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For a reasonable, objective and complete vision of our external world a connection between our senses, our minds and nature is necessary. In order to make this connection, we generally make use of abstract structures known as “Models”. The mathematical diagramation of these models in formal systems or maps allows us to infer causality and make deductions about our environs. A simple representation of the components of a system are input/output block diagrams. [1] Where ƒ the output.
Clearly, B could be the input for another process
and the system can thus be visualized as a network (Figure
1) with all the possible interactions between the
inputs and the outputs (Oltvai and Barabási,
2002). This interrelated system
allows us to study a phenomenon from a holistic perspective instead
of the separate analysis of each of its constituent parts, where
(in the holistic perspective) each element of the process ƒB represents
a bi-functional unit whose activity is defined by its context within
the model and at the same time contributes to that context and thus
the environs or superstructure as a whole. From a conceptual point
of view it is a self-referential system, which means if the context
or the system is severely disrupted, the superstructure, as defined
by its functional component or components, will cease to exist and
the functional component will also disappear. Thus the whole is more
than the sum of its parts, each part having its own semantics and
its own context (Mikulecky, 2005).Metabolism may be understood as a series
of interrelated chemical-enzymatic reactions that are continually operating to maintain
vital functions far from the thermodynamic equilibrium. Metabolism is a natural
phenomenon that may be studied from the perspective of ƒ being a metabolic process in an organism and .
These branched and interconnected reactions (that generate several specific
products) represent the metabolic pathways. The presence or absence of an
enzyme along one of these pathways varies with the type of cell, tissue or
organism and depends on its physiological state (nutritional condition, state
or stress). The behavior of the metabolic network after specific variations
along some of its pathways and flows is difficult to infer, as the cellular
response to perturbations (genetic or environmental) may involve other
networks. (For example: cell signaling and transcriptional cascades) (Schmidt
and Dandekar, 2002; Qi and Ge, 2006). Thus, for the study of metabolism, its
development and applications in biotechnology and metabolic engineering, it is
fundamental to elucidate the organization and the biological significance of
the metabolic pathways (Yang et al. 1998). This review aims to elucidate the
importance of the study of metabolism, based on the concept of the network and to
describe the major mathematical approaches that permit modeling the metabolic
networks. B
As has already been explained, in order to gain a complete vision of metabolism, it is necessary to go beyond a simple description of its pathways. This global vision could be achieved by the study of metabolism based on the network concept, whereby the characterization of the relationships and interconnections between the elements are taken into account rather than simply looking them as a collection of interconnecting elements (Lazebnik, 2002). The particular way in which the elements are linked should be described thoroughly in order to precisely depict the pattern of connections, which is known as the network topology. In
an abstract sense, a network may be described by a graph V, where i=1,
2, 3… M, and M is the total number of nodes
or vertices (v_{i}∈V). Each pair of nodes define
a link and these L links also form a set E = (e)
| _{j}j = (1,2,3,…,L). Otherwise, a network is the
relation of the Cartesian Product N = V x _{1}Vfor
all pairs of nodes v (v_{2} _{r},v_{s}) | r,s =
1,2,3…,M;
where again M is the total number of nodes network. If the
trajectory between two nodes is important for its description, each
link between two nodes would have a particular orientation (Barabási
et al. 1999). This type of graph is
known as a directional graph or digraph and characterizes
the structure or topology of systems such as the metabolic networks
(Figure 2). The
constitutive relations that describe the network elements and the
topology of the network may be formalized independently and then
combined to provide a solution to the network (Almaas
and Barabási, 2006). A network is said to have a
solution when everything observed within it can be specified. When
the network is directed (digraph) the nature of its solution is a
system of pathways that can be formulated as a set coupled kinetic
differential equations that describe the relationships within the
network (Mikulecky, 2005).An incidence matrix I =
K = K + _{i}K_{o}.
