Control and inhibition analysis of complex formation processes
© Saitou et al.; licensee BioMed Central Ltd. 2012
Received: 26 April 2012
Accepted: 18 June 2012
Published: 3 August 2012
Proteolytic degradation of the extracellular matrix (ECM) is a key event in tumour metastasis and invasion. Matrix metalloproteinases (MMPs) are a family of endopeptidases that degrade most of the components of the ECM. Several broad-spectrum MMP inhibitors (MMPIs) have been developed, but have had little success due to side effects. Thus, it is important to develop mathematical methods to provide new drug treatment strategies. Matrix metalloproteinase 2 (MMP2) activation occurs via a mechanism involving complex formation that consists of membrane type 1 MMP (MT1-MMP), tissue inhibitor of matrix metalloproteinase 2 (TIMP2) and MMP2. Here, we focus on developing a method for analysing the complex formation process.
We used control analysis to investigate inhibitor responses in complex formation processes. The essence of the analysis is to define the response coefficient which measures the inhibitory efficiency, a small fractional change of concentration of a targeting molecule in response to a small fractional change of concentration of an inhibitor. First, by using the response coefficient, we investigated models for general classes of complex formation processes: chain reaction systems composed of ordered steps, and chain reaction systems and site-binding reaction systems composed of unordered multi-branched steps. By analysing the ordered step models, we showed that parameter-independent inequalities between the response coefficients held. For the unordered multi-branched step models, we showed that independence of the response coefficients with respect to equilibrium constants held. As an application of our analysis, we discuss a mathematical model for the MMP2 activation process. By putting the experimentally derived parameter values into the model, we were able to conclude that the TIMP2 and MMP2 interaction is the most efficient interaction to consider in selecting inhibitors.
Our result identifies a new drug target in the process of the MMP2 activation. Thus, our analysis will provide new insight into the design of more efficient drug strategies for cancer treatment.
Metastasis and invasion are a major cause of death in cancer patients, and thus preventing this secondary spread of the tumour is an important aspect of cancer therapy. A prerequisite for the migration of endothelial cells through the extracellular matrix (ECM) is the initiation of a biochemical pathway responsible for the proteolytic degradation of these structural barriers. Matrix metalloproteinases (MMPs) are a family of endopeptidases that degrade most of the components of the ECM [1, 2]. Several broad-spectrum MMP inhibitors (MMPIs) have been developed, some of which have been used in clinical trials for cancer treatment . However, these MMP inhibitors have had little success due to side effects, implying that these clearly lacked selectivity in their action. Most MMPs are inhibited by MMPIs that bind to the active sites of MMPs. Similarities in active sites of MMPs pose obstacles to the design of specific inhibitors. It is thus important to find alternatives to these approaches to increase specificity. Therefore, it is worth developing mathematical methods in analysing biochemical reaction pathways to provide new drug treatment strategies.
Matrix metalloproteinase 2 (MMP2) was proposed as a potential therapeutic target, based on its high-level expression in many human tumours and its ability to degrade type IV collagen . The activation of MMP2 proenzyme is processed by membrane type 1 matrix metalloproteinase (MT1-MMP) [5–7]. Under physiologic conditions, MMP2 is secreted as a latent form, pro-MMP2, and it has been established that its activation occurs via a mechanism involving a complex formation that consists of MT1-MMP, tissue inhibitor of matrix metalloproteinase 2 (TIMP2) and pro-MMP2. Here, focusing on the MMP2 activation process, we develop a mathematical method which quantifies a response of systems to an inhibitor and classifies interactions in order of the inhibitory efficiency. The goal of this study is to identify the most efficient interaction to consider in selecting inhibitors.
In order to achieve this goal, we use control analysis, a method of sensitivity analysis which is widely used in many fields (see for example ). Methods of obtaining quantitative measures of control in metabolic pathways were developed by Kacser and Burns  and Heinrich and Rapoport [10, 11] (for a review, see ). The essence of the analysis is to define the response coefficient which measures the inhibitory efficiency, a small fractional change of concentration of a targeting molecule in response to a small fractional change of concentration of an inhibitor. First, using the response coefficient, we investigate models for general classes of complex formation processes: chain reaction systems composed of ordered steps, and chain reaction systems and site-binding reaction systems composed of unordered multi-branched steps. By analysing the ordered step models, we show that parameter-independent inequalities between response coefficients hold. In the unordered multi-branched step models, we assume there are no cooperative reactions, i.e. the equilibrium constant for binding a site of molecule B to a site of another molecule A is independent of whether any of the other sites of molecule A are occupied. This assumption satisfies the detailed balance condition, but it puts a stronger constraint on the system (the importance of the detailed balance condition in kinetic modelling was discussed in [13–15]). Under this assumption, we show that independence of the response coefficients with respect to the equilibrium constants holds. These results indicate that the inhibitory efficiency depends on the topology of pathway networks. As an application of our analysis, we investigate a mathematical model for the MMP2 activation process . In the model, MT1-MMP is dimerized, bound to TIMP2 and forms a quadruple complex by binding proMMP2, MT1-MT1T2M2, which contributes to activation of proMMP2. We try to identify the most efficient interaction to consider in selecting inhibitors by putting the experimentally derived parameter values into the model.
Therefore, our method can serve as a tool for quantifying the inhibitory efficiency and allows us to determine the most efficient method of selecting the inhibitor in complex formation processes. Application of the method to the MMP model may provide a new tool for designing more efficient drug strategies for cancer treatment.
