Each diagram is the average of 100 statistically independent trajectories. The residency color scales are log 10. Row 1, 2, 3 and 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low TSS and high TSS respectively. In high TSS, the messenger RNA sub-manifold is essentially at rest. The lines on panels A and B are separatrix for infinite TSS (“2D”) and finite low TSS (“4D”), see text for details. For panels C through H, the original X₁X₂X₃ view is first rotated about the outward pointing eigenvector of the central fixed point in the system to a yield a new coordinate system: X₁*X₂*X₃* in which X₃* is in the direction of this outward pointing eigenvector. The reference is also shifted to the central fixed point. Data are then histogrammed in the X₁*X₂* plane by integrating over the third coordinate X₃* to produce a view from above. More details in Methods. The numbers on panels E, F, G and H are the various fixed points in the systems. On panels E and F, 9 and 1 are stable attractive fixed points. On panel G, 1 and 4 are saddle nodes. 2, 3 and 5 are stable attractive fixed points. The location of #5 is approximate because of sparse sampling. On panel H, 1 is saddle node; 2 and 3 are stable attractive fixed points.

Each diagram includes 100 statistically independent trajectories plotted every 500^th step. The contribution by components 1, 2 (top row) and 1, 2, 3 (rows 2, 3 and 4) to the local curl modulus is indicated by the color on a linear common scale. Rows 1, 2, 3 and 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low TSS and high TSS respectively. In high TSS, the messenger RNA sub-manifold is essentially at rest. All color bars have the same range. This facilitates comparison across conditions.

Each diagram includes 100 statistically independent trajectories plotted every 500^th step. The contribution by components 1, 2 (top row) and 1, 2, 3 (rows 2, 3 and 4) to the local curl modulus is indicated by the color ranging from each figure’s minimum curl to its maximum; the scale is linear. Row 1, 2, 3 and 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low TSS and high TSS respectively. In high TSS, the messenger RNA sub-manifold is essentially at rest. Contrasting to Figure 2, each color bar is different; each scale ranges from the shown track’s lowest to highest curl modulus. This facilitates focusing on each set of conditions.

Propagation is displayed by sequential colored line segments joining successive locations along the stochastic track. Only a relevant part of the track is displayed; details in the text. The contribution by components 1, 2 (top row) and 1, 2, 3 (rows 2, 3 and 4) to the local curl modulus is indicated by the color ranging from each figure’s minimum curl to its maximum. The curl scale is linear. Rows 1, 2, 3 and 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low TSS and high TSS respectively. In high TSS, the messenger RNA sub-manifold is essentially at rest. The X₁, X₂ and X₃ axes are omitted on panel C through H because they would force an adversely distant viewpoint making details of the tracks imperceptible. See Table S1.

The stochastic potential was computed from the corresponding residency histograms. Rows 1 through 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low and high TSS, respectively. Each panel carries its own linear scale, from the minimum to the maximum of the respective data.

Each panel comprises 100 stochastic tracks. The curl was computed along each track. Rows 1 through 4 are for DS1, DS2, DS3 and DS4 respectively. Left and right columns are for low and high TSS, respectively. Each panel carries its own linear scale ranging from the minimum to the maximum of the respective data.

(A) The part of one typical stochastic track in DS3 at very high TSS, as it passes through the location of attractive fixed point #9. The color linearly indicates the local curl modulus in the 1-2-3 axes. The dotted arrow indicates the general direction of the trajectory. Although fixed point #9 is a stable (attractive) fixed point, the dynamics in this region of phase space is dominated by curl forces. Hence, the motion is spiraling around the general direction of motion drifting through (but not converging to) the fixed point. This is because, locally, the gradient forces effectively vanish. Further from the fixed point, gradient forces relatively increase in strength and coerce the dynamics to dwell in the basin of attraction of fixed point #9. For the purpose of comparison and asserting the commonality of this behavior, several similar tracks are shown on Figure S4. (B) Most of the track is shown in white, rotated to the view discussed in the text, and overlaid on top of the stochastic potential gradient modulus underlying the attractive fixed point #9. Because the curl forces in the neighborhood of the fixed point dominate the dynamics, the track meanders in the neighborhood of the fixed point. But as the track repeatedly approaches the boundary where grad forces become significant again, the process is veered back. (C) Normalized gradient modulus and normalized 123 curl modulus profiles in the X₁ and X₂ directions crossing at the stochastic potential center. The labels b, g, r, and y stand for blue, green, red and yellow. The curl modulus is essentially constant whereas the gradient modulus falls nonlinearly to zero at the fixed point located at the origin. (D) Stochastic track started at (X₁, X₂, X₃, X₄)/ = [30, 30, 30, 30] in a far region of negligible curl as indicated by the linear color bar. As shown by the inset, the track is headed for the fixed points (red X at origin) driven by the gradient force. In this very low curl region however, the track despite being noisy, is mostly linear, lacking the helix nature seen on panel A. The scales of panels A and D (main) are similar.

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