Selective robustness is a key feature of biochemical networks. It confers a fitness benefit to organisms living in dynamic environments. The (in-)sensitivity of a network to external perturbations results from the interplay between network dynamics, structure and enzyme kinetics. In this work, we focus on the subtle interplay between robustness and control (fragility). We describe a quantitative method for defining the fragility and robustness of system fluxes to perturbations. We find that for many mathematical models of metabolic pathways, the robustness of fluxes vis-à-vis perturbations of all the enzyme activities is captured by a broad distribution of the robustness coefficients. We find that in cases where a metabolic pathway flux is made less robust with respect to the perturbation of a particular network step, the average robustness may still be increased. We then show that fragility is conserved upon a perturbation of network processes and equate fragility with control as defined in metabolic control analysis. This highlights the non-intuitive nature of the interplay between fragility and robustness and the need for a dynamic network understanding.
Biological organisms are constantly subjected to changes in their internal and external environments and yet, in most cases, they are able to maintain their functionality. This robustness to perturbations is a key feature of the fitness of organisms and has been the focus of numerous studies, some of which have led to changes in the way in which we view biological systems [1-5]. For example, Barkai and Leibler discovered that bacterial chemotaxis adaptation is a robust network property observed across a wide variety of genotypes rather than being the result of fine-tuning of system parameters [6, 7]. It has also been noted that many diseases can be thought of as failures in the mechanisms responsible for robustness, with pathogens often taking advantage of those very mechanisms in order to maintain and even promote the disease state [8-10]. Such studies have highlighted the importance of understanding biological robustness and the mechanisms that are responsible for it in order for us to be able to manipulate the function of biological systems successfully.
While there has been a great deal of research in recent years into the robustness of a diverse range of biological properties, the term ‘robustness’ is one that is hard to define precisely and generally. In its most general form, robustness can be described as being the maintenance of the characteristic behaviour of a system when subjected to perturbations or conditions of uncertainty. It should be noted that robustness is the maintenance of specific functionality, rather than the system remaining unchanged in all its properties, which would be homeostasis. Robustness takes two different forms, the first being the return of the function in question to the original state following perturbation, the second being a transition to a new state that allows functionality to be maintained in the new conditions . In this second form robustness is most distinct from homeostasis and stability, which can be considered as being subsets of robustness [11, 12]. It is also important to realize that robustness properties can be dramatically altered by differing specifications and, while a system may be robust according to one definition, this may not hold for other definitions . In addition, robustness is not necessarily consistent across all levels of organization. For example, a robust genotype with an unchanging gene expression profile does not always result in a robust phenotype, and vice versa, with robustness at the phenotype level actually facilitating the changes in genotype required for evolvability [14-19]. Similarly, a species that is considered to be robust is not necessarily part of a robust ecosystem, and conversely a robust ecosystem can be composed of non-robust individuals . As such, when discussing robustness it is imperative to describe which property of the system in question is considered to be robust and which type of perturbations or uncertainties it is robust against.
In both biological and man-made systems there are costs associated with making the system robust to perturbations. The two main costs of increased robustness are increasing complexity and, perhaps surprisingly, fragility to unexpected perturbations. The balance between these factors in biological systems has received much attention in recent years [8, 12, 13, 20]. Carlson and Doyle proposed an architecture which they named highly optimized tolerance (HOT) which describes how systems can be designed for high performance in uncertain environments. The HOT architecture emphasizes the fact that limited resources necessitate trade-offs in robustness and highlights the inextricable link between robustness and complexity [1, 21-23]. Using Bode's integral formula, Csete and Doyle showed that for a feedback system net robustness and net fragility are constrained properties, with any increase in robustness at a particular frequency being paid for by an equal fragility at other frequencies . It should be noted, however, that this conservation of total fragility/robustness does not apply to nonlinear systems as, although such systems still demonstrate a trade-off between robustness and fragility, there is not necessarily an exact balance between the two.
Previous studies of robustness in biological organisms have often focused on the effect of deletion mutations to the function of the system [2, 24, 25]. This type of robustness is termed ‘structural robustness’  and although this has a role to play in the overall robustness of an organism, it is not the only type of robustness that is important. The deletion of a step in a pathway is likely to be a rare event. Although such a deletion may occur in a single cell, the population or tissue as a whole is likely to be unaffected. Under natural conditions there are very few reversible inhibitors that will inhibit a reaction by anywhere near 100% and pharmaceutical products are generally only applied at levels that partially inhibit the target reaction. The most serious threats to biological systems often come from malfunctioning components rather than the loss of components .
