A Metabolic Entropy Approach for Measurements of Systemic Metabolic Disruptions in Patho-Physiological States Kirill A. Veselkov,† Valeriy I. Pahomov,‡ John C. Lindon,† Vladimir S. Volynkin,‡ Derek Crockford,† George S. Osipenko,‡ David B. Davies,† Richard H. Barton,† Jung-Wook Bang,† Elaine Holmes,† and Jeremy K. Nicholson*,† Biomolecular Medicine, Department of Surgery and Cancer, Faculty of Medicine, Imperial College London, Sir Alexander Fleming Building, London SW7 2AZ, United Kingdom, and Department of Physics, Sevastopol National Technical University, Streletskaya Bay, Crimea, Ukraine, 99053 Received January 21, 2010
Multicellular organisms maintain the stability of their internal environment using metabolic and physiological regulatory mechanisms that are disrupted during disease. The loss of homeostatic control results in a complex set of disordered states that may lead to metabolic network failure and irreversible system damage. We have applied a new statistical entropy-based approach to quantify temporal systemic disorder (divergence of metabolic responses) in experimental patho-physiological states, via NMR-spectroscopy generated metabolic profiles of urine. A recovery (R-) potential metric has also been developed to evaluate the relative extent to which defined metabolic processes are perturbed in the context of a global system in terms of multiple changes in concentrations of biofluid components accompanying the disrupted functional activity. This approach is sensitive to physiological as well as pathological interventions. We show that global disruptions of metabolic processes, lesion reversibility, and disorder in metabolic responses to a stressor can be visualized via metabolic entropy metrics, giving insights into biological robustness and thus providing a new tool for assessing deviation from homeostatic regulation. Keywords: metabonomics • NMR spectroscopy • metabolic entropy • systemic disruptions • biological robustness • hydrazine • arginine • R-naphthylisothiocyanate • caloric restriction • pancreatic toxicity • hepatotoxicity
Introduction Complex organisms possess multiple cellular, tissue and organismal mechanisms that provide inherent flexibility to manage topographically localized and systemic perturbations of physiology and metabolism.1,2 This flexibility is manifested in “biological robustness”, which characterizes the propensity of a living system to maintain effective functioning in the face of internal and external stressors and perturbations.2,3 In any disease state there is some loss of homeostatic control that generally results in disordered metabolic activity4-7 and system recovery depends on the capacity to adapt to the imposed metabolic stress. The disease processes and underlying biological robustness must, therefore, be understood in the context of the “global” system, in which metabolic networks, distributed in space and time across many cell types, interact cooperatively to maintain functional integrity of the whole system. Typically, systems biology approaches such as control-driven5,8 and kinetics-based9 methodologies require comprehensive information on metabolic processes, such as topologies of metabolic networks, flux distributions, metabolite concentrations and * To whom correspondence should be addressed. E-mail: j.nicholson@ imperial.ac.uk; phone: +44 207 594 3225; fax: +44 207 594 3226. † Imperial College London. ‡ Sevastopol National Technical University. 10.1021/pr1000576
2010 American Chemical Society
enzyme catalytic properties to allow meaningful interpretation. As a result of the inherent complexity of mammalian metabolism, these approaches are best suited to understanding in vitro cellular metabolic behavior.10-12 However, overall disease processes are system-level phenomena involving multiple cell types and the emergent properties of a system’s robustness are manifested at increasing levels of bio-organization. Cell-based metabolic failure models cannot capture this complexity.13 We have focused on evaluating integrated system homeostatic activity and on developing global measures of perturbations that occur in physiological and pathological states. First, we demonstrate that homeostatic regulation of healthy animals leads to constrained metabolic functional activity behavior.14 Second, we introduce the “R-potential” function to provide a measure of deviation from, and recovery of, homeostatic regulation of generalized metabolic processes following stressor-induced perturbation. Finally, we propose that the concepts of configurational and relative entropies may be employed, respectively, to assess the extent of scatteredness (diffusiveness) of metabolic phenotypes and their collective divergence from homeostasis in unperturbed and perturbed system states. We have used sequential, timed 1H NMR metabolic profiles of urine samples15 from rats exposed to a model steatotic hepatotoxin (hydrazine), to a cholestatic hepatotoxin (RJournal of Proteome Research 2010, 9, 3537–3544 3537 Published on Web 04/27/2010
