Bacterial small RNAs (sRNA) are non-coding RNA molecules produced by bacteria. The role of sRNA in bacterial physiology is extremely diverse; they can either bind to protein targets, and modify the function of the bound protein, or bind to mRNA targets and regulate gene expression. Antisense sRNAs can be categorised as cis-encoded sRNAs, where there is an overlap between the antisense sRNA and the target gene, and trans-encoded sRNAs, where the antisense sRNA gene is separate from the target gene.

Relevant Biology

The model is based on sRNAs that bind to the chaperone protein, Hfq. Hfq binds to sRNA, forming a complex. This complex then binds to mRNA and promotes degradation of both the mRNA and sRNA in a stoichiometric manner. Mechanistically, the Hfq-mRNA-sRNA complex is broken down by a degradosome, a complex of proteins where the protein RNAse E is the centerpiece~\cite{aiba2007mechanism}. The important thing to note here is that the order is compulsory.

We can also assume that binding of mRNA to sRNA doesn't happen on its own, which Professor Scott and myself talked about. Some papers seem to suggest that it does, others note the requirement for Hfq.

In some cases Hfq is actually part of the degradosome, for example in SgrS regulation, and sometimes its not, in the case of RyhB. Both SrgS and RyhB are names for specific sRNA that regulate different metabolic pathways; RyhB is responsible for regulating iron metabolism in E. coli, SrgS is responsible for handling glucose-phosphate stress (a rapid increase in glucose-6-phosphate, a precursor to glycolysis). This changes the mechanism quite a bit, however, for the purposes of this model, I'm going to assume that our sRNA suppression style is more akin to RyhB - although we really should look into this.

Our previous models haven't considered the fact that sRNA gets degraded with the mRNA by the degradosome simultaneously. This new formulation is that assumptions' reckoning.

Model Formation

The model of chemical network is shown below. Before writing this out as a system of equations, we want to describe what's happening first. We are tracking the concentrations of seven species: s, m, M, h, H, Hs and Hms, representing the sRNA, the mRNA, the target protein, Hfq mRNA, Hfq, Hfq-sRNA complex, and Hfq-sRNA-mRNA complex respectively.

Production of a species, be it RNA or protein, is denoted with an α and a subscript indicating which species is being produced. Degradation is given by a β with the same subscript convention. Aside from the translation of RNA to protein, everything is based on mass action - with the exception of J, which we've used to denote a flux from H_ms to H, the degradosome destroying the sRNA and mRNA.

This model somewhat lends itself well to the idea of simplification, primarily between all of the complexes. In particular, we will apply a quasi-steady state assumption to the movement between all of the complexes. Recall the dynamics of the complexes seen in the network diagram, we can simplify this a little by first considering the dynamics of the complexes as a single, isolated system, showcased in the figure below.

If we assume no source and no degradation of H (for now), the above system is modeled as:

One simplifying assumption would be if, rather than a Michaelis-Menton term, we had a mass-action term. We don't need to make this assumption, but it makes the rest of the calculations a whole lot easier (as in, will fit between the margins easier). Consider the first-order approximation to the Michaelis-Menton Term:

With this approximation (which is valid for small H _ms - something we can look into), the model reduces nicely. In particular the flux term is approximated as:

The significance of this flux is that it represents the steady state breakdown rate of mRNA and sRNA. What we would really like is this flux term as a function of m and s so that we don't have to keep track of the entire Hfq-complex pathway in our model.

Since we are assuming that flux through this pathway happens much faster than transcription and translation, let's also consider the total amount of Hfq to be constant and given by H_T = H + H_s + H_ms, then we apply the Quasi-Steady State Assumption to our model:

Written more compactly:

The Least Squares Solution to this equation is:

Which means that the flux term is approximated as:

Which means that the flux term is approximated as:

Where K = k₃/(k₁k₂) and note that k₃H_T plays a similar (identical, actually) role to V_max in the Michaelis-Menton formula. Note, however, that J(0,0)is not defined. Observe that:

So a more precise definition of J is:

With all of these simplifying assumptions, the model reduces to:

Here, we've lumped all of the Hfq protein complexes (and, indeed, Hfq) into one species. Notice that the last two differential equations are independent of the rest. Even better - they're Linear! And even better, it's a 2x2 system! Focusing on the last two equations, and rewriting them in vector-matrix form:

Focusing on the homogeneous part, notice that the matrix is lower triangular. Thus, the eigenvalues appear on the diagonal, λ₁ = - β_H; λ₂=-β_h. The corresponding eigenvectors are:

and thus, the fundamental matrix is:

With the Fundamental Matrix, we can apply Variation of Parameters to get the general solution for H_T. Since the transcription for Hfq is under the control of our wonderful experimenters, we could chose to have α_h be some arbitrary function of time. Regardless, the general solution is:

Where h⁽⁰⁾, and H_T⁽⁰⁾ are initial conditions on h and H_T.

This looks pretty crazy, but it gives us a way of solving for H_T, written explicitly:

The input is how α_h varies with time. If we take the initial conditions to be zero for each we get:

At any rate, the point here is that we can solve for exactly how H_T varies with time. We can also see that if our forcing term is constant, then we'll settle down to a steady state concentration of H_T. Additionally, our original model then reduces to three equations:

Notice also that the last equation doesn't really influence the dynamics of the first two. Additionally, if H_T is constant, then it's possible to observe a phase-portrait of the system. It's also possible to look at possible steady states. In other words, we can actually do some analysis on this model without physically being supercomputers.

Parameters

We identified parameters in the literature. The identified parameters and their sources are given in the table below.

sRNA Parameters from Literature

Parameter	Value	Description	Reference
αm	1/600 (nM*s)-1	transcription mRNA	Fender et al.
αs	1/600 (nM*s)-1	transcription sRNA	Fender et al.
αM	1/600 (nM*s)-1	translation of the mRNA	Fender et al.
k-1	0.7*10-4s-1	dissociation constant of Hs to Hfq and sRNA	Fender et al.
k1	106 s-1	association constant of Hfq and sRNA	Fender et al.
k2	3.5*106 (M*s)	association constant of Hs and mRNA	Fender et al.
k3	0.7*10-4 s-1	dissociation constant of Hms, assuming that the Hfq-sRNA binding to the MicC region is independent of the sRNA, mRNA binding. Therefore, k-1=k3	Fender et al.

Sensitivity Analysis

To get a better handle on the dynamics of the system we ran a local sensitivity analysis. This determined what parameters the sRNA system is most sensitive to. The flux control coefficients for the sRNA system can be seen in the figure to the right.

The most notable of these are the flux control coefficients for the following parameters: β M(the degradation rate of YFP), αm (the transcription rate of the YFP mRNA transcript). These large flux control coefficient values demonstrate the system is highly sensitive to changes in these variables. So if we were able to influence these rates we would be able to dramatically change the level of flux through the system. In order for the sRNA system to be most efficient, we would want the flux of YFP through the system to be as low as possible. The most direct way to affect this would be to alter these values. Since we want to use this sRNA system to control a protein whose degradation rate and transcription rate we could not alter, we need to turn our gaze to some of the other factors at play.

Some other notable flux control coefficients are αS, αM, K-1. These rates are respectively the rate of transcription of sRNA, the rate of translation of YFP, and the rate of dissociation of Hfq-sRNa to Hfq and sRNA. These rates have the least impact on the system and are not good targets for optimization of sRNA.