When a step in a reaction is only partially inhibited, network function may not be impaired proportionally. Various robustness mechanisms may come in to play, different from those responsible for topological robustness. For example, the network may bring about robustness by enabling the substrate or regulator of the inhibited step to increase in concentration, or by increasing the expression of the inhibited enzyme . This is known as ‘dynamic robustness’ .
If in this sense a network flux is robust to malfunctioning of a specific step, this implies that that step is not the rate-limiting step of the pathway, or, more subtly, exerts little flux control. This suggests that there is a relationship between dynamic robustness and metabolic control. Metabolic control analysis (MCA) was originally developed independently by Kacser et al.  and Heinrich and Rapoport  and received solid experimental and computational foundation in the work by Groen et al.  and Bakker et al.  respectively. A set of terminologies, which we shall follow here, has been agreed upon [32, 33], extensions to the theory have been developed [34-36] and it has been the subject of a number of reviews [37-39]. It provides a powerful quantitative method for the analysis of relationships between the steady-state properties of biochemical networks as a whole and the properties of their component reactions.
In this paper we present the findings of our investigations into the dynamic robustness of fluxes in biochemical pathways. We discuss how the control coefficients of MCA can be thought of as measures of dynamic fragility, and describe the development of MCA-based flux robustness coefficients (FRCs), which provide a fully quantitative measure of flux robustness. We use these coefficients to analyse a variety of in silico models. The results of these experiments are used to examine whether the links between robustness and fragility that have been observed in other studies of biological robustness are also seen for the measures of dynamic flux robustness that we define here. We highlight the fact that the laws of MCA can be used to demonstrate an important conservation property for fragility of fluxes, and that such conservation does not hold for robustness as we defined it.
In this section we will come to a quantitative definition based on an existing, more qualitative, concept of robustness. This definition is formulated for the issue of robustness per se. Subsequently, we shall see how robustness relates to a concept from an already existing field in biochemistry, i.e. control, and how a methodology used in that field helps in pinpointing properties of the robustness of networks.
A quantitative measure of dynamic flux robustness
The initial challenge we faced was how one can measure the robustness of flux in a biochemical pathway. Flux is a system property and, as such, perturbations to any part of a network are likely to have an effect on flux. It was therefore clear that in order to quantify the robustness of a particular flux, the precise perturbation would have to be specified. Also, biological systems are nonlinear, meaning that the magnitude of response to a perturbation depends upon the magnitude of that perturbation and on the current state of the system. Since most biological systems operate close to steady state , we decided to define the robustness of fluxes in terms of their response to a small perturbation in enzyme activity. Therefore, we first examined a preliminary definition for FRCs () as the maximum percentage decrease in activity of a specific enzyme (ei) allowed when at most a 1% change in a particular flux (Jj) is tolerated. This preliminary is negative when decreasing enzyme activity causes an increase in flux, as may occur if the enzyme in question is part of a different branch in the biochemical pathway to the flux being measured. In this definition, the activity of one particular process in the entire network is interfered with, the concentrations of all other enzymes remaining constant, and the effect is then measured in terms of the flux, which is the functional property of interest to this paper.
The availability of a quantitative measure of the robustness of fluxes allows us to study and compare the robustness of metabolic pathways. Experimental measurement of FRCs of a large number of pathways is impractical, requiring years of laboratory work, and measurement of the robustness of some steps may not yet be achievable. However, the availability of detailed true-to-life in silico kinetic models allows for the simulation of such experiments. Using Systems Biology Markup Language (SBML) models downloaded from jws online, a repository of published kinetic models , we carried out simulations to examine the robustness of fluxes to perturbations of enzyme activity using the definition given above (see 'Experimental procedures').
The results of these initial investigations revealed that the above definition could indeed be used to probe the dynamic flux robustness of models of biological pathways. However, we quickly realized that the definition had some limitations. First, in some cases it was not possible to achieve a 1% inhibition of a particular flux, even upon complete inhibition of a particular enzyme. This might lead one to conclude that the flux in question was completely robust to perturbations of that particular enzyme and, while this could be the case, it is possible that the flux was affected by < 1% but that even such a small change could have an effect on system function. Second, the iterative procedure required to calculate the FRCs was computationally slow, especially on models of larger metabolic networks. While this is acceptable for the analysis of individual models, it makes the analysis of large sets of models prohibitively time-consuming. Another issue is that if one were to calculate the FRCs in the same way but calculating the change in enzyme activity required to get a 1% increase in flux or metabolite concentration, rather than a decrease, then the results could be significantly different. Also, since the relationship between steady-state pathway flux and enzyme activity is not linear, the robustness coefficients calculated will depend upon the change in flux tolerated. This last problem is unavoidable but it raises the question of how one determines what percentage change in function is significant when examining the robustness of metabolic pathways.