The average degree of a homogeneous undirected graph is Z = < K > =
2 L/M, since by definition each link is joined to two
nodes. Nevertheless, in heterogeneous graphs, where the nodes do
not have the same number of links, this value is not valid (Fernández
and Solé, 2005).
Mean path length or graph radius (l). This
is defined as the average distance between any two vertices ( [2]
This equation also describes the average number of steps that must be taken from any node in order to pass through the entire graph.
[3] Where
_{)} which
describes the fraction of nodes that have a certain degree k for
a graph N, i.e. it indicates the
probability p that a randomly selected node has k links.
For a random graph (RG) the probability p_{(k}_{)}of
linking two nodes is:[4] When [5] Many biological phenomena do not adjust to random graphs (where the links are uniformly distributed with probability p about a mean < K >), since natural networks have been observed to show a heterogeneous degree distribution where a small number of nodes has a large number of links (Wodak et al. 2004). Natural networks, such as those described from genetic or metabolic protein-protein interactions show a degree distribution that adjusts to the Bounded Power Law as shown in equation 6: [6] Where The value of γ determines several
network properties and the smaller γ is the more important the hubs nodes
are. For values of γ > 3 hubs nodes are not important. If 3 > γ
> 2, a hierarchical organization of hubs nodes appears, where the more
connected hubs are linked to a small fraction of the total number of nodes.
When γ = 2, a network emerges where the most connected hubs are in contact
with a large fraction of the total number of nodes. The properties of scale
free networks are only observed when γ < 3, explicitly, when the
dispersion of the distribution An
important property associated with scale free networks occurs when
new nodes [7] Where
In general small world networks are characterized by the following properties (Newman, 2001): 1. The local neighborhood of each node is preserved as for regular networks. 2. The network diameter (Shortest average distance between pairs of nodes) increases logarithmically with the number of nodes. 3. Thus, small world networks are those that permit the connection of two nodes with very few links (Amaral et al. 2000).
Metabolic networks are examples of scale
free and small world models. In these networks the nodes represent the
metabolites and the links the enzymatic reactions that transform certain
metabolites into others. These networks are bipartite digraphs: a sub group of
nodes Studies undertaken of the metabolic networks of 43 organisms (Alm and Arkin, 2003) have found that: a) They are scale free networks and non random. b) They are small world networks. c) They are modular. The metabolic networks studied showed a
fixed diameter that varied very little between organisms. Paradoxically, each
hub node was linked with a low probability to other hub nodes (property known
as Although these findings have helped disentangle general principles in biological design, the risk of simplifying natural processes that are dynamic, evolutionary and subject to selective pressure remains. It is thus necessary to apply tools that allow the study of these networks over time (Schilling et al. 1999; Holme and Huss, 2003). In recent years a huge mass of data has
been obtained from the “omics” technologies, such as genomics, transcriptomics
y metabolomics. This information has been stored in extensive data bases
available on the Internet. The study and integrative analysis of these data
bases is supported by mathematical instruments such as cellular models (Fiehn,
2001). The evolution of these tools is an iterative process whereby the models
are compared with the experimental data, which both validate and improve them, leading
to the formulation of new Mathematical models can vary in complexity and focus, nevertheless the model should be judged on the basis of the objective that it pursues, because metabolic networks are dynamic entities with flows of material produced by the biological cellular activity and subject to hierarchical regulatory interactions under cellular direction or canalization (Jørgensen et al. 2005). Thus, the description of metabolic phenotypes should include the composition of the metabolites, the measurement of their flows and the dynamics of the network. The mathematical and computational models most commonly used for the clarification of the structure and dynamics of metabolic networks are discussed in the next section including an analysis of their most important virtues and limitations (Figure 7).