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Veselkov et al. context of a global system in terms of multiple changes in relative concentrations of biofluid components. It is calculated based on changes in a subset of metabolic markers that characterize the perturbed functional activity of an organism, for example, a combination of changes in the urinary levels of trimethylamine-N-oxide, N,N-dimethylglycine, dimethylamine, and succinate indicate renal papillary damage.22 The homeostatic response using this approach is qualitatively compared to nonmechanical work performed on a system by a driving force in a thermodynamic process to attain a given state in order to consider that there is certain metabolic cost associated with function recovery. With multiple deviations of solution components, the work is performed until the difference in the chemical potentials ∆µi of each component is zero,23 that is, the component concentration becomes equal to (or comparable with) the homeostatic value
Figure 1. 1H NMR spectroscopic analysis of metabolic fingerprints. Typical NMR spectra of urine collected two days after initiation of the study from Sprague-Dawley rats subjected to vehicle control, to high dose of steatotic hepatotoxin hydrazine, to high dose of cholestatic hepatotoxin R-naphthylisothiocyanate (ANIT) and to complete caloric restriction for 24 h.
naphthylisothiocyanate (ANIT)), to arginine-induced exocrine pancreatitis and to a more subtle physiological perturbation in caloric-restricted rats (Figure 1). Metabolic profiles of urine are particularly informative outputs of diverse regulatory/ signaling events from an organism that are often disrupted during disease processes.16 The urine samples of all studies were collected over time-periods of 24 h capturing the functional activity of an organism during that time where diurnal and functional changes on a shorter time scale are integrated. Hydrazine-induced toxicity is of particular interest in this study because, in addition to its hepatotoxic effects,17 it is neurotoxic and has effects on several major biochemical pathways, thus modeling a complex disease process.18 In fact, most drugs and toxic agents exert multiple target organ effects depending on the dose and timing of treatment.16
Methods Data Processing. The details of experimental design, sample collection and spectroscopic data preparation are provided (Supporting Information Appendix, “Experimental design and metabolic spectroscopic data preparation”). All data sets were normalized to account for the variable sample dilution by applying the probabilistic quotient normalization method.19 Some NMR signals exhibited positional variations across spectral profiles, and these variations were removed using a recursive segment-wise peak alignment.20 A curve-fitting method was applied to quantify the relative concentrations of metabolites from 1H NMR profiles based on nonoverlapped metabolite peaks as derived from spectra of pure components.21 The originally proposed algorithm was accelerated to handle large number of samples by means of Fast Fourier transform cross correlation. The robustness and reproducibility of quantification of defined biomarker changes via the curve fitting algorithm has been evaluated elsewhere with respect to the toxicological data sets used in this paper.21 Definitions of the R-Potential and Entropy Measures. R-Potential. The R-potential is a measure of the extent to which defined metabolic functions are perturbed by a stressor in the 3538
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Anonmechanical )
Ci
∑ ∆µ ≈ ∑ log C i
i
i
(1)
i0
where Ci and Ci0 are biofluid concentrations of system component ith in “perturbed” and “unperturbed” states, respectively. The qualitative thermodynamic analogy is used to transform concentration changes of solution components into an additive logarithmic scale. This enables characterization of disrupted functional activity by the sum of changes in concentrations of multiple solution components. Assuming that metabolic activity is perturbed irrespective of the direction (up or down) of deviation in concentrations of solution components from homeostasis, the recovery R-potential is defined as
R)
∑ w |log C
Ci
i
i∈M
i0
|
(2)
where M denotes a subset of metabolites characterizing the perturbed function, | denotes the absolute value, wi is a weighting factor for the ith metabolite being inversely proportional to the variability of its log-transformed relative concentration in an unperturbed state, calculated by the four times of the standard deviation of metabolite log-transformed relative concentration, Ci0 and Ci are the ith metabolite concentrations in the unperturbed and perturbed states, respectively. The former is defined to be homeostatic stability (i.e., mean predose). It is assumed that metabolites exhibiting large variability in an unperturbed state contribute to a smaller extent in the R-potential measured perturbation of metabolic function. For comparison purposes, the R-potential values were normalized (divided) by the number of metabolites used. An increase of the R-potential is indicative of perturbed functional activity. The direction of perturbed pathway activity is not considered in the R-potential composite index since pathway disruption is generally accompanied by both increases and decreases in metabolite concentrations. The type of biofluid and biomarkers used to characterize perturbed metabolic functions need to be specified for this measure, using a data-generated hypothesis or prior biological knowledge. Configurational Entropy. The configurational entropy is a measure of the uncertainty of a system state in terms of the extent of scatter (diffusiveness) of metabolic phenotypes (within a group of subjects). The differential statistical entropy of d-dimensional random variable X is