In our initial investigations into the robustness coefficients of various models we noticed an inverse correlation between the FRCs and the flux control coefficients of MCA. This makes sense since, if an enzyme has only a small amount of control over a particular flux, any change in the activity of that enzyme will have little effect on the flux and thus the flux will be robust to changes in that enzyme. The control coefficients of MCA can therefore be considered as coefficients of flux fragility, and FRCs can be defined as their inverse. This overcomes the problems identified with our initial definition of robustness. The coefficients of MCA give a value derived from a linearization of the initial response to a change in enzyme activity, thus eliminating the problems of what percentage change should be considered important and which direction of change in the flux or metabolite concentration should be used, and also allowing for the FRCs to have an infinite range of values. Also, there are methods of calculating control coefficients that remove the need for a time-consuming, iterative process [38, 42]. Thus, we defined coefficients of fragility and robustness based on the control coefficients of MCA [28, 39]:
where f is the system function of interest (i.e. a particular flux, Jj). Here f is seen as a function of all enzyme activities in the network. In a regular metabolic network the latter correspond to all enzyme activities.
While the fragility coefficients defined here are identical to the control coefficients of MCA, it is useful to use the nomenclature given above in the context of the discussions here since we are focusing on the fragility and robustness of biological pathways and addressing different questions from those normally examined by MCA. We thus introduce the term ‘flux fragility coefficient’ (FFC). The routines that we developed based on these definitions (see 'Experimental procedures') allow for the rapid analysis of the robustness of any model of a biological pathway, if in the appropriate SBML format, so long as that model satisfies the conditions required for calculation of control coefficients. We shall refer to the type of robustness/fragility described by the robustness and fragility coefficients described above as MCA-based robustness/fragility in order to distinguish these from other definitions of robustness and fragility. Models were downloaded from jws online, a repository of published kinetic models , and the MCA-based robustness and fragility coefficients were calculated. Table 1 lists the models that were used in our analyses.
Table 1. Models from jws online that were used in our flux robustness analyses
Epidermal growth factor
Bistability double phosphorylation
How robust are biochemical pathways?
Table 2 shows the results of our analysis of the robustness of the flux through alcohol dehydrogenase in the Saccharomyces cerevisiae pathway, comparing the results of the 1% method and the MCA-based method. The model developed by Teusink et al. was used for this analysis. It condenses the many reaction steps between acetaldehyde and succinate via the glyoxylate cycle into a single step . The second column of Table 2 shows the results of calculations where we determined the percentage to which we could reduce the enzyme activity mentioned at the same row in the first column for the flux through alcohol dehydrogenase to ethanol to be reduced by 1%. The robustnesses were not uniform, ranging in magnitude between 1 and 86, with the flux being robust to a value > 10 for more than two-thirds of the perturbed enzymes. Some robustnesses were negative, reflecting that the inhibition of the enzyme results in an increase in flux to ethanol.
Table 2. Flux robustness coefficients for the flux through alcohol dehydrogenase in the S. cerevisiae glycolysis pathway – comparison of the results given by our preliminary 1% change and subsequent MCA-based definitions, calculated from the Teusink model from jws online
(1% decrease method)
A 76% reduction of fructose-1,6-bisphosphate aldolase activity had not resulted in a 1% decrease in flux through alcohol dehydrogenase. Further reduction of enzyme activity resulted in a failure to obtain a steady state.
The third column of Table 2 shows the results of the application of the MCA-based method for calculating FRCs (cf. above). Robustness coefficients derived using this method were also distributed heterogeneously over the network. As expected, the results are similar in terms of the rank order of the FRCs. However, there is a large difference in the values of the FRCs obtained using the two different methods for the enzymes that are most robust.
Surprises may be found in the very high robustness of the flux with respect to perturbations in phosphofructokinase, the supposed key enzyme in the glycolytic pathway, and perhaps in the relatively low robustness with respect to glucose transport. Glucose transport, being on the outside of the cells, could well be the step that is most subject to actual perturbations.
