contours (colors represent $W$ values), optimal paths (black solid lines), and steepest descent paths (dotted red lines). The insets show $W$ along the optimal path. Panels (c) and (d) show $- \log (P)$ from Langevin simulations (colors selected to better match the $W$ values; see the Supplemental Material [31]). The insets show the mean (blue dots) and standard deviations (error bars) of $- \log (P)$ along each CE contour (e.g., contours illustrated as brown dashed lines, see Sec. 8 of the Supplemental Material [31]). Linear fitting was performed on the data points for the contours within the top left and the bottom right basins. The slope of the linear fitting coincides with $ϵ^{- 1}$ , as expected from the theory. In panel (d), some contour lines are shifted and the middle basin is undersampled because of large shot noise.
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