Solution of some equations in Biochemistry

J.P Bennett and J.H. Davenport  H.M. Sauro
School of Mathematical Sciences  Institue of Grassland and Animal Production
University of Bath and Poultry Division
Claverton Down  Roslin,
Bath BA2 7AY  and Midlothian
England  Scotland

Abstract It is possible to write down the equations governing a one-stage enzyme-catalysed reaction (according to Michaelis-Menton kinetics) quite easily, and deduce information about the steady-state flow in such a system. The situation is somewhat more complicated if several such reactions form a linear chain. We have applied Gröber-basis techniques to solve such systems.

Introduction. If we consider an enzyme-catalysed reaction in steady-state, such as

X_{0}⟶^{v}X_{1}

(with the underlying first-order mechanism

X_{0} + E ⇀ ↽_{k_{2}}^{k_{1}} X_{0}E ⇀ ↽_{k_{4}}^{k_{3}} X_{1} + E

in which we have written

K_{eq} = \frac{k_{1}k_{3}}{k_{2}k_{4}}  (overall equilibrium constant)
K_{m,f} = \frac{k_{2} + k_{3}}{k_{1}}  (forward Michaelis constant)
K_{m,r} = \frac{k_{2} + k_{3}}{k_{4}}  (reverse Michaelis constant)
V_{max} = [E_{T OT}]k_{3}  (maximum flux)
[E_{T OT}] = [E] + [X_{0}E]  (conservation of enzyme)

as is standard) the rate of conversion of X_{0} to X_{1} (flux) can be written as

v = \frac{\frac{V_{m}}{K_{m}} ([X_{0}] − [X_{1}]K_{eq})}{1 + \frac{[X_{0}]}{K_{m,f}} + \frac{[X_{1}]}{K_{m,r}}}.  (1)

When the reactions are more complex, e.g.

X_{0}⟶^{v_{1}}S_{1}⟶^{v_{2}}X_{2},

it becomes harder to analyse the situation, even under the steady-state assumption that [S_{1}] is constant. We would like to eliminate [S_{1}] from the two equations (analogous to (1)) which determine the fluxes in the two stages. This has been done [...], and the result is that [S_{1}] satisfies a quadratic equation. As far as the authors are aware, the case of three-stage linear chains has not been solved.

Our solution.

The three-stage chain is

X_{0}⟶^{v_{1}}S_{1}⟶^{v_{2}}S_{2}⟶^{v_{3}}X_{3}.

From the point of view of Gröbner-bases, the problem is, in principle, trivial. We have the equation v_{1} = v_{2} = v_{3} at steady state, and all that is required is to eliminate the variables [S_{1}] and [S_{2}] from this. We used REDUCE-3 [Hearn, 1983] running on a Sun 3/160 and a Gröbner-basis package written by the second author. The only delicate point is the ring in which the Gröbner-basis is to be calculated: this is L[s_{1}, s_{2}] where s_{1} and s_{2} are [S_{1}] and [S_{2}] (trying to mix biochemical notation with that of modern algebra is confusing at best!), where L is the field generated over the rational by all the other indeterminates that appear in the equations. The REDUCE input to solve this problem is surprisingly simple:

v1:=(v1m/k1mf)*(x0-s1*k1eq)/(1+x0/k1mf+s1/k1mr);
grobner!-constant v1m,k1mf,k1mr,k1eq;
v2:=(v2m/k2mf)*(s1-s2*k2eq)/(1+s1/k2mf+s2/k2mr);
grobner!-constant v2m,k2mf,k2mr,k2eq;
v3:=(v3m/k3mf)*(s2-x3*k3eq)/(1+s2/k3mf+x3/k3mr);
grobner!-constant v3m,k3mf,k3mr,k3eq;
grobner!-constant x0,x3;
grobner(num(v1-v2),num(v1-v3));

