Conceived and designed the experiments: NJ BOP. Performed the experiments: NJ. Analyzed the data: NJ. Wrote the paper: NJ BOP.
The authors have declared that no competing interests exist.
The study of dynamic functions of large-scale biological networks has intensified in recent years. A critical component in developing an understanding of such dynamics involves the study of their hierarchical organization. We investigate the temporal hierarchy in biochemical reaction networks focusing on: (1) the elucidation of the existence of “pools” (i.e., aggregate variables) formed from component concentrations and (2) the determination of their composition and interactions over different time scales. To date the identification of such pools without prior knowledge of their composition has been a challenge. A new approach is developed for the algorithmic identification of pool formation using correlations between elements of the modal matrix that correspond to a pair of concentrations and how such correlations form over the hierarchy of time scales. The analysis elucidates a temporal hierarchy of events that range from chemical equilibration events to the formation of physiologically meaningful pools, culminating in a network-scale (dynamic) structure–(physiological) function relationship. This method is validated on a model of human red blood cell metabolism and further applied to kinetic models of yeast glycolysis and human folate metabolism, enabling the simplification of these models. The understanding of temporal hierarchy and the formation of dynamic aggregates on different time scales is foundational to the study of network dynamics and has relevance in multiple areas ranging from bacterial strain design and metabolic engineering to the understanding of disease processes in humans.
Cellular metabolism describes the complex web of biochemical transformations that are necessary to build the structural components, to convert nutrients into “usable energy” by the cell, and to degrade or excrete the by-products. A critical aspect toward understanding metabolism is the set of dynamic interactions between metabolites, some of which occur very quickly while others occur more slowly. To develop a “systems” understanding of how networks operate dynamically we need to identify the different processes that occur on different time scales. When one moves from very fast time scales to slower ones, certain components in the network move in concert and pool together. We develop a method to elucidate the time scale hierarchy of a network and to simplify its structure by identifying these pools. This is applied to dynamic models of metabolism for the human red blood cell, human folate metabolism, and yeast glycolysis. It was possible to simplify the structure of these networks into biologically meaningful groups of variables. Because dynamics play important roles in normal and abnormal function in biology, it is expected that this work will contribute to an area of great relevance for human disease and engineering applications.
The network of interactions that occur between biological components on a range of various spatial and temporal scales confer hierarchical functionality in living cells. In order to determine how molecular events organize themselves into coherent physiological functions, in silico approaches are needed to analyze how physiological functions emerge from the evolved temporal structure of networks. Time scale decomposition is a well-established, classical approach to dissecting network dynamics and there is a notable history of analyzing the time scale hierarchy in metabolic networks and matching the events that unfold on each time scale with a physiological function
The dynamics of biological networks characteristically span large time scales (8 to 10 orders of magnitude), which contributes to the challenge of analyzing and interpreting related models. However, there is structure in this dynamic hierarchy of events, particularly in biochemical networks in which the fastest motions generally correspond to the chemical equilibria between metabolites, and the slower motions reflect more physiologically relevant transformations. Appreciation of this observation can result in elucidating structure from the network and simplifying the interactions. The reduction in dynamic dimensionality is based on such pooling and the analysis of pooling is focused in the underlying time scale hierarchy and its determinants. Understanding the time scale hierarchy and pooling structure of these networks is critical to understanding network behavior and simplifying it down to the core interactions.
Top-down studies of dynamic characteristics of networks begin with fully developed kinetic models that are formal representations of large amounts of data about the chemistry and kinetics component interactions. Network properties can be studied by numerical simulations (that are condition-specific) or by analysis (that often yield general model properties) of the model equations. Since comprehensive numerical simulation studies become intractable for larger networks and the identification of general model properties are needed for the judicious simplification of models, there is a need for analysis based methods in order to characterize properties of dynamic networks. In this study we present an in silico analysis method to determine pooling of variables in complex dynamic models of biochemical reaction networks. This method is used to study metabolic network models and allows us to identify and analyze pool formation resulting from the underlying stoichiometric, thermodynamic, and kinetic properties.
