In general, a metabolic pathway can be thought of as a network of biochemical reactions whose function is to take in substrates from the environment and transform them into something that is required by the organism. Thus the organism may take in fuel molecules which are used to produce ATP which is, in turn, used as the main energy 'currency' of the cell. Or alternatively, the organism may take in substrates which it uses as the building blocks for different subcellular components. Thus we can often think of a metabolic pathway as having a source substrate (such as the fuel and building block molecules) and an end product or sink (such as ATP or the subcellular components).

If the sources or sinks of a metabolic pathway do not change significantly in concentrations, or if their concentrations do not significantly effect the rates of the reaction in the metabolic pathway, then the pathway can often develop a steady state. This occurs when the concentrations of all the metabolites between the source(s) and the sink(s) do not change over time. Thus the idea of a steady state in a metabolic pathway is similar to the idea of the steady state in enzyme kinetics, a concept that was introduced in Chapter 2. Mathematically, the steady state of a metabolic pathway is described by setting the right-hand side of Eqn [4.3] to 0, i.e.,

The idea of a steady state is really a mathematical abstraction that never completely occurs in reality. One reason for this is that there is never a situation when the concentrations of the sources or sinks are completely constant or have absolutely no effect on the reaction rates. A second reason is that in reality, it would take an infinite amount of time to reach a true steady state. This is because as the system moves closer to a steady state, the rate at which the system approaches the steady state becomes slower. Notwithstanding these limitations, the idea of steady state is still a very useful one. For example, routine measurements of the glycolytic metabolites in red blood cells and other cell types show that as long as the cells are incubated under the same conditions, the assayed concentrations of the metabolites remain remarkably constant over time.

To determine the steady state of a metabolic system, it is necessary to solve the set of nonlinear algebraic equations that are defined by Eqn [4.9]. This process can be tedious to program de novo, so we have provided the following function in the MetabolicControlAnalysis package, which carries out the operation.

SteadyState[S, N, v, p] UsestheMathematica function

Solve to determine the solution to Eqn [4.9], Note that likeNDSolveMatrix, the inclusion of the parameter table p is optional

Q: Consider the linear three-reaction sequence described in the worked example in Section 4.3, but now assume that the concentrations of S1 and S4 are fixed at 1 mM by external processes. What are the steady-state concentrations of S2 and S3 ?

A: The first step in answering this question is to define an equation list of the reaction sequence that is given in Section 4.3.

eqns = { {si [t] ^ s2 [t]} , {s2[t] ^ ss[t]}, {s3[t] ^ s4[t]}};

Having done this, we are now in a position to define the appropriate matrices and vectors of the system. Note that when doing this we define S1 and S2 to be external parameters. So the relevant substrate list is

S := SMatrix[eqns, {s1[t] , s4[t]}] ; S // MatrixForm

and the relevant stoichiometry matrix and reaction list are

N = NMatrix[eqns, {sift] , s4 [t]}] ; N // MatrixForm

Km |
l,1,S1 |
(1+.miirsV) |

Vma x , |
2 , f s 2 [ t ] | |

Km |
i,2,S2 |
( 1+s ) |

Vma x , |
3 , f S 3 [ t ] |

So if we specify the parameters, we can then use SteadyState to determine the steady state.

P = {si [t] , s4 [t] , Vmax,1,f , Vmax,2,f , Vmax,3,f , Km,1,s1 , Km,2,s2 , Kn,3,s3 }

pv = {1, 1, 1, 1, 1, 1, 1, 1}; SteadyState[S, N, V, Transpose[{p, pv}]]

Thus the function SteadyState returns a replacement rule which can be used in the usual manner to obtain the concentration of each metabolite.

Alternatively, NDSolveMatrix can be used to follow the time course of [S2] and [S3 ] from an initial concentration of 0 mM to their respective steady-state values. The graphical output shows that the steady state is attained after ~30 h.

{s2 [0] == 0, s3 [0] == 0} , {t, 0, 30} , Transpose [{p, pv}]] ;

AxesLabel -> {"Time (h)", "Concentration (mM)"}];

Concentration (mM)

Concentration (mM)

By leaving out the ymax and Km from the parameter table it is possible to obtain expressions for the steady-state concentrations of S2 and S3 written in terms of ymax and Kn. This enables visual inspection of the expressions which may help obtain an idea of which parameters most influence the steady-state concentrations. This analysis can be a prelude to the quantitative MCA that is discussed in Chapter 5.

We can implement this analysis by not including the ymax and Km parameters in the parameter table that was defined at the beginning of this example. Thus, p2 = {si[t] , s4[t]} ; pv2 = {i, 1};

Vmax,1,f - Vmax,2,f - Km,1,s1 Vmax,2,f _Km,3 , s 3 Vmax , 1 , f_ = =

By omitting p altogether, the steady state can be expressed in terms of ^max, Km, S1, and S4.

SteadyState[S, N, v]

Km,1,si Vmax,2,f + Vmax,1,f Si[t] - Vmax,2,f Si [ t ] '

Km,1,si Vmax,3,f + Vmax,1,f S1 [ t ] - Vmax,3,f S1 [ t ] '

The SteadyState function will often be unable to locate a solution for metabolic models that are more complicated than the one described in the above question/answer. In large metabolic models there may be no analytical solutions or the solution may be so complicated that the algorithm simply cannot determine it. In the latter cases, the function NSteadyState can be used to find an approximate numerical solution. This algorithm is based on Mathematical FindRoot function.

NSteadyStatefS, N, v, p, init] Usesthe Mathematica function FindRoot to determine an approximate numerical solution to Eqn [4.9] where init contains initial estimates of the steady state concentations in the form of areplacement rule. Note that the inclusion of the parameter table p is optional.

Solving metabolic steady states numerically.

Q: Calculate the steady-state concentrations of S2 and S3 for the metabolic scheme described in the previous question, using NSteadyState.

A: To use NSteadyState we must first set up a replacement rule containing initial estimates of the steady-state concentrations of S2 and S3. We can use the simulation values at 20 h that were obtained in the previous question/answer to do this.

rrule = Table[Spj -> ssvaluesp] , {i, Length[S]}]

{s2[t] ^ 0.999664, s3[t] ^ 0.997363} Note that in the above input we have relied on the Part or [[.. .]] function.

With the initial estimate, we can refine the answer by using the function, NSteadyState.

In conclusion, we see that the numerical estimate is the same as that returned in the previous example.

listpij returns the ith element of a list

Selecting elements of lists.

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