The calls to grobner!-constant declare that the indeterminates mentioned belong in L, rather than to the polynomial extension of L. The call to grobner passes in two polynomials (the numerators of v_{1} − v_{2} and v_{1} − v_{3}), and computes a lexicographical-order Gröbner-basis for the resulting ideal of L[s_{1}, s_{2}]. The calculation took about five seconds, compared with one minute if all three equations are given to grobner (if only v_{1} −v_{2} and v_{2} −v_{3} are given to grobner, the calculation runs for about 15 minutes before exceeding the 2Mb heap available in our version of REDUCE). It turns out that s_{1} satisfies a cubic equation, reproduced in the appendix, and s_{2} is linear in s_{1} (the details are also in the appendix). REDUCE is capable of generating equations in FORTRAN form, and machine-readable copies of these equations are obtainable from the first author.

Conclusions.

Acknowledgements. The Gröbner-basis package used was written while the second author was visiting the Royal Technical Highschool, Stockholm, and the author would like to acknowledge the hospitality and helpful suggestions of Stefan Arnborg and Ian Cohen.

References

Appendix.

The cubic satisfied by s_{1} can be written as

as_{1}^{3} + bs_{1}^{2} + cs_{1} + d = 0,

where the coefficients are (in the order a, b, c, d)

−v_{max,1}^{3}k_{eq,1}^{3}k_{m,1,r}^{3}k_{m,2,r}k_{m,3,r} − v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}k_{m,3,r}
−2v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}k_{m,3,r} − 2v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}
−v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − k_{m,1,f}^{3}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r},

x_{3}v_{max,1}^{3}k_{eq,1}^{3}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f} − x_{3}v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}

+x_{3}v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f} − x_{3}v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}

+x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r} + x_{3}k_{m,1,f}^{3}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

+2v_{max,1}^{3}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,r}k_{m,3,r} − v_{max,1}^{3}k_{eq,1}^{3}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}k_{m,3,r} + v_{max,1}^{3}k_{eq,1}^{3}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f}k_{m,3,r}

+v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}k_{m,3,r} + 3v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}k_{m,3,r}

−v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}k_{m,3,r} − v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,r}v_{max,3}k_{m,3,r}

−v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r} − 2v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r}

+v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f}k_{m,3,r} − v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}k_{m,3,r}

−v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

+v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 3v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

−v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r}

−v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 3v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

−v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 2k_{m,1,f}^{3}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}

−2k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r},

−2x_{3}v_{max,1}^{3}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f} + x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}

−2x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f} + x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} − x_{3}v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} − x_{3}v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r}

+x_{3}v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f} + x_{3}v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}

+2x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}

+2x_{3}v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r}

−x_{3}v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

+2x_{3}k_{m,1,f}^{3}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r} + 2x_{3}k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

−v_{max,1}^{3}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,r}k_{m,3,r} + 2v_{max,1}^{3}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}k_{m,3,r}

−2v_{max,1}^{3}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f}k_{m,3,r} − v_{max,1}^{2}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}k_{m,3,r}

+v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}k_{m,3,r} + v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}k_{m,2,r}v_{max,3}k_{m,3,r}

+v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r}+3v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r}

−2v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f}k_{m,3,r} − v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r}

−v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}k_{m,3,r}

+v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,r}v_{max,3}k_{m,3,r} − v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r}

−v_{max,1}^{2}x_{0}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r} +v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} + v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r} +v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

−v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r}

−v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}
+v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}
−2v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}
−v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r}

−v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − v_{max,1}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

−k_{m,1,f}^{3}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 2k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}

−k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}

and

x_{3}v_{max,1}^{3}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f} + x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f}

+x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} + x_{3}v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r}

+x_{3}v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} + x_{3}v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} +x_{3}v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f} + x_{3}v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

+x_{3}v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r}

+x_{3}v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}
+x_{3}v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}^{2}k_{eq,3}k_{m,3,r}
+2x_{3}v_{max,1}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r}