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Measurements of Systemic Metabolic Disruptions H(X) ) -
∫ f(x)log f(x)dx
(3)
where X represents a set of metabolic phenotypes of subjects in a given experimental condition with unknown density function f(x). The entropy is the average value of -log f(x) and having unbiased estimators log f(xi) leads to an unbiased estimator of entropy. The former can be estimated by considering the probability distribution for the distance between xi and its k-th nearest neighbor sample points.24 The resulting k nearest neighbor (k-NN) estimate of differential entropy using a random sample (x1, x2, x3 ... xn) of n realizations of a d-dimensional variable X with unknown density function f(x) can be expressed as follows
Hk ) ψ(n) - ψ(k) + log(cd) +
d n
n
∑ log ε
k i
sample points in multidimensional metric space. A higher entropy value indicates a larger scatter (diffusiveness) of metabolic phenotypes within a group of animals in metabolic space leading to a larger product of all pairwise sample distances. As a consequence, there is a higher probability of observing a given combination of concentrations and identities of biofluid metabolites in a given experimental condition just by chance (see Supporting Information for calculations of entropy measures). Relative Entropy. The relative entropy is a measure of the extent of the uncertainty of metabolic behavior with respect to homeostasis in terms of collective divergence of metabolic phenotypes in a perturbed state from metabolic phenotypes in an unperturbed state. The relative entropy (Kullback-Leibler divergence) of ddimensional random variable X is
(4)
i)1
D(X) ) Hk is the nearest neighbor entropy estimator, xi is the ith metabolic phenotype (e.g., 1H NMR urine sample) of an animal in an experimental condition, εik is the distance between the xi sample and its k-NN sample in some d-dimensional metric space, n is the overall number of samples, ψ(a) is the digamma function and cd is the volume of the d-dimensional unit ball depending on a metric space (e.g., cd ) πd/2/Γ(1 + d/2) for the Euclidean norm used in our application). The derivation of k-NN entropy estimators of differential entropy, and the proof of their asymptotic unbiasedness and consistency has been provided independently by several authors elsewhere.24,25 The mean over all k nearest neighbor estimators has been shown recently26 to possess desirable properties in terms of smaller estimator variance and so was used in this application. n-1
Hmean )
∑
1 H ) const(n, d) + n - 1 k)1 k
∑
d log ||xi - xj || (5) n(n - 1) i,j;i*j where | | denotes a distance (Euclidean in our application) between two metabolic samples. To ensure comparable contribution of minor and major spectral intensity variables into the distance measure, the metabolic profiles were standardized by applying the logarithm transformation as follows (see Supporting Information for further details on the entropy calculations) ||xi - xj || )
∑
(log(xiS/xoS) - log(xjS/xoS))2
(6)
S
where xo is the mean predose biofluid profile. A similar distance measure has recently been shown to possess a comparative advantage in reflecting the regulatory similarity in gene expression studies.27 For comparison purposes, the entropy value at the predose time point was subtracted from all estimated entropy estimates and these values were divided by the number of variables used (spectral data points). In this application, this removes the bias adjustment constant, which depends on sample size and dimensionality, yielding changes in the mean nearest neighbour entropy estimate that are proportional to the logarithm of the product of all pair-wise distances between
dx ∫ q(x)log q(x) f(x)
(7)
where q and f are probability density functions of metabolic phenotypes (NMR biofluid profiles) in perturbed and unperturbed states (conditions), respectively. Using random samples (x1, x2, x3 ... xn) and (y1, y2, y3 ... ym) of perturbed and unperturbed states, respectively, the k-NN relative entropy estimator26 is
Dk )
d n
n
k vm (i)
∑ log F (i) + log n m- 1 i)1
(8)
k n
where vkm(i) and Fnk(i) is the distance of unperturbed metabolic phenotype (biofluid sample) xi to its k-th nearest neighbor sample in perturbed and unperturbed states, respectively. The mean nearest neighbor estimator26 (when m is equal to n-1) is Dmean )
d ( n(n - 1)
∑ log ||x - y ||- ∑ log ||x - x ||) i
i,j
j
i
j
i*j
