I'm having trouble solving for the steady-state solution for the following simple biomolecular reactions describing the reversible binding of a ligand (in concentration $L$) to its receptor (in concentration $R$) to form a complex (in concentration $C$): $$L'=-k_fLR+k_rC\qquad R'=v-aR-k_fLR+k_rC\qquad C'=k_fLR-k_rC$$ where $k_f$ and $k_r$ represent the forward and reverse rate constants (respectively); $v$ represents the production rate constant of the receptor; and $a$ represents the degradation rate constant of the receptor.
Now, let's solve for the steady-state concentrations $L_s$, $R_s$ and $C_s$ of the ligand, receptor, and complex by setting the left-hand sides of the three equations above as zero. Considering the steady-state first equation, I calculate $L_s$ as:
$$L_s=k_rC_s/(k_fR_s)$$
Considering the steady-state of the second equation, I calculate $R_s$ as:
$$R_s=v/a$$
Substituting these results into the steady-state solution of the second equation:
$$ 0 = (k_fk_rC_sR_s)/(k_fR_s) - k_rC_s$$
which gives me the result:
$$C_s = 0$$
This is one of the steady states (the trivial solution), but I'm wondering why I'm not also finding the non-trivial solution?
Observe that the first and third steady-state equations are, aside from a change of sign, identical, so the system is underdetermined. If you start by solving for $C$ and substituting into the second equation, it becomes $v_R-a_RV=0$, so you get a steady-state value for $R$, but $L$ remains a free variable. Is there an additional constraint that you can introduce (or perhaps you’ve written the equations down incorrectly)?