+x_{3}v_{max,1}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r} + x_{3}k_{m,1,f}^{3}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r} +2x_{3}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r} + x_{3}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}^{2}k_{eq,3}k_{m,3,r}

−v_{max,1}^{3}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}k_{m,3,r} + v_{max,1}^{3}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,f}k_{m,3,r}

−v_{max,1}^{2}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r}

+v_{max,1}^{2}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,f}v_{max,3}k_{m,3,f}k_{m,3,r} + v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r} +v_{max,1}^{2}k_{m,1,f}x_{0}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r} +v_{max,1}^{2}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r} +v_{max,1}k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} + v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{m,3,f}k_{m,3,r}

+v_{max,1}k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}.

When REDUCE is asked to factorise these expressions as much as possible (on factor), the results become somewhat more manageable, with a, b, c and d becoming:

−(2v_{max,2} + v_{max,3})v_{max,1}k_{m,1,f}^{2}k_{eq,1}k_{m,1,r}k_{m,2,r}v_{max,3}k_{m,3,r}
−(v_{max,2} + 2v_{max,3})v_{max,1}^{2}k_{m,1,f}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,r}k_{m,3,r} − v_{max,1}^{3}k_{eq,1}^{3}k_{m,1,r}^{3}k_{m,2,r}k_{m,3,r}
−k_{m,1,f}^{3}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r},

−(((((2k_{m,2,r} − k_{m,3,f})k_{m,2,f}v_{max,3} + v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}) + (v_{max,2} + v_{max,3})k_{m,1,r}k_{m,2,r})k_{eq,1} −(v_{max,2} + 3v_{max,3})x_{0}k_{m,2,r})k_{m,1,f} + (v_{max,2} + v_{max,3})x_{0}k_{eq,1}k_{m,1,r}k_{m,2,r})v_{max,1}^{2}k_{eq,1}k_{m,1,r}^{2}k_{m,3,r}

−((((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + (3v_{max,2} + v_{max,3})k_{m,1,r})k_{eq,1} − (v_{max,2} + v_{max,3})x_{0})k_{m,1,f}

+(3v_{max,2} + v_{max,3})x_{0}k_{eq,1}k_{m,1,r})v_{max,1}k_{m,1,f}k_{m,1,r}k_{m,2,r}v_{max,3}k_{m,3,r}
−((k_{m,2,r} − k_{m,3,f})k_{eq,1}k_{m,2,f} − 2x_{0}k_{m,2,r})v_{max,1}^{3}k_{eq,1}^{2}k_{m,1,r}^{3}k_{m,3,r}

+(v_{max,1}k_{eq,1}k_{m,1,r}k_{m,2,f} − k_{m,1,f}v_{max,2}k_{eq,2}k_{m,2,r})(v_{max,1}k_{eq,1}k_{m,1,r}k_{m,3,f} − k_{m,1,f}v_{max,3}k_{eq,3}k_{m,3,r})×

(v_{max,1}k_{eq,1}k_{m,1,r} + k_{m,1,f}v_{max,3})x_{3}

−2k_{m,1,f}^{3}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 2k_{m,1,f}^{2}x_{0}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r},

(((((3k_{m,2,r} − 2k_{m,3,f})k_{m,2,f}v_{max,3} + v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}) + (v_{max,2} + v_{max,3})k_{m,1,r}k_{m,2,r})x_{0}k_{eq,1}

−(v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3})k_{eq,1}^{2}k_{m,1,r}k_{m,2,r} − x_{0}^{2}k_{m,2,r}v_{max,3})k_{m,1,f}

−(v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3})x_{0}k_{eq,1}^{2}k_{m,1,r}k_{m,2,r}
+(v_{max,2} + v_{max,3})x_{0}^{2}k_{eq,1}k_{m,1,r}k_{m,2,r})v_{max,1}^{2}k_{m,1,r}^{2}k_{m,3,r}

