The Dynamics of Inducible Genetic Circuits

TL;DR

This paper makes a departure from the conventional dynamical systems view of these regulatory motifs by using statistical mechanical models to focus on endogenous signaling knobs rather than on the convenient but more experimentally remote knobs such as dissociation constants, transcription rates and degradation rates that are often considered.

Abstract

Genes are connected in complex networks of interactions where often the product of one gene is a transcription factor that alters the expression of another. Many of these networks are based on a few fundamental motifs leading to switches and oscillators of various kinds. And yet, there is more to the story than which transcription factors control these various circuits. These transcription factors are often themselves under the control of effector molecules that bind them and alter their level of activity. Traditionally, much beautiful work has shown how to think about the stability of the different states achieved by these fundamental regulatory architectures by examining how parameters such as transcription rates, degradation rates and dissociation constants tune the circuit, giving rise to behavior such as bistability. However, such studies explore dynamics without asking how these quantities are altered in real time in living cells as opposed to at the fingertips of the synthetic biologist's pipette or on the computational biologist's computer screen. In this paper, we make a departure from the conventional dynamical systems view of these regulatory motifs by using statistical mechanical models to focus on endogenous signaling knobs such as effector concentrations rather than on the convenient but more experimentally remote knobs such as dissociation constants, transcription rates and degradation rates that are often considered. We also contrast the traditional use of Hill functions to describe transcription factor binding with more detailed thermodynamic models. This approach provides insights into how biological parameters are tuned to control the stability of regulatory motifs in living cells, sometimes revealing quite a different picture than is found by using Hill functions and tuning circuit parameters by hand.

Figures (38)

• Figure 1: Gallery of examples of regulatory circuits participating in the genetic decisions of animal development. (A) A three-node network thought to be relevant to the control of digit formation, adapted from zuniga2014turing. (B) An example involving vulval development in C. elegans, where epidermal growth factor (EGF) and Notch induce cells toward one of three possible fates, adapted from corson2012geometry and schindlermorphogenesis. (C) Transcription factors compete and maintain cell pluripotency unless sufficiently induced to reprogram a cell to a differentiated fate, adapted from liu2008yamanaka.
• Figure 2: The auto-activation regulatory circuit. (A) Schematic of the operation of the circuit. Polymerase binding at the promoter (blue) transcribes the gene (encoded in the light green region), producing a protein that can activate its own expression at a sufficient concentration. In our model, an activator can bind at one of two possible sites to enhance gene transcription. (B) Thermodynamic states, weights, and rates for the circuit in the traditional model without induction. The parameter $\omega$ denotes the cooperative strength of two activators binding. (C) Thermodynamic states, weights and rates for the case in which the effector tunes the fraction of active activators. Note that in both of these cases the parameters $K_d$, $\omega$, $r_0$, $r_1$ and $r_2$ are effective parameters that have hidden dependence upon the number of polymerases and the strength with which it binds the promoter. The explicit definitions of these effective parameters are worked out in Appendix \ref{['Section:CoarseGraining']}.
• Figure 3: Tuning genetic circuits. The schematic shows different knobs which are available to the cell, the theorist and the experimentalist, namely (A) effector concentration (and by consequence the number of active activator or repressor molecules), (B) binding affinity $K_{d}$, (C) protein production rate $r$, and (D) cooperativity $\omega$.
• Figure 4: States and weights for an allosteric transcription factor with an effector that can bind at two sites on the protein. Effectors can bind to both the active and inactive forms of the transcription factor with different dissociation constants $K_A$ and $K_I$, which determine whether the effector stabilizes the protein more strongly in its active or inactive configuration. The sum of the thermodynamic weights for the active and inactive conformations are shown at the bottom.
• Figure 5: Activity of a transcription factor as a function of effector concentration $c$. Parameters used throughout the paper, unless otherwise stated, are: $K_A = 140 \, \mu M$, $K_I = 530 \, nM$, $\varepsilon = 4.5 \, k_BT$. (A) Probability of active transcription factor as a function of effector concentration, defined by Eqn. \ref{['eqn:MWC2site']}. The half maximal effective concentration EC$_{50}$, defined as the effector concentration $c^*$ such that $p_{\text{act}}(c^*) = (p_{\text{act}}^{max} + p_{\text{act}}^{min})/2$, is plotted in purple. (B) The effective dissociation constant $K_d^{\text{eff}}$ (dimensionless with respect to $K_{d}$) as a function of effector concentration. Saturation of $p_\text{act}(c)$ corresponds to minimal $K_d^{\text{eff}}$, and leakiness of $p_\text{act}(c)$ corresponds to maximal $K_d^{\text{eff}}$.