Generating rate equations for complex enzyme systems by a computer-assisted systematic method
© Qi et al; licensee BioMed Central Ltd. 2009
Received: 26 January 2009
Accepted: 4 August 2009
Published: 4 August 2009
While the theory of enzyme kinetics is fundamental to analyzing and simulating biochemical systems, the derivation of rate equations for complex mechanisms for enzyme-catalyzed reactions is cumbersome and error prone. Therefore, a number of algorithms and related computer programs have been developed to assist in such derivations. Yet although a number of algorithms, programs, and software packages are reported in the literature, one or more significant limitation is associated with each of these tools. Furthermore, none is freely available for download and use by the community.
We have implemented an algorithm based on the schematic method of King and Altman (KA) that employs the topological theory of linear graphs for systematic generation of valid reaction patterns in a GUI-based stand-alone computer program called KAPattern. The underlying algorithm allows for the assumption steady-state, rapid equilibrium-binding, and/or irreversibility for individual steps in catalytic mechanisms. The program can automatically generate MathML and MATLAB output files that users can easily incorporate into simulation programs.
A computer program, called KAPattern, for generating rate equations for complex enzyme system is a freely available and can be accessed at http://www.biocoda.org.
Since Haldane's analysis of a simple enzyme mechanism , kinetic analysis has been central to our quantitative understanding of enzyme mechanisms [2, 3]. In conventional applications, kinetic data from initial-rate experiments are used to evaluate enzyme mechanisms based upon derived mechanistic rate expressions. Such rate expressions are important in building integrated models of metabolic systems which involve a number of enzymatic reactions [4, 5]. In principle, the rate equations for a given discrete-state reaction mechanism can be derived by solving a system of simultaneous nonlinear algebraic equations that result from the steady-state expressions for the concentrations of all of the enzyme intermediates. This approach was first applied successfully by Botts and Morales  to some enzymatic systems. However, when the system involves multiple substrates, enzyme complexes, and products , deriving rate equations based on steady-state equations may be too complex to be of practical interest and also can be liable to human errors. Therefore, systematic approaches, as reviewed by Huang , are desirable.
King and Altman  introduced a graphical/schematic method for facilitating derivation of steady-state rate equations in enzymatic systems. Modifications introduced by Volkenstein and Goldsein  and Cha  added substantial power to the King-Altman method by applying graph theory and allowing for the assumption that one or more of the reversible steps in the enzyme mechanism is maintained in rapid equilibrium . Other alternative methods include those described by Fromm , Orsi , Ainsworth [13, 14], Indge and Childs , and Chou and Forsen .
Even when using graphical methods, manually deriving the steady-state rate equations for non-trivial enzyme mechanisms can be cumbersome and error-prone. Therefore, computer-assisted methods are useful. Applying the method of King-Altman, Pring  and Rhoads  developed two programmes, K and D, which perform logical operations essential for generating rate equations based on the strictly steady-state assumption with respect to a certain class of species present. Lam and Priest  introduced an algorithm based on graph theory that is computer programmable. Cornish-Bowden  presented a computer implementation of Cha's method using an exhaustive search. A computer program developed by Kinderlerer and Ainsworth  is restricted to enzyme mechanisms involving up to 10 enzyme intermediates. Straathof and Heijnen  and Fromm and Fromm  introduced methods to derive rate equations for enzyme systems using the symbolic algebra packages Maple and Mathematica. However, these programs derive only strictly steady-state rate equations and cannot obtain rate equations involving irreversible steps. Varon et al.  developed a program called Albass that overcame many of the limitations of earlier programs. Several years later, the Varon group developed two new programs written in C++, called Referass  and WinStes , which can derive rate equations for mechanisms with up to 255 intermediate states with up to 255 reactions. The algorithms and software developed by Varon and colleagues represent the most powerful and flexible previously developed tools for deriving enzyme rate expressions. Yet, like other previous packages, it does not appear to be currently available.
There is no limitation on the size of the system other than that imposed by the available memory and CPU resources.
The program can output the results (the generated rate equations) as a MathML file or a MATLAB .m file which may be integrated into simulation program. (For instance, it can be used in conjunction with a simulation package such as BISEN .)
The program provides visualization of all the valid KA patterns.