+((((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + (v_{max,2} + v_{max,3})k_{m,1,r})x_{0}

−((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + k_{m,1,r}v_{max,2})k_{eq,1}k_{m,1,r})k_{m,1,f}^{2}

−(((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + 2k_{m,1,r}v_{max,2})k_{eq,1} − (v_{max,2} + v_{max,3})x_{0})k_{m,1,f}x_{0}k_{m,1,r}

−x_{0}^{2}k_{eq,1}k_{m,1,r}^{2}v_{max,2})v_{max,1}k_{m,1,r}k_{m,2,r}v_{max,3}k_{m,3,r}

+(2(k_{m,2,r} − k_{m,3,f})k_{eq,1}k_{m,2,f} − x_{0}k_{m,2,r})v_{max,1}^{3}x_{0}k_{eq,1}k_{m,1,r}^{3}k_{m,3,r}

−(2v_{max,1}^{2}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,3,f} − v_{max,1}k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} −v_{max,1}k_{m,1,f}x_{0}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}

+v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} + v_{max,1}x_{0}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f} +v_{max,1}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} − 2k_{m,1,f}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r} −2k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r})(v_{max,1}k_{eq,1}k_{m,1,r} + k_{m,1,f}v_{max,3})x_{3}k_{m,1,r}

−k_{m,1,f}^{3}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r} − 2k_{m,1,f}^{2}x_{0}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}−

k_{m,1,f}x_{0}^{2}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}v_{max,3}^{2}k_{m,3,r}

and

(((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3})k_{eq,1}k_{m,1,r}k_{m,2,r} − (k_{m,2,r} − k_{m,3,f})x_{0}k_{m,2,f}v_{max,3})k_{m,1,f}

+(v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3})x_{0}k_{eq,1}k_{m,1,r}k_{m,2,r})v_{max,1}^{2}x_{0}k_{m,1,r}^{2}k_{m,3,r}

+(v_{max,1}x_{0}k_{m,2,f} + k_{m,1,f}v_{max,2}k_{eq,2}k_{m,2,r} + x_{0}v_{max,2}k_{eq,2}k_{m,2,r})

(v_{max,1}x_{0}k_{m,3,f} + k_{m,1,f}v_{max,3}k_{eq,3}k_{m,3,r} + x_{0}v_{max,3}k_{eq,3}k_{m,3,r})(v_{max,1}k_{eq,1}k_{m,1,r} + k_{m,1,f}v_{max,3})x_{3}k_{m,1,r}^{2}

+(k_{m,1,f} + x_{0})(v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3})v_{max,1}k_{m,1,f}x_{0}k_{m,1,r}^{2}k_{m,2,r}v_{max,3}k_{m,3,r}

−(k_{m,2,r} − k_{m,3,f})v_{max,1}^{3}x_{0}^{2}k_{eq,1}k_{m,1,r}^{3}k_{m,2,f}k_{m,3,r}.

Once s_{1} is determined, s_{2} satisfies a linear equation in s_{1}, so we have s_{2} = e/f where e and f (in that order) are:

\begin{matrix}s_{1}^{2}k_{m,2,r}k_{m,3,r}( & v_{max,1}^{2}k_{eq,1}^{2}k_{m,1,r}^{2} + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,2} \\ + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,3} + k_{m,1,f}^{2}v_{max,2}v_{max,3}) \end{matrix}

+s_{1}( − x_{3}v_{max,1}^{2}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,3,f} + x_{3}v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}

+ x_{3}v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} − x_{3}k_{m,1,f}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r} − v_{max,1}^{2}x_{0}k_{eq,1}k_{m,1,r}^{2}k_{m,2,r}k_{m,3,r} + v_{max,1}^{2}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,2,r}k_{m,3,r} − v_{max,1}^{2}k_{eq,1}^{2}k_{m,1,r}^{2}k_{m,2,f}k_{m,3,f}k_{m,3,r} − v_{max,1}k_{m,1,f}x_{0}k_{m,1,r}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}k_{m,3,r} + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r} + v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r} + v_{max,1}x_{0}k_{eq,1}k_{m,1,r}^{2}v_{max,2}k_{m,2,r}k_{m,3,r} + k_{m,1,f}^{2}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r} + k_{m,1,f}x_{0}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r})