If we consider the simplest possible biochemical pathway, with a single enzyme, the flux through that enzyme will be proportional to the activity of that enzyme, as parameterized by the kcat or by the enzyme concentration . Thus, a 1% increase in the activity of the enzyme would lead to an increase in flux of 1%. As such, we can see that the FRC of a process in isolation will always be equal to 1:
Re-inspection of Table 2 reveals that all robustness coefficients except for that corresponding to the glucose transporter were higher than would be expected for enzymes in isolation. Clearly, this network, or at least the mathematical representation of the network, has increased the flux robustness as we have defined it. The average absolute value of the FRCs calculated using the MCA-based method was 6344 ÷ 17 ≈ 373.
Can robustness be increased or is it conserved?
It would seem that the robustness of a flux with respect to inhibition of an enzyme in a metabolic pathway could be increased by creating an excess of that enzyme. We examined this in a system of particular medical relevance, i.e. the glycolytic pathway of Trypanosoma brucei, the causative agent of sleeping sickness, using the in silico model developed by Bakker et al. . The second column of Table 3 gives the MCA-based FRCs for the ATP synthesizing flux with respect to perturbations of various glycolytic enzymes. Again, all of the robustness coefficients exceeded 1 quite substantially, with the possible exception of the flux robustness with respect to perturbations of the glucose transporter. We expected that by doubling the expression level of the glucose transporter we should be able to make the flux somewhat more robust vis-à-vis perturbations in this step. The third column of Table 3 lists all the MCA-based FRCs for this organism genetically engineered in silico. Indeed, the flux robustness with respect to this perturbation increased, but to our surprise it increased not just a little but more than 80-fold. By altering the expression level of its genes, an organism may be able to greatly adjust the flux robustnesses with respect to its individual gene products.
Table 3. Effect of doubling glucose transporter activity on the flux fragility and robustness coefficients for the vital (ATP synthesizing) flux of the Trypanosoma brucei glycolysis pathway. Robustness and fragility coefficients were calculated from the model published by Helfert et al. 
Doubled Glucose transporter activity:
Doubled Glctr activity:
Total of absolute values
The FRCs defined in this paper are system properties, not merely properties of the perturbed enzymes. This became strongly apparent upon further inspection of Table 3. Amplification of the glucose transporter did not only affect the flux robustness with respect to perturbation in this pathway step, but also robustnesses with respect to other even quite distant steps. Even for pyruvate kinase, at the other end of the pathway, the robustness coefficient was reduced (in silico) by a factor of almost 10 by the twofold overexpression of the glucose carrier.
When the flux robustness with respect to the glucose transporter was increased, the robustness with respect to pyruvate kinase decreased, and so did all the other MCA-based robustness coefficients. This phenomenon was reminiscent of the trade-off of robustness that is sometimes suggested: an increase in robustness with respect to one type of perturbation might necessarily be at the cost of a decrease in robustness with respect to other perturbations.
If this trade-off was symmetric, the sum of the FRCs should be constant. Then there would be no way to increase the overall robustness of a system. The bottom two rows of Table 3 show that this is not the case: both the total robustness and the sum of the absolute values of the FRCs decreased with the in silico amplification of the glucose transporter. Apparently, the trade-off in robustness is incomplete. Total robustness is not conserved and average robustness of a flux with respect to all enzyme perturbations can be increased (in this case by reducing the expression level of the glucose transporter genes).
A conserved property after all: fragility
Unlike MCA-based robustness, MCA-based fragility is a conserved property and the sum of the FFCs is unchanged upon amplification of the glucose transport, as shown in the fourth and fifth columns of Table 3 . MCA has a number of laws, including the summation law for flux control coefficients :
Because of the equality of fragility and control coefficients, this explains our observation that the sum of the FFCs always remains the same and thus MCA-based fragility is a conserved property.
A robustness screen of different metabolic models
To determine whether the above observed increased robustness in the networks we analysed might be a generic principle we examined all models in jws online that met the requirements for analysis by MCA. Some of these FRCs were again negative. However, since it is the magnitude of the coefficient that determines how robust a particular flux is to a given perturbation, the absolute values of the FRCs were considered. The dots in Fig. 1A give the absolute values of all the robustness coefficients for all the pathways in jws online (i.e. all ). Clearly, examination of the FRCs obtained for all the pathways in jws online reveals that in the vast majority of cases the individual FRCs were much larger than 1, also reflecting a small flux control by those enzymatic steps.