Functions available in KAPattern may help the end-users to obtain insights on catalytic mechanism (e.g., structural properties, topological features, stoichiometric matrix etc.) that may be useful for other applications.
Foremost, the package is freely available for download and use by the community.
Results and Discussion
The King-Altman method
Here Σ i represents the sum of the 12 terms associated with the state i and Σ is the sum over all 5 sets of 12 terms for all states, and E o is the total enzyme concentration.
As described above, the graphical method of King and Altman is based on determining a set of KA patterns that are subsets of the graph of the enzyme mechanism. Each KA pattern contains the maximal number of edges possible while not containing any closed loops. Each enzyme state (each vertex in the graph) has associated with it a directional diagram for each KA pattern. Enumerating all directional diagrams becomes more difficult as the enzyme mechanism becomes more complex.
Previous applications of the theory of graphs to the solution of enzyme kinetic problems have been aimed at developing algorithms that are easy to program and allow users to rapidly calculate the steady-state concentrations of enzyme states, and thereby obtain expressions for the rate of product accumulation [9, 11]. Unlike using symbolic algebra packages to solve a set of nonlinear algebraic equations based on steady-state and mass conservation, these approaches take advantage of the similarity between complex enzyme mechanisms and electrical networks. Specifically, it has been proved that the method used to generate trees from linear graphs can be applied to complex enzymatic reaction mechanisms .
where c is a element which has been operated. We will see below how this property can be applied as an algebraic representation of a requirement for valid KA patterns.
Each element ℒ ij is the index of the link between node (enzyme state) i and j.
Application of the Lamb-Priest algorithm starts with randomly selecting n - 1 nodes from the linear graph, and determining the links connected to the n -1 nodes. (It makes no difference which node is excluded; the same final results are obtained for any arbitrary choices.) It is easy to carry out this operation on ℒ by deleting a row (column) and then listing separately all the nonzero entries from the remaining n - 1 rows (columns). In the next step, using the Wang Algebra described above, the links listing obtained in previous step are alphanumerically multiplied. Here the alphanumeric multiplication of elements (integers or other symbols) is defined as a list rather than numerical values. For example, multiplying alphanumerically 1 and 2 is equal 12 rather than 2. The result of this operation is the set of all valid KA patterns, expressed by a set of link index array. The Wang Algebra principle guarantees that no invalid or redundant patterns are generated through these steps.
Thus, an alphanumerical multiplication of two lists, such as (1 2 3) and (2 5), yields a list of all of the entry-by-entry products, (1–2 1–5 2–5 2–3 3–5) (Recall that the product of identical elements, such as 2-2 is discarded. The product is discarded if it shows up more than once.). The result lists all the valid KA patterns represented by 12 lists of link indexes. The graph representation of all the valid KA patterns is shown in Figure 2A. The algorithm ensures that only one link shows up in one single pattern only once (Equation (2) and Equation (3)), and that there are no redundant patterns.
We use the enzyme-catalyzed reaction mechanism of fumarase to illustrate the usage of the KAPattern program. Given a simple input file for this complex enzymatic reaction, this program produces the link matrix ℒ and kinetic matrix as well as generates all the valid KA patterns, outputs each pathway corresponding to each enzyme form based on the generated valid patterns, and outputs the results (i.e., the generated rate equations) as a MathML file or a MATLAB .m file, which can be used in a simulation program of an integrated metabolic system model.
In this program, the input file for an enzyme-catalyzed reaction mechanism is a simple .txt file that lists every pseudo-first-order rate constants in the enzyme catalytic system. Below is the input file for the fumarase reaction mechanism; the first column and the second column are the indexes of the enzyme forms, the third column is the pseudo-first-order rate constants connecting the corresponding two enzyme forms (transferring from the first to the second):
1 3 k1*A
3 1 k_1
1 2 k_6*P
2 1 k6
1 5 k3*B
5 1 k_3
2 4 k_5
4 2 k5*C
3 4 k2*B
4 3 k_2
4 5 k_4
5 4 k4*A
To clarify, the first line of the input.txt file is 1 3 k1*A, which means the rate constant for the enzyme conversion from form 1 (E1) to 3 (E3) is k1*A. The functions ReadInput and GetLink in our KAPattern program, read the input file and generate the matrices ℒ (Equation (4)) and (Equation (5)). The function Wang is used to generate the valid KA patterns. (For detailed description of functions and the full example, see the additional file 1: Appendix.)