The qualitative approach fundamentally uses graph theory and describes metabolic networks as bipartide digraphs with start nodes (precursor metabolites), end nodes (end products) and connections that link both nodes in a directional manner (chemical-enzymatic reactions) (Alterovitz and Ramoni, 2006). The enzymatic reactions are governed by chemical and thermodynamic laws. According to Xia et al. (2004) the most important rules of metabolic restriction are: · Stoichiometry: Refers to the number of molecules that participate in the chemical reaction. This property does not vary for the same reaction in different organisms and does not change with pressure, temperature or other physicochemical parameters. Chemical Stoichiometry appears as a topological property of metabolic networks. · Relative reaction rate: This property is determined by the thermodynamics of the reaction and is dependent on the physicochemical conditions of the environs. · Absolute reaction rate: This property is determined by the genetic function and is specific for each enzyme. If changes that modify the enzymes occur in the genome, these will show up in the chemical reactions. The
properties mentioned above provide limited information on the functional
state of any metabolic network restricted to a temporal or spatial
interval and to the physicochemical nature of the extra and intra
cellular environment (Schilling et al. 1999; Fiehn,
2001). Biological
systems are directed by a cellular organization which determines
the temporal nature of the interactions. This behavior is linked
to properties of a superior order peculiar to cellular networks
-self-organization and self-assemble Another quality inherent to biological networks is the bi-linearity and high interconnectivity between its components, thus cellular networks behave as hypergraphs (Wagner, 2001). Hypergraphs have topologically non linear properties, with a higher growth of functional states than is expected from the number of components in the network. In other words, the number of phenotypic functions that derive from a genome is not linearly correlated with the number of genes present in that genome. This fundamental characteristic promotes the expression of different possible functional states (similar phenotypic behaviors). Nevertheless, an organism does not exploit or completely use all of its similar functional states. At a particular moment in time and space, only a limited subgroup of these states will be selected and expressed by the cells (Palsson, 2006). Thus, metabolic networks are under inviolable restrictions (hard restrictions) associated with the thermodynamics of the organism (balance of mass and energy), their evolutionary history (genetic and regulatory restrictions) and restrictions related to the environment, allowing a profile of solutions (phenotypic structure ) in a given interval of space and time (Förster et al. 2003). Why is it necessary to study the topology of metabolic networks? How can structural information help to clarify the role of individual nodes and their relationships within the network? Which of the functions or solutions of the network will be exploited by the cells, given the existence of equivalent, multiple states of the network or silent phenotypes? How do the responses of the cells change over time as a result of selective pressure (Evolutionary history)? (Vitkup et al. 2006). These questions must be answered in the
future with the support of these tools. An approach to finding the volume and
shape of possible phenotypes or solutions for the optimization of biological
traits under genetic, environmental or evolutionary restrictions, is performed
by the group of Braunstein et al. (2008) where they suggest that the best
technique that allows this characterization is based on the method of Monte
Carlo sampling (MCS) of the area of metabolic flux of the network under steady
state, given the complexity of metabolic networks and the enormous
computational cost involved in these calculations. They propose a computational
strategy known as message-passing algorithm derived from the field of
Statistical Physics and Information Theory. This algorithm based on conjecture
or Bethe approximation (Bethe Ansatz) allows calculating the volume of a convex
polygon of incomplete higher-order dimension. Was successfully used in the
characterization of a metabolic network of 46 reactions and 34 metabolites in
blood and red blood cells in predicting the effect on the disruption of some
genes of central metabolism (gene knock-out) in On the other hand, the need for a
system-wide approach to network construction in plants has been recognized and used to construct networks in In a gene to metabolite study, Rischer et
al. (2006) use a holistic approach to characterize