We also calculated the total robustness of the fluxes, which we defined as the sum of the absolute values of the MCA-based FRCs of the flux with respect to perturbations of all the individual enzymes, represented by the dots in Fig. 1B. These total robustnesses were almost all higher than 100. They tended to increase strongly with the size of the pathway, an increase that was much less clear for the robustnesses with respect to individual perturbations. However, the total robustness of a flux would be strongly affected by a single enzyme with a high robustness coefficient. Figure 1C shows that there is a very strong relationship between the highest FRC for a flux and its total robustness, bringing into question the validity of total robustness as a measure of a system property. For each pathway we calculated the mean of the FRCs for a particular flux that are between the 25th and 75th percentile of the FRCs when ranked in order of magnitude (i.e. those that lie within the interquartile range). Figure 1D shows a strong correlation between these interquartile range means and the total robustness, suggesting that, although the total robustness could be strongly affected by a single high robustness coefficient, it can still be used as an indicator of the robustness of the system. The means of the interquartile ranges of the FRCs were also generally much larger than 1.
Given the summation law for flux control coefficients, it is simple to demonstrate that MCA-based flux robustness is not conserved by considering a simple two-enzyme system. As this would be a linear pathway, the flux through both enzymes would be the same. Thus, given Eqn (4),
Therefore, since the MCA-based FRCs are the inverse of the MCA-based FFCs, the sum of the robustness coefficients of this system can be given by
Figure 2 shows a plot of the sum of the MCA-based FRCs for this system for varying values of , demonstrating that the sum of the FRCs is not a conserved property. Figure 2 also shows that, the more homogeneous the distribution of the FFCs, the less effect on the total robustness a small change to one of the enzymes would have. It is also worth noting that the total robustness is lowest at this point, suggesting that heterogeneity promotes robustness.
To further investigate the relationship between heterogeneity and robustness, we calculated the coefficient of variation for the robustness coefficients of each flux of the models from jws online, and plotted this against the total flux robustness. The coefficient of variation is a normalized measure of dispersion which expresses the standard deviation as a percentage of the mean (cv% = (σ/μ) × 100). Figure 3 shows a strong correlation between the heterogeneity of the FRCs and the total flux robustness.
Trade-offs exist but not between robustness and fragility
Having demonstrated trade-offs in MCA-based robustness and in fragility, albeit an incomplete trade-off in the case of robustness, we wanted to know if such trade-offs occurred between total robustness and total fragility. Since an increase in a particular FRC must be accompanied by a decrease in the corresponding FFC, one might expect a change in total robustness to be accompanied by a change in the opposite direction for total (absolute) fragility.
Examination of the results of the in silico models from jws online revealed that there is no trade-off between total robustness and total fragility, with cases being observed where the two measures changed in the same direction and in opposite directions. Table 4 shows the changes in the MCA-based FRCs and FFCs corresponding to the flux through hexokinase, and the total robustness and fragility of this flux, in the S. cerevisiae glycolysis pathway that result from doubling the activity of either hexokinase or pyruvate kinase. When the activity of hexokinase is doubled the total robustness increases while the total absolute fragility decreases, whereas when pyruvate kinase activity is doubled there is an increase in both total robustness and total absolute fragility.
Table 4. Example of increase in total robustness accompanied by a decrease in total fragility: effect of doubling hexokinase (HK)/pyruvate kinase (PYK) activity on the flux robustness and fragility coefficients for the flux through hexokinase of the S. cerevisiae glycolysis pathway, calculated from the Teusink model from jws online
Doubled HK activity:
Doubled PYK activity:
Doubled HK activity:
Doubled PYK activity:
Glycerol 3-phosphate dehydrogenase
Total of absolute values
It was also interesting to note that doubling of hexokinase activity resulted in a 5% increase in flux through hexokinase, thus demonstrating that increased total robustness does not necessarily come at the cost of performance, although this does not discount the possibilty of trade-offs in fragility or performance in some other part of the system.
