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Editorial Reviews. From the Back Cover. Linear and integer programming are fundamental toolkits for data and information science and technology, particularly .
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- Linear and Integer Programming Made Easy : T. C. Hu :
- Linear and Integer Programming Made Easy
Then the problem becomes to identify the optimal enzyme set whose inhibition eliminates the target compounds and at the same time, incurs minimum damage based on the given metabolic network and a set of target compounds. We denote such problem as enzyme combination identification ECI. Sridhar et al. Furthermore, two filtering strategies are proposed to prune the search space to guarantee an optimal solution.
However, their algorithm is complicated and impractical for us to use it. In this paper, we develop an efficient method based on Integer Linear Programming [ 20 — 22 ] which has a wide application in solving NP-hard problems. Furthermore, all instances of the original problem are needed to convert into integer programming formalization so as to apply the existing free solver called CPLEX [ 23 ].
This paper has two main contributions. First, we formulate a new biological problem based on Boolean metabolic network to identify the drug target with minimum damage. By integrating the human metabolic networks, we have shown that the proposed approach can accurately identify the target enzyme set for known drugs in the literature.
Furthermore, experiments have shown that our proposed method is extremely efficient which can effectively solve the problem in seconds. The remainder of the paper is organized as follows. Finally, conclusions are given in the last section. Let C , R , and E denote the set of compounds, reactions, and enzymes, respectively. The subscript m,n,l denote the number of compound nodes , reaction nodes and enzyme nodes , respectively.
Source nodes are those nodes having no incoming edges and they are the seed compounds. A metabolic network can be defined as follows:. The main problem Enzyme Combination Identification ECI in a Boolean network is first described with a simple example and then followed by its mathematical formalization.
A small hypothetical metabolic network is shown in Fig. For instance, R 1 is a reaction, its substrates are C 1 and E 2 and its product is C 2. Here C 5 is the target compound in this figure which shall be stopped. To stop the production of C 5 , reaction R 2 must be prevented from taking place. One of the possible solution is to disrupt one of its catalyzing enzymes E 1 for instance. Another is to stop the production of its reactant compounds i. If C 2 is stopped, then two possible ways can achieve this effect i.
It is noted that C 2 , R 2 , C 7 and R 1 forms a cycle, the inhibition of E 1 will result in inactivating reaction R 2 which will further makes C 2 inproducible. The other way is inhibiting enzyme E 2 which makes R 1 inactive and further stop the production of compound C 2. Therefore, the inhibition of enzyme E 1 or E 2 can result in stopping the production of the target compound.
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For simplicity, we assume there are no external inputs to all reactions and all the input compounds related to reactions are shown in the network. We note that different compounds and enzymes may have different levels of importance in the metabolic networks. Here we assume that all the compounds and enzymes are of equal importance. We assign binary value i. Then G can be regarded as a valid assignment if the following conditions are satisfied:.
From Fig. Then we denote the number of non-target compounds knocked out as the damage , which is caused by the manipulating the enzyme set in the metabolic network. It can be seen that the damage of inhibiting E 2 is 2 i. Compound C 7 is still producible because it can be produced by R 3 even after the inhibition of E 2. The damage effect of inhibition of E 1 is 4 i. Both E 1 and E 2 are potential drug targets since they can achieve the effect of disrupting the target compound C 5. However, E 2 is a better drug target than E 1 owing to the fact that it causes less damage.
In this section, we introduce integer programming-based methods for ECI. Integer programming, in particular, Integer Linear Programming ILP is set to minimize or maximize a linear objective function under linear constraints with all the variables taking integer values.
In the following, each variable takes including the binary value i. The ILP formulation for the network in Fig. Here all variables including the value of reaction compound and enzyme nodes take either 1 or 0. Similarly, TCi and FCi are used to represent the values of compound nodes. The objective function 1 means that the damage should be minimized. Equation 2 means that the target compound v c 5 should be 0 after the assignment converges.
Thus Eq. Similarly, Eqs.
For a compound node with indegree is 1 which indicates the node has only one incoming edge, the value of the predecessor is just copied. For instance, since v c 2 has only one predecessor v r 1 , v c 2 is just copied from v r 1 as shown in Eq. Similarly, v c 4 is just copied from v r 2 which is shown in Eq. Thus, Eq. Equation 6 - Eq. X in Eq. ECI is NP-complete problem with the maximum indegree and outdegree being bounded by 2. Obviously, the problem is in NP, it suffices to show that it is NP-hardness. The proof is by a polynomial time reduction from minimum edge cover MEC , which is a problem for a given graph to find the minimum number of edges so that each node is incident to at least one of the selected edges.