the terpenoids indol alkaloid (TIA) biosynthesis in the Madagascar periwinkle In principle, the structure affects the function, a fact which can be used to discover or predict new functions or even the evolutive origin of the metabolic network (Solé and Valverde, 2008). The topology dependent approach has contributed to the understanding of general aspects of metabolic networks (scale free, small world effect, preferential attachment, modularity, network motifs) (Jin et al. 2007). The development of metabolic maps along with the progress of implementation of computer platforms to simulate cellular functions under the tutelage of tools such as graph theory, have been an important first step in understanding the contribution of cellular metabolism (Ishii et al. 2004). Nonetheless, this perspective has not been able to resolve questions inherent to variations in time, is ambiguous in its representation for detailed discussions of structure and is limited as regards of the interpretation and comparative analysis of experimental metabolic data associated with other biological networks (For example: transcriptomic networks or networks of protein interactions) (Gross, 2005), due to the hypergraphic nature of metabolism. Then it is important to incorporate information relevant to flows within the network or changes of the metabolic mass over time, for a complete analysis of metabolic systems (Steuer, 2007; Domijan and Kirkilionis, 2008) due to the static nature of the topological models. We continue in this review with the conceptual models that incorporate the kinetic parameters of metabolism. Although the development of dynamic models that permit a complete simulation of the cellular system have been attempted, this has proved to be a difficult task due to the lack of unambiguous information on the kinetic and regulatory properties of metabolic reactions (Tomita et al. 1999). Nevertheless, even in the absence of kinetic data, relevant information on the theoretic capacities and operative modes of metabolic systems may be obtained using different Stoichiometric approaches, such as: Flux Balance Analysis (FBA) and Metabolic Flux Analysis (MFA), and others applications: Elemental Mode Analysis (EMA) and Extreme Pathway Analysis (EPA) (Ciliberto et al. 2007; Rios-Estepa and Lange, 2007). The Flux Analysis is based on the description of the flow reactions of a metabolic network under certain restrictions or boundary conditions of biological networks: a) physicochemical and thermodynamic restrictions, b) topological restrictions c) environmental restrictions and d) regulatory restrictions (Kauffman et al. 2003). FBA translates enzymatic reactions into flow differential equations such as: [8] where = (X _{i}x_{1}, x_{2}, . . . , x), for _{m}I = 1,2,3,…,m metabolites, describes the metabolic flow vector: V= (V _{j}v_{1},
v_{2}, . . . , v) for _{n}j = 1,2,3,…,n reactions
and the Stoichiometric matrix S as the linear transformation of the
vector in V. The general dynamic mass balance equation
for a metabolic network (Equation 8) is represented by a set or system of
differential equations (Varma and Palsson, 1994): X[9] Each equation represents the sum of all
the individual flows x. There are m metabolites (_{i}x_{i})
and n reactions (v_{i}) in the network, thus the dimensions (dim)
of the vectors , X and of the matrix VS are:dim( m, dim() = Vn, dim(S) = mxn [10]The
information on the Stoichiometric of the reactions is represented
by these equations and by the Assuming that metabolic networks operate between successive quasi-stationary states (the macroscopic variables - metabolite concentration and flux - are maintained within a tolerance interval defined within a time period) where a 0 net flux balance exists, in accordance with the law of conservation of mass (Roscher et al. 1998). The distributions of the flows that
satisfy this condition belong to the null space of the [11] Due
to the fact that the system is indeterminate, since m < n, which
permits infinite solutions _{j}) is obtained, such
as α ≤ V_{j} ≤ β (Segrè et
al. 2002). Thereafter,
with the use of linear programming the optimum solutions for the system can be
found (Bro et al. 2006; Nielsen, 2007). This reduction in the space of
admissible flux provides the basis for FBA (Reed and Palsson, 2003) (Figure 9).Several metabolic networks have been
analyzed in yeast, plants and mammalian cell culture using these tools. Among
them the photosynthetic prokaryote MFA
is a powerful methodology for the determination metabolic pathway
fluxes ( In a study of the metabolic reconfiguration
in response to oxidative stress Ralser et al. (2007) showed that there is a dynamic rerouting of the
metabolic flux to the pentose phosphate pathway, with the concomitant
generation of the reduced electron carrier nicotinamide adenine dinucleotide
phosphate (NADPH), which is a conserved post transcriptional response to
oxidative stress. After Grant (2008) this study also demonstrates the need to integrate
genomic, biochemical and In addition to quantification of pathway fluxes, metabolic flux analysis can provide additional insights about other important cell physiological characteristics. As shown here: (1) Identification of branch point control (nodal rigidity) in cellular pathways, (2) Identification of alternative pathways, (3) Calculation of non measured extracellular fluxes and (4) calculation of maximum theoretical yields (Stephanopoulos et al. 1998). In MFA, kinetic parameters are not required, all that is needed are the stoichiometric data and the data for the metabolic demand of the network, however, additional available information could be incorporated in the future (for example thermodynamic restrictions) (Liebermeister and Klipp, 2006). One of the main disadvantages of this method is that it assumes the metabolic network is in steady-state. Computationally, steady-state models are easiest to manage