(9) where n is the sample size, d is the number of dimensions (spectroscopic data points). For comparison purposes, the values of relative entropy were divided by the number of variables used. The relative entropy is estimated by comparing the product of pairwise distances between metabolic phenotypes of an unperturbed (healthy or control) state with the product of pairwise distances between metabolic phenotypes of perturbed and unperturbed states. A higher value of relative entropy indicates an increased uncertainty of metabolic behavior with respect to homeostasis resulting in a larger product of all pair wise distances between metabolic samples of unperturbed and perturbed states.
Results and Discussion Homeostasis-Based Approach to Disease Modeling. Metabolic network activity requires tight coordination of energy, anabolism and catabolism to satisfy the dynamic physiological needs of an organism.1,22 Homeostatic regulation inevitably imposes constraints on the pathway flow of metabolites during normal functional activity, as illustrated by a subset of functional end points exhibiting relative stability in control animals (Figure 2A). In the face of toxin-induced metabolic stress the Journal of Proteome Research • Vol. 9, No. 7, 2010 3539
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Figure 2. System metabolic responses to various perturbations. Visualization of the system-level metabolic responses as a surface map for (A) the control case and (B) the hydrazine high dose case, (C) 2D representation of the 3D surface map only for hydrazine high dose where the color map corresponds to a gradient of upward (red) and downward (blue) deviations from homeostasis, expressed as the logarithm of time-related changes of a metabolite concentration relative to its regulated (predose) value. (D) Normalized R-potential time-related changes of generalized TCA metabolic function for various treatments. The R-potential data points are summarized by means of box plots; the box edges are the 25th and 75th observation percentiles, whiskers extend to the most extreme observations and notches indicates the confidence interval around the median.
activity of multiple pathways may be affected at different sites leading to considerable changes in metabolic phenotypes of an organism. This is illustrated by the changes of a subset of biofluid metabolites, known to alter during hydrazine toxicity,28,29 that are related to perturbations in various functional processes targeted by the toxin (Figure 2B). The toxicity or disease is usually a consequence of disrupted pathway activity that propagates to a higher level of bio-organization by overwhelming various adaptive mechanisms and manifests itself at the level of tissues/organs.4,30 R-Potential: The “Metabolic Cost” Associated with the Recovery of Stressor-Induced Metabolic Disruptions. Both during and after stressor-induced injury an organism responds actively by attempting to compensate and to adapt. The recovery of homeostasis comes at the expense of various 3540
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functional resources, involving energy expenditure, orchestrated by sophisticated regulatory activity, that is, there is a metabolic cost associated with the recovery of normal functional activity.13 The homeostatic response can thus be qualitatively compared by analogy with the nonmechanical work performed on a system by a driving force in a thermodynamic process to attain a given state, for example, to regulate a chemical reaction in the direction satisfying metabolic demands of various functional processes. As a result, a representative R-potential metric has been introduced to assess the extent to which defined metabolic functions are perturbed in the context of a global system (see Methods). The R-potential is calculated on the basis of a subset of metabolites in defined metabolic processes such as found in the tricarboxylic acid cycle (TCA) or urea cycles since changes
Measurements of Systemic Metabolic Disruptions in concentrations of multiple pathway metabolites in biofluids generally accompany the disruption of metabolic activity. This is due to the complex architecture of metabolic regulatory networks, requiring coordination of activities and concentrations of multiple enzymes for the regulation of metabolic processes.5,8 On the basis of simultaneous changes of a group of key metabolites such as citrate, 2-oxoglutarate, fumarate, succinate etc., it can be concluded that a stressor causes systemic disruptions of TCA metabolism, which clearly has implications for the global energy status of an organism. Indeed, hydrazine can irreversibly react with 2-oxoglutarate to form 1,4,5,6-tetrahydro-6-oxo-3-pyridazine carboxylic acid,29 while ANIT can alter mitochondrial membrane depolarization31 and thus compromise the mitochondrial function directly. To estimate the extent to which defined metabolic functions are disrupted by a stressor, the R-potential was calculated using the changes in concentrations of urine biomarkers, identified and interpreted in the context of the relevant metabolic processes in our previous