Systems Biology of Bacterial Immune Systems: Regulation of Restriction-Modification and CRISPR-Cas Systems

Abstract

Restriction-modification (R-M) and CRISPR-Cas are bacterial immune systems which defend their prokaryotic hosts from invasive DNA. Understanding how these systems are regulated is necessary for both biotechnology applications, and for understanding how they modulate horizontal gene transfer (including acquisition of virulence factors). We here review results on modeling these systems which point to common general principles underlying their architecture and dynamical response, with particular emphasis on modeling methods. We show that the modeling predictions are in a good agreement with both in vitro measurements of promoter transcription activity and the first in vivo measurements of gene expression dynamics in R-M systems. Modeling induction of CRISPR-Cas systems is challenging, as signaling which leads to their activation is currently unknown. However, based on similarities between transcription regulation in CRISPR-Cas and some R-M systems, we argue that transcription regulation of much simpler (and better studied) R-M systems can be used as a proxy for CRISPR-Cas transcription regulation, allowing to in silico assess CRISPR-Cas dynamical properties. Based on the obtained results, we propose that mechanistically otherwise different bacterial immune systems, presumably due to a common function, share the same unifying principles governing their expression dynamics.

Appendix

Derivation of the Boltzmann Distribution

Consider a system (s) in contact with a thermal reservoir (r), which together constitute an isolated system with fixed total energy E = E ^(s) + E ^(r). According to the second law of thermodynamics, such an isolated system evolves toward that partition of energy between a system and a reservoir, which can be established through a largest number of a whole system microstates (Phillips et al. 2012). Therefore, the probability that a system has energy ${E}_i^{\left(\mathrm{s}\right)}$ is proportional to the number of corresponding microstates of the overall system, $\Omega \left(E,{E}_i^{\left(\mathrm{s}\right)}\right)={\Omega}^{\left(\mathrm{s}\right)}\left({E}_i^{\left(\mathrm{s}\right)}\times {\Omega}^{\left(\mathrm{r}\right)}\left(E-{E}_i^{\left(\mathrm{s}\right)}\right)\right)$. System degeneracy is directly related to its entropy S = k _B ln(Ω), where k _B is the Boltzmann constant, so the probability that a system has energy ${E}_i^{\left(\mathrm{s}\right)}$ reads

$$ {\displaystyle \begin{array}{c}P\left({E}_i^{\left(\mathrm{s}\right)}\right)\propto \exp \left({S}^{\left(\mathrm{s}\right)}\left({E}_i^{\left(\mathrm{s}\right)}\right)/{k}_B\right)\cdot \exp \left({S}^{\left(\mathrm{r}\right)}\left(E-{E}_i^{\left(\mathrm{s}\right)}\right)/{k}_B\right)\\ {}\approx \exp \left({S}^{\left(\mathrm{s}\right)}\left({E}_i^{\left(\mathrm{s}\right)}\right)/{k}_B\right)\cdot \exp \left(\left({S}^{\left(\mathrm{r}\right)}(E)-\frac{dS^{\left(\mathrm{r}\right)}}{dE}\cdot {E}_i^{\left(\mathrm{s}\right)}\right)/{k}_B\right)\\ {}\propto \exp \left({S}^{\left(\mathrm{s}\right)}\left({E}_i^{\left(\mathrm{s}\right)}\right)/{k}_B\right)\cdot \exp \left(-{E}_i^{\left(\mathrm{s}\right)}/\left({k}_B\cdot T\right)\right)\end{array}} $$

(13)

where in the second step, reservoir entropy is expended about S ^(r)(E) (note that this approximation is valid when a reservoir is much bigger than the system, so ${E}_i^{\left(\mathrm{s}\right)}\ll E$), while in the third step the thermodynamic definition of temperature (∂S/∂E)_V,N = 1/T is used. The first term in Eq. (13) gives the number of microstates of a system with energy ${E}_i^{\left(\mathrm{s}\right)}$ (i.e., Ω^(s) $\left({E}_i^{\left(\mathrm{s}\right)}\right)$), while the second term, called the Boltzmann factor, represents the unnormalized probability of selecting one particular system microstate at energy ${E}_i^{\left(\mathrm{s}\right)}$, i.e., it represents a statistical weight of that one particular microstate (Sneppen and Zocchi 2005).