Networking enhances robustness
Numerous studies have discussed the link between robustness and complexity, with increased robustness necessitating increased complexity [8, 12, 13, 20]. Although increased complexity can come in a number of forms, one of the most fundamental ways in which it can be achieved is through networking, with components interacting with each other to form a pathway. Given this, we hypothesized that networking would enhance MCA-based flux robustness. One would expect this to be the case since any perturbation to part of a network could be compensated for, at least partially, by the action of other components of the network. We can liken this to the difference between a single-engined aeroplane and a twin-engined one. If there was an interruption to the fuel line to the engine of the single-engined aeroplane then the reduction in fuel delivery would lead to a decrease in air speed, with only a certain reduction being tolerated before the air speed would be too low and the aeroplane would crash. However, if the same thing occurred in a twin-engined aeroplane then the effect of the fuel interruption on the air speed would not be so great since the other engine would still be functional, allowing a greater reduction in fuel delivery to be tolerated.
Any discussion of the effect of networking on robustness requires one to consider how to extend the definition of robustness presented here for individual step perturbations to the robustness of a pathway as a whole. This is not a trivial matter since to quantify robustness one needs to be specific about the type of perturbation and response in question. In this paper we have used the total and the mean of the absolute values of the MCA-based FRCs as summary measures of the pathway robustness. For the investigations into flux robustness presented here, these summary values provide one way of comparing the robustness of pathways with their fragility, and for examining the changes in robustness brought about by changes to parts of a pathway. However, such summary values raise the issue of what is considered to be a robust pathway. If we consider two unbranched 10-enzyme pathways, one having an equal distribution of control with FFCs of 0.1 for each enzyme, the other having one extremely fragile step with an FFC of 0.91 and the other enzymes all having FFCs of 0.01, then which would be considered more robust? Using the measures used in this paper of mean and total robustness, the latter pathway would be considered to be much more robust. We argue that this is a fair representation since there is only a 10% chance of a mutation affecting the fragile enzyme and, although such a mutation would have a large effect on the pathway flux, there is a much greater chance that the mutation would occur in some other part of the pathway and have very little effect on the flux. Thus, on average, the robustness of the second pathway to a random perturbation is higher. However, one could argue that in reality the first pathway is more robust as there is no particularly fragile step and, although perturbation of any enzyme would have a greater effect on the pathway flux than a perturbation of the non-fragile steps in the second pathway, there is no extreme fragility in the system (i.e. it is better to have a small but non-lethal effect resulting from each individual perturbation than to have a system where most perturbations have little effect but perturbations of the fragile step have a very large, possibly lethal effect).
Another issue with using the mean and total robustness as indicators of pathway robustness is that the presence of a near-equilibrium reaction, which would have a very small FFC and hence an extremely large FRC, would dominate in the calculation of the mean or sum of the FRCs. Let us consider a system in which control is evenly distributed amongst the majority of the enzymes but there is a single enzyme with a low FFC of 0.01 (). One could increase the total robustness of the system by reducing the activity of that enzyme to the point where it had an FFC of 0.001, thus increasing the total robustness by 900 while having little effect on the FRCs of the other enzymes in the pathway. Thus, the total robustness of the pathway would be much higher despite the distribution of fragility being very similar.
We have made some steps to address these issues with regard to the analyses performed here, demonstrating that there is a strong correlation between the total flux robustness and the mean of those FRCs that lie within the interquartile range of the robustness coefficients for each flux. This suggests that the trends we have discussed are unlikely to be affected by the presence of extremely high robustness coefficients for individual pathway steps. Although these are issues which need to be examined further, and highlight the importance of examining the individual FRCs when studying the robustness of a pathway, but we feel that the total and mean robustness are both adequate and suitable measures for the discussions presented in this paper. One of the functions of defining robustness, mean robustness and total robustness is that the limitations of the concepts also become clear.
Our analyses of the models from jws online, which are all multi-enzyme pathways, have shown that 97% of the individual MCA-based FRCs of these pathways have a magnitude of > 1, and in all cases tested the sum of the FRCs for a particular flux was > 1. This leads us to conclude that networking increases robustness in biological systems. Although in some networks not every robustness coefficient will be much greater than that of an isolated enzyme, one also has to take into account that, if the mutation/perturbation rate of the system is the same, then the chances of a particular enzyme being affected are reduced as pathway size increases.
We demonstrated mathematically that the total of the MCA-based FRCs of a two-enzyme system is always greater than that of an enzyme in isolation, as is the mean of the FRCs. However, while it is clear that moving from a one-enzyme system to a two-enzyme system increases MCA-based flux robustness, it is not so clear what the effect of extending the network further would be. There will be cases where a three-enzyme system is more robust than a two-enzyme system, and this will be largely dependent on the kinetic properties of the individual enzymes.