studies,29,32,33 such as fumarate, 2-oxoglutarate, succinate and citrate representing the TCA metabolism disrupted in all case studied; taurine and creatine representative of general hepatotoxicity;34 2-aminoadipate, which is a key product of lysine catabolism, and N-acetylcitrulline and argininosuccinate representing the urea cycle disrupted by hydrazine toxicity, as shown in Figure 2C.28,29 As expected, the time-related changes of the R-potential over the study period are essentially constant for the control case implying the stability of functional activity (Figures 2D and S1, Supporting Information). The R-potentials calculated for the effects of toxins and caloric restriction change with time and their relative values indicate the extent to which defined processes are disrupted. At doses selected to cause no or little toxicity, defined by classical histopathological assays (Supporting Information, Tables S1-S4) the initial progressive disruptions of TCA metabolism and other characterized metabolic functions recover promptly, that is, the self-regulatory mechanisms of an organism enable it to overcome stressor-induced metabolic stress efficiently (Figures 2D and S1, Supporting Information). The extent of these perturbations is comparable to the effect of caloric restriction, where metabolic pathways can be modulated in a way that extends the lifespan of animals.35 The global activity of energy metabolism clearly exhibits adaptive changes in response to caloric restriction. After initial progressive changes over two days, it achieves stability in animals undergoing mild and severe caloric restriction over the course of the study or subsequent recovery to normality in animals subjected to a complete caloric restriction for the first 24 h (Figure 2D, caloric restriction). By contrast, the overt toxicities caused by high doses of all toxins studied (Supporting Information, Tables S1-S4) results in progressively more severe and pronounced disruptions of TCA metabolism that are nonreversible within the 7 day time course in the case of hydrazine and arginine toxicities (Figure 2D). The more severe disruptions and slower recovery are also evident for other metabolic functions in the high-dose case of hydrazine (Figure S1 Supporting Information), though the characteristics and magnitude of the reversibility of dysfunction is pathway or process specific. The latter confirms the modular organization of metabolic networks. For example, perturbations of lysine catabolism and the urea cycle recover in about 7 days, whereas the more general liver dysfunction and dysregulation in energy metabolism exhibit slower and incomplete recovery in the high-dose case of hydrazine (Figure S1 Supporting
research articles Information). Disease mechanisms and processes are commonly conceptualized at cellular or biochemical pathway level, but the impaired recovery of generalized metabolic functions is an indication of compromised robustness of the whole system. Hydrazine is a good example of a toxin, which has a primary metabolic effect in the liver, disrupting metabolism of lysine leading to accumulation of 2-aminoadipate, which in turn has a secondary effect in the hippocampus of the brain by blocking the conversion of kynurenine to kynurenic acid.29 The metabolic disruptions need to be considered in the physiological context of a dynamically interacting multiorgan system, which can never be interpreted using in vitro approaches. Metabolic Entropy Measurements of System Level Disorder in Physiological and Pathological States. As a result of minor genetic and physiological differences among the animals in all cases studied in this work, homeostatic regulation is also expected to result in a variety of constrained metabolic phenotypes (concentrations and identities of biofluid metabolites) within a group of animals in an unperturbed (healthy) state. However, metabolic responses to a stressor can be substantially altered by the slight difference in the activity of drug metabolizing enzymes, for example, which cause divergence in subject-specific systemic metabolic behavior. Such sensitivity to initial conditions is a characteristic of unpredictable (“disordered”) behavior of a system36 and it needs to be assessed as an additional feature of stressor-induced metabolic injury. We have thus introduced the concept of metabolic configurational entropy to measure the scatter (diffusiveness) of metabolic phenotypes within a group of animals and the concept of relative entropy to characterize the collective divergence of their metabolic phenotypes from homeostasis (see Methods). Both entropies are estimated based on a pairwise distance configuration of sample points (standardized 1 H NMR biofluid spectra) in multidimensional metric space. This configuration can be partially represented in a twodimensional space by means of distance-preserving mapping (e.g., Sammon mapping37), which minimizes the information loss of pairwise distance configuration of sample points, when samples are mapped to a lower dimensional space (Figure 3A). Under the given experimental conditions, homeostatic