and steady-state systems have been used productively in plant studies over the last dozen years to yield extensive and detailed flux maps of central metabolism (Saito, 2009). An alternative to estimate flows and develop predictive models in non-plants systems is the use of the dynamic labeling methods. Dynamic labeling traces the change in the level of a metabolite in a metabolic pathway after the application of a stable isotope-labeled compound. The time-dependent decrease in the labeling level of the precursor as well as the increase in that of a down-stream metabolite in the pathway is monitored as a dynamic process. The value of metabolic flux is determined by fitting the model describing the labeling dynamics of the pathway to the observed labeling data (Matsuda et al. 2007). In addition, a number of studies have used
batch cultures in which plant cell suspensions are grown for several days on a
medium that contains
In the same way as for Graph analysis, the
Stoichiometric approach does not permit the direct description of the dynamic properties
of the systems. Specifically, does not permit conclusions about the dynamic
behavior for a future metabolic state, for example, if we wish to use the flow
distribution function For this purpose, kinetic models that
permit the satisfactory description of the dynamics of metabolic networks have
been developed. The metabolic control analysis (MCA) allows the clustering of
metabolic pathways as a set of ordinary differential equations assuming spatial
homogeneity (Cronwright et al. 2002). To better understand this mathematical
model we shall use the following example, suppose a small metabolic route,
which for simplicity is assumed that contains no elements or regulatory
structures and assume the transformation of the species [12] The Several dynamic properties, such as the
dynamic response to perturbations, the stability of a metabolic state and the
transition of an oscillating behavior can be obtained using a local linear
approximation of the system. This local approximation is obtained by means of
the expansion of a Taylor Series of the metabolic system in a stationary state
(not necessarily unique or stable). The linear term for the expansion is the
Jacobian Matrix X_{(t)} (equation
13). This matrix represents the dynamic response of the system in the neighborhood
of a stationary state.[13] In contrast to the mass balance equation (equation
11), which does not provide information on the stability of the metabolic state
when disrupted, the real part of the eigenvalue (characteristic root) of the Jacobian
describes the possible scenarios of stability and their transitions from the
metabolic state, for example: Hopf type bifurcations (Steuer, 2007). If we take
the metabolic pathway of Figure 10 and define { [14] The Jacobian matrix of equation 13 becomes: [15] Now that we know the elements of indicates a stable
solution for the system, which converges to a stationary state. On the other
side, the positive real part of the eigenvalue of J describes an
unstable solution and the system does not converge to a stationary state. In Figure
11 it is shown the different dynamic behaviors of the system depending on
the eigenvalues of the Jacobian matrix evaluated around a stationary state JXº.Let us consider the 3 x 3 ; If we assume integer numbers for A defining characteristic of living cells is the ability to respond dynamically to external stimuli while maintaining homeostasis under resting conditions. Capturing both of these features in a single kinetic model is difficult because the model must be able to reproduce both behaviors using the same set of molecular components. A novel approach differs critically from metabolic flux analysis and previous genome-scale metabolic network reconstructions because it accommodates nonlinear terms that describe the dynamic behavior of each reaction in the system. Previous large scale network reconstructions typically use a stoichiometric matrix to represent the gross flux of metabolites in the system. Purvis et al. (2009) using a modified kinetic modeling (models are built in a stepwise fashion, beginning with small ‘‘resting’’ networks that are combined to form larger models with complex time-dependent behaviors), not only reduce the computational cost of fitting experimental time-series data but can also provide insight into limitations on system concentrations and architecture. They have preserved the mathematical form of each kinetic rate equations as reported in the literature, allowing models to be built from existing data in a ‘‘bottom-up’’ fashion while still allowing calibration to whole-system experimental data. This feature will substantially improve the accuracy of dynamical system simulation and parameter estimation. This method was illustrate showing how 77 reactions from 17 primary data sources were integrated to construct an accurate model of intracellular calcium and phosphoinositide metabolism in the resting and activated human platelet. Although the explicit kinetic models quantitatively describe the dynamics and complex properties of metabolic systems (oscillations, multi-stability or irreversible commutations), their computational construction becomes complicated when the components of the network are numerous and the kinetic parameters incomplete (Cronwright et al. 2002).
