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.