With all the valid KA patterns generated, it is straightforward to enumerate all of the directional diagrams using the information from the and ℒ matrices. For the enzyme form E i , the program checks each non-zero entry of the i th column in the ℒ matrix against all links in the link list of one pattern and finds every link that points to the enzyme form E i . Based on the next end point, the process is repeated until no links is left out in the list. Finally, multiplying all the pseudo-first-order rate constants, we can get the expression corresponding to one pattern. Repeating this procedure for each KA pattern and for each enzyme form, we obtain the concentration of each enzyme form E i relative to the total concentration of enzyme E o .
The cost of the rate equation generation depends not only on the size of the problem, but also on the complexity of the problem. For most small-sized enzyme systems we tested, the program gives results less than 1 second. For the moderate-size problem example we present in the additional file (See the additional file 1: Appendix.), the program generates 288 valid KA patterns in about 4.5 seconds (on Intel Pentium IV, 2.0 GHz, 2 GB RAM).
Below is the MATLAB output file generated from the function BuildFile of the KAPattern program for the fumarase reaction mechanism. The output has one or two input parameters, depending on whether there are pseudo-first-order rate constants, and three output variables. If there are pseudo-first-order rate constants, which include substrates or products concentrations, the two parameters will be two arrays K and Con, corresponding to reaction rate constants and substrate and product concentrations, respectively. The three output variables are N, F, and D, where N and F are vectors listing the numerators Σ i and the fractions Σ i /Σ for Equation (1). The output variable D is the denominator D.
function [E, F, D] = Expression(K, Con)
%% concentration values
A = Con(1);
B = Con(2);
C = Con(3);
P = Con(4);
%% rate constant k values
k1 = K(1);
k2 = K(2);
k3 = K(3);
k4 = K(4);
k5 = K(5);
k6 = K(6);
k_1 = K(7);
k_2 = K(8);
k_3 = K(9);
k_4 = K(10);
k_5 = K(11);
k_6 = K(12);
E(1) = k6*k_1* k_3*k5*C + k6*k_1*k_3*k_2 + k6*k_1*k_3*k_4 + k6*k_1*k5*C*k4*A + k6*k_1*k_2*k4*A + k6*k_3*k5*C*k2*B + k6*k_3*k_4*k2*B + k6*k5*C*k2*B*k4*A + k_1*k_3*k_2*k_5 + k_1*k_3*k_4*k_5 + k_1*k_2*k_5*k4*A + k_3*k_4*k_5*k2*B;
E(2) = k_6*P*k5*C*k_1*k_3 + k_6*P*k_1*k_3*k_2 + k_6*P*k_1*k_3*k_4 + k_6*P*k5*C*k_1*k4*A + k_6*P*k_1*k_2*k4*A + k_6*P*k5*C*k_3*k2*B + k_6*P*k_3*k_4*k2*B + k_6*P*k5*C*k2*B*k4*A + k5*C*k2*B*k1*A*k_3 + k5*C*k4*A*k3*B*k_1 + k5*C*k2*B*k4*A*k1*A + k5*C*k2*B*k4*A*k3*B;
E(3) = k1*A*k6*k_3*k5*C + k1*A*k_2*k6*k_3 + k1*A*k6*k_3*k_4 + k1*A*k6*k5*C*k4*A + k1*A*k_2*k6*k4*A + k_2*k_5*k_6*P*k_3 + k_2*k4*A*k3*B*k6 + k_2*k_5*k4*A*k_6*P + k1*A*k_2*k_3*k_5 + k1*A*k_3*k_4*k_5 + k1*A*k_2*k_5*k4*A + k_2*k_5*k4*A*k3*B;
E(4) = k_5*k_6*P*k_1*k_3 + k2*B*k1*A*k6*k_3 + k4*A*k3*B*k6*k_1 + k_5*k4*A*k_6*P*k_1 + k2*B*k4*A*k1*A*k6 + k_5*k2*B*k_6*P*k_3 + k2*B*k4*A*k3*B*k6 + k_5*k2*B*k4*A*k_6*P + k_5*k2*B*k1*A*k_3 + k_5*k4*A*k3*B*k_1 + k_5*k2*B*k4*A*k1*A + k_5*k2*B*k4*A*k3*B;
E(5) = k3*B*k6*k_1*k5*C + k3*B*k6*k_1*k_2 + k3*B*k_4*k6*k_1 + k_4*k_5*k_6*P*k_1 + k_4*k2*B*k1*A*k6 + k3*B*k6*k5*C*k2*B + k3*B*k_4*k6*k2*B + k_4*k_5*k2*B*k_6*P + k3*B*k_1*k_2*k_5 + k3*B*k_4*k_1*k_5 + k_4*k_5*k2*B*k1*A + k3*B*k_4*k_5*k2*B;