Linearizing the mass conservation equations for a chemical reacting system around the steady state yields the linear form of the dynamic mass balances,
We can apply a similarity transformation (see
The modal matrix separates the dynamics of the network into a series of dynamically independent motions
(A) The Jacobian acts as a linear operator mapping the dynamics onto the deviation variable. (B) The Modal Matrix maps network dynamics onto independent time scales. Panels C and D illustrate two approaches to understanding the interactions between metabolites on the different time scales. (C) Beginning from the fastest time scale and moving forward, components that move together on subsequent time scales are lumped into an aggregate pool variable. The pooling pictorially for three different time scales in glycolysis and the Rapoport-Leubering shunt in the red cell. The large blue dots indicate pool formation between two metabolites, signifying that these two metabolites become coupled or correlated on slower times scales. In this case, glyceraldehyde 3-phosphate and dihydroxyacetone phosphate pool together after the first time scale, the hexose phosphates pool together after the fourth time scale, and the triose phosphates pool together after the eighth time scale. (D) Each time scale is analyzed independently and the interactions are defined in terms of the coefficients in the model and their contribution to the cumulative sum of the modal coefficients. Analyzing all of the modes in this manner allows the identification of variables that are dominant across multiple modes and identifying the time scales across which they are most active. Four fundamental subspaces are associated with
A column of the modal matrix describes the participation of a concentration in each of the linearly independent modes. When two concentrations (
The set of dynamic equations are linearized about a particular steady state. Applying a similarity transformation to the Jacobian, enables the calculation of the modal matrix (depicted to the right). The rows of the matrix correspond to different time scales. When the ratios between two entries are constant, the two metabolites pool together. After the second time scale, metabolites
Employing a geometric interpretation for this determination, one can explicitly identify pool formation by calculating the angle between columns (
In order to identify the time scales at which pool formation occurs, we compute the angle between two columns as a function of an index
The pools can be described as a sum of matrix products over the time scales of the network:
One can quantitatively ascertain the contribution of each metabolite to each mode by rank ordering the normalized mode and keeping only the largest weights that add up to the specified cutoff percentage. At low cut-off ranges, all metabolites with small contributions to the mode will be zeroed out. The interactions across the modes can be mapped on top of the interactions defined by the stoichiometric matrix in order to compare and contrast the topological connectivity versus the dynamic connectivity at time scales of interest.
The models studied here exhibit a significant span of time scales (
The lower left triangle indicates the modes after which pooling occurs between the corresponding metabolites (one being the fastest time scale). The upper right triangle are plots of the slopes between the two metabolites for the remaining time scales after pool formation (the origin is always included in these approximations of the slopes), color coded according to the time scale at which pooling occurs. Some of the metabolites such as the phosphoglycerates have fairly constant ratios once they join aggregate pools. Others, such as ATP and ADP have varying ratios. These ratios change when interactions with other pathways dominate on subsequent time scales, for example when glycolytic intermediates dominate on one time scale and nucleotide salvage metabolites dominate on another, the respective interactions between ATP and ADP are affected differently. The cutoff value for cos(θ) was 0.9. Abbreviations: G6P, Glucose-6-phosphate; F6P, Fructose-6-phosphate; FDP. Fructose-1,6-bisphosphate; DHAP, Dihydroxyacetone phosphate; GAP, Glyceraldehyde-3-phosphate; DPG13. 1,3-bisphosphoglycerate; DPG23, 2,3-bisphosphoglycerate; PG3, 3-phosphoglycerate; PG2, 2-phosphoglycerate; PEP, Phosphoenolpyruvate; PYR, Pyruvate; LAC. Lactate; NADH, Nicotinamide adenine dinucleotide (reduced); GL6P. 6-Phospho-
System | Dimension | Rank | Pooling | Conservation Pools | Effective Dimensionality | Time Scale Span |
34 | 34 | Complete | 0 | 17 | 1.30E+10 | |
10 | 9 | Complete | 1 | 6 | 4.88E+04 | |
20 | 20 | Fragmented | 0 | 13 | 7.50E+06 |
Pooling of the tiled modal arrays can be classified as complete (in which all elements pools together eventually) or fragmented. The number of conservation pools is equal to the size of the left null space of the Jacobian. The effective dynamic dimensionality is the number of different time scales at which pooling occurs. The time scale span is the ratio of the largest to smallest eigenvalue for each of the networks.
Jacobian Analysis | Dynamic Simulations |
Generalized results | Conditions specific results |
Scaleable | Intractable as the number of variables increase |
A single set of calculations will characterize a particular steady state | Resource intensive, requires many simulations in order to characterize network pooling |
Linear regime near a particular steady state | May move from one steady state to another |
The approach presented here for analysis of the Jacobian in order to characterize network dynamics allows generalized, comprehensive elucidation of dynamics around a particular steady state directly and in a scaleable manner. In contrast, although the approach using dynamic simulations is not restricted to a particular steady state, it is resource intensive and for larger networks it is infeasible to characterize all of the different possible initial conditions, due to combinatorial growth of the possible combinations.
The lower left triangle of the tiled array indicates the time scale
The upper right triangle of the tiled array contains plots show the ratios of the modal coefficients (
The pooling structure observed in
Pooling on fast time scales define the chemical equilibrium pools and on slower time scales the physiological pools.
The time scale (
The thermodynamic equilibrium pools of the network can be seen in
Two striking features of the tiled array are (1) the pooling between the majority of the compounds occurs on the slowest time scales and (2) the slopes for many of these are horizontal or vertical lines, implying dynamically independent behavior. This dynamically independent behavior may result from a lack of connectivity (topological decoupling) or from independent kinetics (e.g., kinetic decoupling resulting from a zero order rate law). Thus, if compounds are detected to be topologically decoupled in the tiling array they are expected to dominate a particular mode.
A kinetic or topologically decoupled compound will undergo the largest changes in concentration and interactions with other compounds during those time scales on which it plays a dominant role the modes. After these time scales have been passed, concentration changes are likely to be less significant and the compounds could be viewed as joining an aggregate pool, but may not be in a fixed ratio as would be dictated through strictly thermodynamic or kinetic coupling.
