Transcriptional regulatory cascades in development: initial rates, not steady state, determine network kinetics

Abstract

A model was built to examine the kinetics of regulatory cascades such as occur in developmental gene networks. The model relates occupancy of cis-regulatory target sites to transcriptional initiation rate, and thence to RNA and protein output. The model was used to simulate regulatory cascades in which genes encoding transcription factors are successively activated. Using realistic parameter ranges based on extensive earlier measurements in sea urchin embryos, we find that transitions of regulatory states occur sharply in these simulations, with respect to time or changing transcription factor concentrations. As is often observed in developing systems, the simulated regulatory cascades display a succession of gene activations separated by delays of some hours. The most important causes of this behavior are cooperativity in the assembly of cis-regulatory complexes and the high specificity of transcription factors for their target sites. Successive transitions in state occur long in advance of the approach to steady-state levels of the molecules that drive the process. The kinetics of such developmental systems thus depend mainly on the initial output rates of genes activated in response to the advent of new transcription factors.

Fig. 1.

Model for effect of cis-regulatory occupancy on transcriptional output. (A) Cartoon representing a three-factor cis-regulatory occupancy system. Factors A, B, and C can bind in any order (Left), although triple occupancy is needed for function. All factors interact energetically with one another when brought into proximity as well as with the DNA, i.e., bind cooperatively. Relative equilibrium constants (K_R) for interaction with DNA are symbolized in blue, and cooperativity constants (K_q) are symbolized in yellow. The black circle highlights the functional state. (Center) Assembly of triple complex. (Right) DNA looping results in proximity of fully loaded CRM to the BTA (teal blob), activation thereof, and recruitment of polymerase (orange bricks); transcription ensues. (B) Two-factor CRM, model equations, and terms. For derivations, see Supporting Text. Both factors (A and B) must be bound to the CRM at once for activation of transcription to occur. Eq. 1 defines double occupancy of the CRM. Eq. 2 defines the relation between occupancy and initiation rate. Eqs. 3–5 define the kinetics of nRNA, mRNA, and protein output.

Fig. 2.

Simulations exploring effects of cooperativity of specific transcription factor binding affinity, transcription factor concentrations, and strength of activation function. Default settings were for Eq. 1, K_RA = 10⁵; K_q = 10; D_N is taken as ≈20% of genome size. For molar calculations the volume of a sea urchin embryo cell nucleus was taken to be 4 × 10^–15 liter. For 1.6 × 10⁸ sites, D_N is thus 0.07 M. For transcription factor concentration, A = B = 2,000 molecules (molc.) per nucleus. For Eq. 2, I_M = 5.45 initiations per min^–1 per gene; k_b = 0.44 (see Supporting Text). For Eqs. 3–5, k_dn = 0.02; k_dm = 0.002; k_dP = 0.02; Δt₁ was taken as 20 min, and Δt₂ as 7 min; k_t = 2 molecules min^–1 per mRNA. The values are all from Table 1, except that the default of k_b was computed as follows: Y_AB was calculated from the default values (Eq. 1) and k_b was obtained from Eq. 2, such that as Y_AB → 1, I → I_M. The default for K_q is close to that for a –2 KCal cooperativity exchange (K_q = 7.5), as in the case considered in ref. . (A–D) Cooperativity effects. (A) Effects on occupancy of the CRM by both factors, as a function of number of transcription factor molecules per nucleus. (B–D) Effects on outputs of nRNA, mRNA, and protein, respectively, as transcription factor concentration is varied. (E) Effect of K_R on CRM occupancy, as transcription factor concentration is varied (for examples of transcription factors for which these K_R values apply, see Supporting Text). (F) Time kinetics of nRNA and mRNA accumulation, respectively, for different transcription factor concentrations; in E and FK_q was set at 7.5. (H) Effect of varying k_b on the relation between initiation rate (events per min) and occupancy (Eq. 2).

Fig. 3.

Simulations of cascade behavior. (A) Cartoon of three-step cascade and initiation rates. Double occupancy of each CRM is required as before. Only the first 400 min are shown; for default parameter values see Fig. 2. (B) Effects of cooperativity. (C) Effects of a 2-fold difference in k_b, the activation efficiency (the default value calculated as in the legend of Fig. 2 is 0.44). (D) Effects of K_R. In B–D, dashed lines represent upper parameter assumed; solid lines represent lower parameter; the gray areas represent the time after 10 h. These portions of the curves would probably never be observed in life because in the developing embryo given patterns of gene expression in given cells rarely extend into this time domain. Molc., molecules.