The modified kinetic model proposed by Steuer et al. (2006) is an intermediate between the stoichiometric and deterministic models; it describes the stability and robustness of the possible metabolic states of a network using the minimum relevant kinetic information and it identifies the important interactions and the parameters that rule the dynamic properties of metabolic systems, without the knowledge of the explicit functional forms of the differential equations of the enzymatic kinetic rate. For most natural phenomena is unnecessary an explicit kinetic model and the J matrix, expressed in terms of the derived partial equations (equation 13), generally requires the explicit kinetic knowledge of the enzymatic reactions (values of the kinetic parameters k). Nevertheless, it is still possible to specify the structure of the Jacobian matrix, even without this information, if each element in the matrix is restricted to a definite time interval (Strogatz, 1994; Scott, 2007). Just as the stoichiometric balance imposes restrictions and defines a space of flux solution, the Jacobian matrix imposes a set of possible dynamic behaviors for the metabolic network and defines the dynamic capabilities of any metabolic state. The structural kinetic models (SKM) are based on the development of a parametric representation of the Jacobian matrix, in a way that each element in the matrix is defined within a time interval without the need of further kinetic information of the metabolic system under study. The
parametric Jacobian matrix () is defined as the
matricial product, where
the matrix The SKM model proposes the rewriting of the system of differential equations described in equation 12 to: [16] where ^{o} indicates
the concentrations of the i-thmetabolite and V_{j}^{o} = V(X^{o},k)
represents the distribution of the j-th
flux, both parameters being associated with a stationary state (not
necessarily unique or stable). By definition the matrices [17] The Jacobian is re-written with respect to
the [18] The stationary metabolic state in our
example in Figure 10 is then defined by the metabolic concentration With SKM, the system of kinetic equations is replaced by a parametric representation of the metabolic pathway in terms of the Jacobian matrix, becoming in our example: [19] The
elements of the matrix Λ
represent: a) the structure of the system as defined by the vector
V° and the
temporal scale as defined by the vector X°, b) the elements of
the matrix represent the normalized
degree of saturation of each reaction with respect to its substrate,
where ∈ [0,1] for all substrates
and ∈
[0,-1] for all of the metabolic products (Bapat, 2000).