Networks that are tightly connected in terms of stoichiometry and kinetics will result in complete pooling of all metabolites on the slowest time scales, which is seen in large part in the red cell (
The tiled arrays can be used to define the ‘effective dynamic dimensionality’ of the models by counting the number of different time scales during which two or more variables form an aggregate pool. For the networks considered, the effective dynamic dimensionality reduced the dimension by one-third to one-half (see
The tiled pooling array for folate metabolism was computed (
Abbreviations: 5MTHF, 5-methyltetrahydrofolate; THF, tetrahydrofolate; DHF, dihydrofolate; CH2F, 5,10-methylenetrahydrofolate; CHF, 5,10-methenyltetrahydrofolate; 10FTHF, 10-formyltetrahydrofolate; MET, methionine; SAM, S-adenosylmethionine; SAH, S-adenosylhomocysteine; and HCY, homocysteine.
The slowest mode in the folate network (∼30 minute time scale) is shown. The green and red shaded elements reflect the dynamic interactions between the metabolites on the 30 minute time scale (the colors reflect positive and negative entries, respectively). The blue lines indicate topological connectivity (i.e. from the stoichiometric matrix).
(A) A map of the folate network described by
The tiled pooling array for the yeast glycolytic pathway was computed (
Glycolytic intermediates and adenosine phosphates pool together on fast time scales. Fragmented pooling is also observed (i.e. there were 0 entries in the slowest mode, indicating that on the slowest time scale, all of the components in the network do not move together in a lumped pool). GLC, intracellular glucose; GLCX, extracellular glucose; G6P, glucose 6-phosphate; F6P, fructose 6-phosphate; FBP, fructose 1,6-bisphosphate; GAP, glyceraldehyde 3-phosphate; DHAP, dihydroxyacetone phosphate; GLYC, intracellular glycerol; GLYCX, extracellular glycerol; BPG, 1,3-bisphosphoglycerate; PEP, phosphoenol pyruvate; PYR, pyruvate; ACA, intracellular acetaldehyde; ACAx, extracellular acetaldehyde; EtOH, intracellular ethanol; EtOHx, extracellular ethanol; NADH, nicotinamide adenine dinucleotide (reduced form); ATP. adenosine triphosphate; ADP, adenosine diphosphate.
The glycolytic and redox potentials are similar to those in the red cell. The adenosine phosphate potential is only composed of ATP and ADP. The NADH/NAD ratio determines the redox interactions with glycolysis, glycogen, and conversion between acetaldehyde and ethanol.
Taken together, in this study we: (1) developed top-down approaches for the computationally driven delineation of pools, (2) showed how to distinguish between topological, kinetic and thermodynamic basis for pool formation, and (3) applied the methods to analyze the dynamic structure of metabolic network models in yeast and humans. The application of these methods enabled the simplification of the networks based on the dynamic pooling of metabolites on progressive time scales and the identification of the key metabolic interactions on the slower time scales.
There were some observations in the results worth pointing to suggest further areas worthy of investigation. The pooling ratios between metabolites are not always constant and metabolites that pool early on are more likely to have changing ratios on subsequent time scales. Furthermore, metabolites that are connected to multiple pathways are likely to have change ratios even after they begin pooling. This is observed for ATP and ADP in the human red blood cell (
A growing number of large-scale kinetic models of biochemical reaction networks are becoming available (e.g.
The pooling approaches developed here were based on identifying the dynamically independent times scales and their corresponding modes. The principle pooling approach considered here was based on a particular calculation given by the matrix product,
There has been an increased interest in the analysis of the intrinsic characteristics of the growing number of available large-scale kinetic biological network models
Recently, with the continued development of technologies and experimental procedures to calculate cellular fluxes using isotopomer data and to carry out quantitative metabolomic measurements on a larger scale
The method developed above was developed, tested, and implemented in Mathematica (Wolfram Research, Chicago, IL) version 5.2. The models analyzed herein: the model of human red cell metabolism
For each model, a stable steady state was identified by integrating the equations over time until the concentration variables no longer changed (error <1×10−10, see
Temporal decomposition was carried out as described in the
The calculations for the correlations across progressive time scales were carried out as described in
Values for the Gibbs standard free energies of formation for the metabolites in the human red cell model were used from
Illustrative example of time scale decomposition. An Illustrative example of modal decomposition for a toy network. The dynamics of 3 reactions involving 6 metabolites is analyzed in terms of the Jacobian matrix. Time scale decomposition is carried out along with simulations and determination of the pooling structure for this toy example.
(0.09 MB PDF)
Standard free energy of formation ratios for metabolites in the human red blood cell. The lower left triangle of the tiled array depicts a matrix with the ratios of the Gibbs free energy of formation between the metabolites in the red blood cell metabolic network. Ratios below 0.85 or above 1.15 were filtered out and not shown. The remaining entries (blackened entries) indicate expected pools if thermodynamic considerations alone determined the behavior of the network in a closed system.
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Steady state concentrations and fluxes for folate, yeast, and red blood cell.
(0.02 MB XLS)
We thank Dr. Christopher Henry for helpful correspondence and determination of Gibbs free energy of formation calculations for various metabolites.