D = E(1) + E(2) + E(3) + E(4) + E(5);
F(1) = E(1)/D;
F(2) = E(2)/D;
F(3) = E(3)/D;
F(4) = E(4)/D;
F(5) = E(5)/D;
Correspondingly, the numerator part in the output file will be changed to
E(1) = k6*k_1*k_3*k5*C + k6*k_1*k_3*k_2 + k6*k_1*k_3*k_4 + k6*k_1*k5*C*k4*A + k6*k_1*k_2*k4*A + k6*k_3*k5*C*k2*B + k6*k_3*k_4*k2*B + k6*k5*C*k2*B*k4*A;
E(2) = k5*C*k2*B*k1*A*k_3 + k5*C*k4*A*k3*B*k_1 + k5*C*k2*B*k4*A*k1*A + k5*C*k2*B*k4*A*k3*B;
E(3) = k1*A*k6*k_3*k5*C + k1*A*k_2*k6*k_3 + k1*A*k6*k_3*k_4 + k1*A*k6*k5*C*k4*A + k1*A*k_2*k6*k4*A + k_2*k4*A*k3*B*k6;
E(4) = k2*B*k1*A*k6*k_3 + k4*A*k3*B*k6*k_1 + k2*B*k4*A*k1*A*k6 + k2*B*k4*A*k3*B*k6;
E(5) = k3*B*k6*k_1*k5*C + k3*B*k6*k_1*k_2 + k3*B*k_4*k6*k_1 + k_4*k2*B*k1*A*k6 + k3*B*k6*k5*C*k2*B + k3*B*k_4*k6*k2*B;
Graphical User Interface (GUI)
We have described a systematic method and the corresponding computer program, called KAPattern, for generating rate equations for any complex enzyme systems. This program generates complete set of valid King-Altman patterns for complex enzyme-catalyzed reaction mechanisms. Unlike other computer-assisted methods that use symbolic algebra packages to solve the system of nonlinear algebraic equations arising from steady-state mass conservation, our program is developed from the original schematic method of King-Altman  and employs the topological theory of linear graphs . Our program can derive rate equations for both strictly steady-state conditions and those with rapid equilibrium steps. The enzyme mechanism can be either branched or unbranched enzyme mechanisms containing both reversible and irreversible reactions steps. Using a simple, easy-to-understand input file, our program can produce a MATLAB .m file or MathML file that can be integrated into other biochemical system model programs. It can illustrate the visualization of all the valid KA patterns as well. In addition, the generated link matrix ℒ and kinetic matrix , which characterize the enzyme mechanisms here, may be useful for other applications (e.g. to characterize the topological properties and stoichiometric matrix of large-scale networks).
It should be emphasized that in the current version, our program is restricted to systems whose element reactions are association or dissociation of substrates or first-order inter-conversion of enzyme species.
Systems involving allosteric activation and inhibition or other protein-protein interactions should be handled carefully, because our approach still lacks direct connections between the rate constants and the kinetic constants, such as Michaelis-Menten constants. Those connections are important for analyzing enzyme kinetic experimental data.
Availability and requirements
The KAPattern is written in MATLAB and distributed as a standalone GUI-based application for Windows, Mac or Linux/Unix. The MATLAB source codes, and the KAPattern stand-alone program are freely available and can be accessed at http://www.biocoda.org.
We thank Fan Wu and Kalyan Vinnakota for helpful discussions. Special thanks to Brain Carlson for helping us compile the Mac version package. This work was supported by National Institute of Health Grant R01-HL072011.
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