We then construct the corresponding ECI as below. It is noted that the minimum damage is determined uniquely by the inhibition of enzyme set. Then V e can be regarded as virtual nodes and denoted as an empty set in this case. The ECI problem can be converted into the problem of identifying the minimum set of non-target compounds.
Thus the graph for MEC shown in Fig. It is clear that this conversion can be done in polynomial time. To guarantee that the target compound c t is stopped i. If G has an edge cover of size z , then it follows that the minimum number of c i taking 0 should be z. Since there is an edge between c i and r j if and only if v j is incident to e i. The polynomial time reduction from minimum edge cover MEC problem to minimum damage identification problem.
In this section, we verify the biological validity of our proposed method by using known drugs. Besides, the performance of the ILP-ECI algorithm is evaluated by using the execution time which indicates the total time taken by the method. KEGG is database which provides known drug molecules along the the enzymes they inhibit and their therapeutic category.
Then we use drugs at this database as our benchmarks and we report two of them due to the space limitation. The drug we used in this paper is Benoxaprofen D which inhibits arachidonate 5-lipoxygenase E1. This enzyme appears in several networks including arachidonic acid metabolism network hsa According to our graph model, the removal of E1. Furthermore, these four compounds play an important role in the mechanisms of toxic brain damage in acute methanol poisoning in humans [ 27 ]. Thus, we chose them as the target compounds. Apart from that, inhibiting this enzyme also eliminates four more compounds i.
Thus, it is shown that ILP-ECI potentially finds a better solution in this experiment than the existing drugs since the same target compounds are eliminated by the existing drug in addition to other four compounds. Indeed, recent research validated our model since the anti-inflammatory effect of the levels of LTA4H [ 29 ] and LTC4 [ 30 ] has been observed. The computational time in this experiment takes only 5. Another experiment we conducted is the histidine metabolism network hsa The enzyme amine oxidase E.
According to our graph model, the removal of enzyme E.
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It should be noted that the level of pros-methylimidazoleacetic acid is closely related to severity of Parkinson disease in patients [ 31 , 32 ]. It takes only 3. This experiment verifies that Rasgiline targets the optimal enzyme. An important advantage of our Boolean model is its capability of detecting the lack of substrates where the connectivity-based methods fail to handle this. Another advantage of this model is its capability of handling branches and cycles in a pathway from the source compound to the target compound.
However, there are still have some limitations in this method. One of the major drawbacks is the assumption that all the compounds and enzymes are of equal importance. However, different compounds and enzymes may have different levels of importance in the metabolic networks. Our future work will focus on developing other models which include the weight of different nodes.
In this paper, we formulate the optimal enzyme combination identification ECI problem as an optimization problem in Boolean metabolic networks. We have proven that ECI in the Boolean model is NP-complete and the target enzyme set is uniquely determined when the target compounds are given. Considering that the computational time of IP-based method is exponential to the number of variables, to improve the scalability of the developed method, it is vital to reduce the number of variables appearing in IP formalization.
The efficiency and effectiveness are validated by the computational experiments in which datasets were downloaded from the KEGG database. The results demonstrate that the proposed model can accurately identify the target enzymes for known successful drugs in the literature.
Linear and Integer Programming Made Easy : T. C. Hu :
Specifically, ILP-ECI has found a different enzyme set for the target compounds of Benoxaprofen which indicates that our method has a great potential to be better than Benoxaprofen. The experiments also show that ILP-ECI can solve the problem in a short time which confirms the efficiency of our algorithm.
The authors would like to thank anonymous reviewers for your helpful and constructive comments. YQ designed the research. YQ and WKC proposed the methods and did theoretical analysis. YQ and HJ collected the data. All authors read and approved the final manuscript.
Linear and Integer Programming Made Easy
Yushan Qiu, Email: nc. Hao Jiang, Email: nc. Wai-Ki Ching, Email: kh. Xiaoqing Cheng, Email: nc. National Center for Biotechnology Information , U. BMC Syst Biol. Published online Apr Author information Article notes Copyright and License information Disclaimer. Corresponding author. RSS Feed. Sarah Smith. In addition to explaining the periodic structure of optimum solutions, it features an entire chapter on the asymptotic algorithm for integer programming. Suitable for undergraduates and graduates, it can also be used for self-study.
Loaded with examples and figures, it illustrates each concept with numerical examples. An excellent primer for undergraduates and graduates, this volume can also be used for self-study. An introductory chapter is followed by explorations of the dimension of solution space, equivalent formulations, linear programs in matrix notations, complementary slackness, and revised simplex method.