The response coefficient as a measure of the inhibitory efficiency
where [E] T is the total concentration of the enzyme. The concentration [ES] is the enzyme concentration which contributes to P production, while the concentration [E] + [EI] is the enzyme concentration which does not contribute to P production. The parameter K, defined by K = kd/ka, is the equilibrium constant of the enzyme and substrate binding reaction, where kd is the dissociation rate constant and ka is the association rate constant. The coefficient α is written as , where K I is the equilibrium constant of the enzyme and inhibitor binding reaction. Because α is always greater than 1, an addition of the inhibitor is equivalent to an increase of the equilibrium constant K. Therefore, in this paper, instead of adding inhibitors, we perform an inhibition analysis by changing equilibrium constants.
So far, we have discussed behaviours of the response coefficient in a simple model as an example. We would like to generalize this analysis to larger systems which have multiple interactions. From here, we consider models for general classes of complex formation processes: (1) chain reaction systems composed of ordered steps, (2) chain reaction systems composed of unordered multi-branched steps and (3) site-binding reaction systems composed of unordered multi-branched steps.
Analysis of the ordered step models
where R i 123, R i 12 and R i 1(i = 1, 2) denote the response coefficients for [A1A2A3], [A1A2] and [A1] with respect to K i , respectively. As can be seen from (3), the following inequalities between the response coefficients hold: R1123 > R2123, R212 > R112 and R11 > R21. The inequality R1123 > R2123 shows that inhibitory efficiency of the A3 binding reaction is larger than that of the A2binding reaction. The sign of R112 is negative, while that of R212 is positive. This means that the complex concentration [A1A2] increases by the inhibition of the A2 binding reaction.
Thus, we have shown that the parameter-independent inequalities hold in Model Sn. The inequalities indicate that the reactions can be sorted in order of the inhibitory efficiency, which is independent of the values of the equilibrium constants.
Analysis of the unordered multi-branched step models – chain reaction systems
Analysis of the unordered multi-branched step models – site-binding systems
An application - MT1-MMP/TIMP2/MMP2 complex formation model
Experimentally derived parameter values for the MMP model
[MT 1] T
[T 2] T
[M 2] T
k a(MT 1−MT 1)
From Hoshino 2012 
k d(MT 1−MT 1)
From Hoshino 2012 
k a(MT 1−T 2)
From Toth 2000 
k d(MT 1−T 2)
From Toth 2000 
k a(T 2−M 2)
From Olson 1997 
k d(T 2−M 2)
From Olson 1997 
Summary of equilibrium constants
K MT 1−MT 1
K MT 1−T 2
K T 2−M 2
Several broad-spectrum MMPIs function by strongly chelating the Zn ion that lies in the MMP active site. Similarities in active sites of MMPs pose obstacles to the design of specific inhibitors . Thus, a new approach to the identification of new drug targets is important. Here, focusing on the MMP2 activation process, we were able to determine that the TIMP2-MMP2 is the most effective interaction. It is reported that TIMP2 interacts with MMP2 through the C-terminal domain of the enzyme that is distinct from the active site [21, 22]. Therefore, our result identifies a new drug target in the process of MMP2 activation. Development of low molecular weight compounds capable of effectively and specifically inhibiting the TIMP2 and MMP2 binding interaction will be the subject of future research. Our result can be validated using cell culture systems.
In this paper, our aim is to quantify the response of a system to the addition of inhibitors and to classify their interactions in order of their inhibitory efficiency. In order to analyse the response systematically, we used control analysis. Using the response coefficients, we revealed that the parameter-independent inequalities between the response coefficients hold in the ordered step models. For the unordered multi-branched step models, we showed that independence of the response coefficients with respect to the equilibrium constants holds. These results indicate that the inhibitory efficiency depends on the topology of the pathway networks. We applied our analysis to a complex formation model describing the formation of complexes of MMP2 and MT1-MMP in the presence of TIMP2 . In the complex formation process between these molecules, there are three interactions, i.e. the MT1-MMP dimerization, the MT1-MMP and TIMP2 binding reaction and the TIMP2 and MMP2 binding reaction. We tried to identify the most efficient interaction to consider in selecting the inhibitors by putting the experimentally derived parameter values into the model. The novel finding of the analysis is that the inhibition of the TIMP2 and MMP2 binding interaction is the most efficient method for suppressing the quadruple complex MT1-MT1T2M2, which contributes to the MMP2 activation. This result identifies a new drug target in the process of MMP2 activation.
Our method can also be applied to other models of complex formation processes. However, there are some weaknesses in the analysis presented here. Throughout our treatment, we have considered only the steady state in a well-stirred environment. For the case that the substrate is a non-diffusible molecule such as ECM, the well-stirred assumption is not valid. Thus, in this case, we should consider a three dimensional compartment model in which the extracellular space is divided into small compartments [16, 23]. This will allow us to simulate a reaction–diffusion system. Furthermore, we analysed models in closed systems, but intra- and intercellular transport of molecules should play important roles in biochemical reactions occurring in cells. Thus, spatio-temporal dynamics associated with a mechanism such as a positive feedback loop  should be considered.
Numerical computation scheme
We employed the fourth-order Runge–Kutta method to solve systems of ordinary differential equations numerically. In all simulations, the time step was taken as dt = 0.001 sec and time evolution was performed up to the time T = 100000 sec. In the calculation of the response coefficients, the small fractional change of the equilibrium constant K was δK/K = 0.01.
This work was supported by the Japan Science Technology Agency (JST), Core Research for Evolutional Science and Technology (CREST), Alliance for Breakthrough Between Mathematics and Sciences.
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