regulation substantially constrains a variety of identities and/or concentrations of biofluid metabolites of control animals whose sample points are located in a closed proximity to each other in space over the time course of the study (Figure 3A). The corresponding values of configurational and relative entropies exhibit changes in well-defined limits (Figures 3B, C, S3 and S4, Supporting Information). This indicates that the adjustment of metabolism in response to various external factors (e.g., daily fluctuations in nutrient uptake) occurs rapidly, in a matter of minutes or hours, resulting in the relative stability of integrated functional end-points of control animals over a longer (24 h) time scale. In the face of minor environmental adversities (low dose toxins), initial divergence of metabolite phenotypes from homeostasis, captured by the relative entropy time course, recovers promptly (Figures 3B and S4, Supporting Information). In the low-dose case of hydrazine a relatively higher impact on metabolism was observed with the development of minimal steatotic changes in hepatocytes of several animals and slower system recovery. The extent of this divergence is however still comparable to the pure physiological effects of caloric restriction (Figure S4, Supporting Information). Moreover, the associated time-related changes of configurational entropy imply little or no scatter in the metabolic phenotypes induced by Journal of Proteome Research • Vol. 9, No. 7, 2010 3541
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Figure 3. Configurational and relative entropies. (A) Sammon’s mapping of metabolic phenotypes (log transformed standardized 1H NMR spectra of urine samples) of animals into a 2-dimensional space with minimal loss of pairwise distances between sample points. Each point represents a sample of an individual animal at a given time point. (B) Time-dependent changes of relative entropies. (C) Time-dependent changes of configurational entropies. The variability of entropy estimators was assessed by leaving one observation out and recomputing the entropy statistic (resampling).
caloric restriction and low doses of toxin (Figures 3B, S2 and S3, Supporting Information). There is growing evidence that such coordinated adaptive responses to physiological disruptions can even condition the affected biological system to possess an enhanced resistance to a subsequent overwhelming toxic challenge.38,39 The administration of high doses of toxin results in progressively more severe metabolic disruptions that eventually overwhelm the homeostatic capacity, leading to pathological changes in all cases studied (Supporting Information, Tables S1-S4). Arginine is a good example of a conditionally essential amino acid playing a major role in a variety of mammalian metabolic pathways. Despite its physiological role, a single large intraperitoneal dose of arginine exceeds the systemic reserve capacity for metabolism and induces acute necrotizing pancreatitis in rats.40 Such disruptions compromise system robustness and necessitate mobilization of additional functional resources orchestrated by more sophisticated regulatory strategies requiring a longer recovery time. The time-related changes of relative entropy show that onset and recovery phases of arginine-induced toxicity are followed by irreversible perturbations of metabolism over the time course of the study (Figure 3B). The latter reflects consequences of impaired pancreatic function.33 Hydrazine toxicity induces irreversible system damage and impairs recovery processes, while complete recovery occurs relatively slowly over six days after the high-dose of ANIT (Figure S4, Supporting Information). All disruptions induced by high doses of toxins are accompanied by substantial divergence in subject-specific systemic behavior, that is, an increased variety of metabolic phenotypes in perturbed states (Figures 3C and S3, Supporting Information). The effects of strong perturbations can thus be sensitive to minor differences in genetic background and/or physiology of animals and be less predictable. Intriguingly, subject-specific systemic divergence occurs during the progres3542
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sion and recovery phases of toxicity indicating the differential susceptibility of animals to the intoxification process and the varied capacity to recover the metabolic disruptions. The actual pathology (maximal toxin-induced injury) however appears paradoxically to result in a highly constrained state with a reduced variety of metabolic phenotypes (something that we have termed metabolic focusing) and levels of configurational entropy comparable to the control case, as illustrated by ANIT and Arginine toxicity examples (Figure 3C, ANIT at 48 h postdose and Arginine at 24 h postdose). This is because in pathological states metabolic biochemistry is “focused” by the severity of the lesion, dominated by cell death and necrosis in the target organ. In other words, common major metabolic effects result from specific massive damage, occurring across a group of animals.