There are likely to be a number of reasons why a network would increase robustness compared with a process in isolation. The presence of other enzymes in the system means that a single enzyme is unlikely to have complete control over the flux and, as such, any perturbation to that enzyme would have less effect. The effect of any perturbation on a particular flux could be reduced through changes in other parts of the system brought about from the resulting changes in substrate or product levels. In addition, in networked systems there is the possibility of having more elaborate regulatory mechanisms such as multiple feedback loops which would act to compensate for changes in an enzyme's activity, or alternative pathways that allow the same function to be performed by different network components. The relative importance of factors such as these in determining exactly how networking increases robustness is a subject for further investigation.
Fragility is conserved, robustness is not
Trade-offs between robustness and fragility have been described in both biological and man-made systems [8, 12, 13, 20]. Csete and Doyle demonstrated that for simple feedback circuits net robustness and net fragility are constrained properties, with an increase in robustness at one frequency coming at the cost of increased fragility at other frequencies .
As such, one would expect similar trade-offs to be observed for MCA-based flux robustness with an increase in robustness of a flux to perturbations of one enzyme being paid for by increased fragility against perturbations of a different enzyme. We first note that this trade-off would be different since it is not a trade-off of the robustness against a perturbation at one frequency versus robustness versus perturbation at a different frequency. In the present paper we consider steady rather than oscillatory perturbations and the possible trade-off considered is between robustness against perturbations of one enzyme versus robustness against perturbations of the other enzymes.
Intuitively it is perhaps hard to say whether or not this would be the case as one can liken the problem to various examples for which such trade-offs may or may not occur. For example, one could liken this to an automobile engine. If the engine was repeatedly stressed by being used at maximum power then this could lead to a fault occurring and to a decreased performance or to failure. However, if the power of the engine were limited to a level below its maximum then it would be more robust and, although there would be a cost of decreased performance, there would not be any obvious reason for fragility to be increased. In fact, the robustness of other components in the system, such as the clutch and gearbox, would also be increased since they would also be under less stress. Alternatively, one might consider a reduction in performance to be an increase in fragility, as would be the case if the engine was powering a hovercraft and reduced performance meant that lift was reduced resulting in the craft striking more obstacles.
Since, when discussing how robust or fragile a system is, we are only really interested in the magnitude of the coefficients and not their sign, one might then expect to observe the trade-off between robustness and fragility that has been suggested for many other systems. Thus, one would expect any increase in the sum of the absolute values of the FRCs of an enzyme to be accompanied by an increase in the sum of the absolute values of its FFCs. However, our in silico experiments showed that some changes to a biological pathway can cause an increase in the sum of the absolute values of the FRCs for a particular flux that is accompanied by a decrease in the sum of the absolute values of the FFCs, and in some cases the rate of flux also increased (see Table 4). This contradicts previous studies which have stated that increased robustness must come at the cost of either increased fragility or reduced performance, although it is possible that such trade-offs may still be present but that they arise in a form that is not measured by the coefficients described here, e.g. through increased resource demand [1, 8, 20].
The summation laws of MCA allow us to state that MCA-based flux fragility is a conserved property. Therefore, any perturbation to an enzyme causing a change in one FFC is compensated for by an equal and opposite change in the other FFCs corresponding to that enzyme. However, despite this, one should be careful when saying that all systems are equally fragile, or that a system is equally fragile in the perturbation of different fluxes. Although this is true, it can also be confusing. This is because fragility coefficients can also have negative values and our intuition may not be fit to comprehend this phenomenon and its implications. An increase in the fragility of a flux with respect to changes in the activity of one enzyme could be counteracted by an increase in the magnitude of the fragility coefficient of that flux to a different enzyme which has a negative value. Thus, although the sum of the fragility coefficients would be the same, the sum of their absolute values would change, as can be seen in Table 3, and it could well be the absolute values of the FFCs that are of interest if one were interested in the size of the response to a challenge when discussing fragility.