+k_{m,1,r}(x_{3}v_{max,1}^{2}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,3,f} − x_{3}v_{max,1}k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}

+ x_{3}v_{max,1}k_{m,1,f}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} + x_{3}v_{max,1}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}k_{eq,3}k_{m,3,r} − x_{3}k_{m,1,f}^{2}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r} − x_{3}k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r}v_{max,3}k_{eq,3}k_{m,3,r} − v_{max,1}^{2}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,2,r}k_{m,3,r} + v_{max,1}^{2}x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}k_{m,3,f}k_{m,3,r} − v_{max,1}k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r}k_{m,3,f}k_{m,3,r} − v_{max,1}k_{m,1,f}x_{0}k_{m,2,f}k_{m,2,r}v_{max,3}k_{m,3,r})

and

v_{max,1}k_{m,1,r}k_{m,3,r}(k_{m,1,f}x_{0}v_{max,2}k_{eq,2}k_{m,2,r} + k_{m,1,f}x_{0}k_{m,2,f}v_{max,3} + k_{m,1,f}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r}

+k_{m,1,f}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3} + x_{0}k_{eq,1}k_{m,1,r}v_{max,2}k_{eq,2}k_{m,2,r} + x_{0}k_{eq,1}k_{m,1,r}k_{m,2,f}v_{max,3}). The results with on factor this time are (for −e and f)

((((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + s_{1}v_{max,3})x_{0}

− ((v_{max,2}k_{eq,2}k_{m,3,f} + k_{m,2,f}v_{max,3}) + k_{m,1,r}v_{max,2})s_{1}k_{eq,1}

− (v_{max,2} + v_{max,3})s_{1}^{2}k_{eq,1})k_{m,1,f} − x_{0}s_{1}k_{eq,1}k_{m,1,r}v_{max,2})v_{max,1}k_{m,1,r}k_{m,2,r}k_{m,3,r} − (v_{max,1}x_{0}k_{m,1,r}k_{m,3,f} − v_{max,1}s_{1}k_{eq,1}k_{m,1,r}k_{m,3,f} + k_{m,1,f}s_{1}v_{max,3}k_{eq,3}k_{m,3,r}

+ k_{m,1,f}k_{m,1,r}v_{max,3}k_{eq,3}k_{m,3,r} + x_{0}k_{m,1,r}v_{max,3}k_{eq,3}k_{m,3,r})× (v_{max,1}k_{eq,1}k_{m,1,r}k_{m,2,f} − k_{m,1,f}v_{max,2}k_{eq,2}k_{m,2,r})x_{3} + (x_{0} − s_{1}k_{eq,1})(s_{1}k_{m,2,r} + k_{m,2,f}k_{m,2,r} − k_{m,2,f}k_{m,3,f})v_{max,1}^{2}k_{eq,1}k_{m,1,r}^{2}k_{m,3,r} − (s_{1} + k_{m,1,r})k_{m,1,f}^{2}s_{1}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r} − k_{m,1,f}x_{0}s_{1}k_{m,1,r}v_{max,2}k_{m,2,r}v_{max,3}k_{m,3,r}

and

((x_{0} + k_{eq,1}k_{m,1,r})k_{m,1,f} + x_{0}k_{eq,1}k_{m,1,r})(v_{max,2}k_{eq,2}k_{m,2,r} + k_{m,2,f}v_{max,3})v_{max,1}k_{m,1,r}k_{m,3,r}. The advantages gained by partial factoring are quite evident in this formulation.