The saturation or effective kinetic order, even with unknown values
is defined for an interval: ∈ [0,1] of the reaction
V2 with respect to the substrate X1 and ∈ [0,-1] of the reaction V3 with respect to the substrate X2. Assuming
similar conditions for V2 and V3 as in the explicit kinetic equations
and [20] The
Jacobian is evaluated in terms of the four generalized parameters
with the purpose of studying the dynamic behavior of a system after
a disturbance around a stationary state Because lacking or incomplete enzyme-kinetic information that is necessary for classic kinetic models, this proposed method create bridges between topology based approaches and explicit kinetic models of metabolic networks. The SKM models permit to make quantitative conclusions about the dynamics of the system based on a minimum of information (qualitative) compared to the explicit kinetic models (quantitative). This approach was used by Grimbs et al. (2007), to represent the metabolic network of the human erythrocyte. In particular, transitions to instability, occurring via a loss of a stable steady state, were previously argued to play a crucial role in senescence and metabolic collapse of erythrocytes and may act as a primary signal for cell removal in patients with hemolytic anemia. While usually an investigation of such transitions necessitates the construction of explicit kinetic models, this approach allows the draw of quantitative conclusions about the stability of metabolic states in response to an increased ATP demand, occurring under conditions of osmotic or mechanic stress. The authors demonstrated that different metabolic states, each satisfying the flux balance equation and thermodynamic constraints, can nonetheless show drastic differences in the ability to ensure stability and maintain metabolic homeostasis.
Functional behavior emerges from the nonlinear interactions between genes, proteins and metabolites within metabolic and regulatory networks. It has also been argued that the kinetic study of the network of enzymatic reactions (fluxomics) is key for the study of the dynamics of metabolic networks. The quantification of metabolic flows using isotopomers- 13Cbased metabolic flux analysis- (Rousu et al. 2003; Sauer, 2006), the probabilistic graphical models (bayesian, continuous differential kinetic, Markovian, master-slave synchronization) (Weitzke and Ortoleva, 2003; Wodak et al. 2004; Ao, 2005; Pécou, 2005) and random Boolean network models (Kauffman, 1969) have tried to resolve this dilemma, although some contradictions do exist as the minimum amount of information required and the relevance of the modularity of the network (Stumpf and Wiuf, 2005; Ingram et al. 2006; Solé and Valverde, 2006; Hormozdiari et al. 2007). In this review several analytical approaches have been examined, the first model, although qualitative, characterizes the topological properties of metabolic networks without making any inferences about their dynamic behavior. The second approach explicitly describes the dynamics of the system once the kinetic parameters of each component are known, although this may be a disadvantage in cases where the systems are incomplete or large, due to the high computational costs of the calculations. Lastly, semi-quantitative models that permit inferences about the stability and dynamics of the system without the need for an exhaustive knowledge of every constituent element were described, although chemical or functional restrictions may be required in order to obtain reasonable solutions. Although these approaches have been successful, they are limited in regard the application of regulatory parameters (feedback loops). In short, these models require iterative processes for the incorporation of new information and the validation of this information using experimental data, thus permitting the evolution and improvement of these mathematical tools (Figure 12). Although the approaches described here have both advantages and limitations for the study of the functionality and evolution of biological systems, all of the models reviewed agree that robustness is a common property of biological networks. The robustness of biological systems lies in the persistence of their functionality in the face of perturbations such as the removal of network links or nodes. In other words, robustness represents the capacity of the systems to maintain their stability in spite of changes to either their internal or external environment. A mathematical definition of robustness (R) of a system (s) measured as a function of the property (a) could be, according to Kitano (2007): Where the function represents the probability of the perturbation of space p and an evaluation function of (p) that determines the extension and degree of the perturbation, defined as: Both metabolic flow and kinetic analyses permit the quantitative evaluation of the redistribution of the flows when enzymatic reactions are perturbed and to clarify the implications of unstable states and oscillations around a stationary state. The dynamic stability of the stationary state is important for homeostatic sustainability and robustness against disturbances. It would be illuminating if, in the future, efforts were directed towards the elucidation of this type of behavior in biological systems. The reconstruction of metabolic networks offers a mathematical tool that permits the recuperation of information inherent to their context, within the Systems Biology approach, thus allowing an approximation to the understanding of the cellular behavior, based on the quantity of information about metabolism accumulated from the advances that have been made in the fields of genomics, transcriptomics and metabolomics (Karp et al. 2002). Thus, mathematical models and simulations can be used as a guide to the design and improvement of biotechnological production systems (Sweetlove and Fernie, 2005). Finally, different approaches have been
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