Although Csete and Doyle showed the sum of robustnesses and fragilities across all frequencies to be conserved for systems with feedback loops , neither net robustness nor fragility were shown to be conserved properties. We have demonstrated that, for our MCA-based definition, net flux fragility is not only constrained, but is a conserved property, with the sum of the FFCs for a particular flux to perturbations in each of the enzymes in the system always being equal to unity (see Eqns (1) and (4)). MCA-based flux robustness is also a constrained property, in the sense that the sum of the inverses of the robustness coefficients for a particular flux must sum to unity across all enzymes, but we have shown that it is not a conserved property. A key difference between our work and that of Csete and Doyle is that we have defined robustness and fragility as two separate measurements of the properties of a pathway, whereas they only used a single measure of fragility/robustness with negative measures being classed as fragility and positive ones as robustness. Our findings also extend to all models of biochemical networks that reach steady state, not just those with feedback loops.
The majority of the pathways used in our analyses were metabolic pathways, although some signal transduction and enzymology pathways were included. However, since the robustness coefficients used here are calculated from the control coefficients of MCA, the robustness of any pathway for which MCA can be performed could be calculated using this method, including gene expression pathways. Since its introduction, a number of extensions to MCA have been developed, including hierarchical control analysis, which allows the control exerted by gene expression and signalling to be quantified , and methods for analysing control in time-dependent systems  including those exhibiting limit-cycle oscillations . Similar extensions could be made to the robustness coefficients defined here in order to analyse such pathways.
Retrieval of data and calculation of steady-state properties
Models were downloaded from jws online (see Table 1) in SBML format and matlab was used to analyse the models. The SBML Toolbox  was used to translate the SBML models into a matlab (Mathworks, Natick, MA, USA) object, allowing vectors of metabolites S, rate equations v and the stoichiometric matrix N to be obtained. This toolbox was also used to generate files containing the system of ordinary differential equations (ODEs) describing the reaction kinetics of each model. These were used as inputs for matlab's ode15s ODE solver for the calculation of the steady-state metabolite concentrations. The ode15s ODE solver was chosen as this implicit method can solve stiff systems of ODEs and has been shown to be optimal for solving the systems of ODEs of biological models . The metabolite concentrations obtained were substituted into the differential equations in order to ensure that a result of zero was obtained for each equation, and thus that the concentrations obtained were those of the system's steady state. If this was not the case then the iteration limits and number of iterations of the ODE solver were increased until either the steady-state concentrations were found or the model was considered not to reach a steady state and discarded. Steady-state fluxes were calculated by substituting the steady-state metabolite concentrations into the rate equations.
Perturbation of models
Perturbations to enzyme activity were simulated by multiplying the appropriate rate equations by the desired prefactor (e.g. 0.99 for a 1% decrease in enzyme activity). The ODE files were then regenerated and the new steady-state properties were calculated as described above.
This technique was used when calculating FRCs based on the perturbation required to obtain a 1% decrease in flux. To obtain a particular FRC, the prefactor for the rate equation of the enzyme in question was decreased until a 1% change was observed in the flux of interest. If the flux increased as enzyme concentration decreased then the FRC was negative.
Calculation of flux robustness coefficients using the MCA-based approach
The unnormalized elasticity matrix was obtained by calculating the partial derivatives of the vector of rate equations with respect to the metabolites ∂v/∂S. The steady-state metabolite concentrations were substituted into the unnormalized elasticity matrix and the metabolic control coefficients were calculated using the matrix inversion method described by Reder. This method uses a structural approach to calculate metabolic control coefficients from the stoichiometric matrix and the unnormalized elasticity matrix . In brief, the metabolic network toolbox for matlab (http://www.molgen.mpg.de/~lieberme/pages/mnt.html) was used to calculate the reduced stoichiometric matrix NR and link matrix L. The unnormalized metabolite concentration and flux control coefficient matrices, and , were then calculated as follows:
where In is an n × n identity matrix, n being the number of reactions.
The normalized flux control coefficient matrix CJ was calculated as described by Hofmeyr  :
with DJ being a diagonal matrix with the steady-state fluxes on its diagonal. The matrix of flux robustness coefficients, ℜJ, was then obtained by calculating the inverse of each element of the matrix of normalized flux control coefficients:
We thank the UK Biotechnology and Biological Sciences Research Council (BBSRC) and the Engineering and Physical Sciences Research Council (EPSRC) for financial support including for the Manchester Centre for Integrative Systems Biology (http://www.mcisb.org/) and the Doctoral Training Centre ISBML (BB/C0082191, EP/D508053/1, BG5302251, BBD0190791) and other grants (http://www.systembiology.net/support/). We are equally grateful to EU's FP7 programs including UNICELLSYS and SYNPOL for support, as to various Euopean funders for SYSMO and an ERASysBio+ grant. We thank the anonymous referees for suggestions.