Conclusions Considering metabolism as a regulated homeostasis-oriented system, a new strategy for modeling disease has been introduced that is based on measuring systemic disruptions and associated “metabolic cost” to recover metabolic functions. As pathological conditions usually have multiple targets and mechanisms, the R-potential measure can be used to evaluate the dysregulation and recovery of one or more specific functional processes. The major advantage of calculating the R-potential is that, in the context of a global system, it can highlight the relative extent to which defined processes (for example, homeostasis in specific organs or pathways) are perturbed by a stressor. The R-potential can also facilitate evaluation of the stage at which the level of disrupted activity of metabolic pathways exceeds the homeostatic recovery capacity. In turn, a clinically significant recovery can be accompanied by the reduction of the R-potential values toward the normal values, without necessarily reaching the unper-
Measurements of Systemic Metabolic Disruptions turbed levels of metabolites. It is well established that metabolic profiling of urine and plasma samples can generate biomarkers of many types of organ dysfunctions.22,41-43 Urine is a particularly useful biofluid because it captures effects integrated over several hours and reflects homeostatic responses noninvasively. The proposed measure of configurational entropy is useful in quantifying the degree of scatter of metabolic phenotypes or divergence in subject-specific behavior in unperturbed and perturbed states. Disruptions that are sensitive to minor difference in genetic background and/or physiology may lead to unpredictable metabolic outcomes. In addition, the relative entropy is useful in assessing the degree of collective divergence of metabolic phenotypes of animals from homeostasis and reveals recovery of integrated systems. Using this approach, we have shown that pathological processes cause progressively severe disruptions of metabolism, impaired recovery of generalized metabolic functions and substantial divergence in subject-specific behavior despite minor differences in genetic background/or physiology of animals. Biochemical measurements on in vivo systems are always incomplete in the sense that it is never possible to measure all metabolite concentrations or interacting pathways of tissues and cells in real time.12,22 Nonetheless, by taking a “top-down” systems biology approach using time-resolved metabolic signatures of biofluids such as urine, we have provided a means to assess the general dysregulated behavior of metabolism and compromised system robustness during toxicity episodes. For an organism with an impaired capacity to recover, even a small additional disruption can lead to a catastrophic system failure.44,45 This entropic approach to medical systems biology leads to an enhancement of the concept of biological robustness to augment existing network methods for visualizing metabolic disorder during diseases. In summary the approaches outlined here using the R-potential and the concept of metabolic entropies should be widely applicable in metabonomic studies of systems biology dysfunction and their recovery under therapeutic regimes and may have a value in other postgenomic disciplines.
Acknowledgment. We thank Technologie Servier, Orléans for financial support. The members of the COnsortium for MEtabonomic Toxicology (COMET) are acknowledged for providing the data sets used in the manuscript. Supporting Information Available: Supplementary Figure 1, the normalized R-potential time-related changes of generalized metabolic functions disrupted by hydrazine; Supplementary Figure 2, the time related changes of configurational entropy; Supplementary Figure 3, the time related changes of configurational entropy for various treatments; Supplementary Figure 4, the time related changes of relative entropy for various treatments; Supplementary Table 1, average serum clinical chemistry results for hydrazine administration; Supplementary Table 2, a summary of histopathological analysis for hydrazine; Supplementary Table 3, average serum clinical chemistry results for ANIT administration; Supplementary Table 4, a summary of histopathological analysis for arginine. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Frayn, K. N. Metabolic regulation: a human perspective, 2nd ed.; Blackwell Science: Oxford, 2003.
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