Figure 1.
Workflow for Monte Carlo based model generation and the subsequent detection of patterns by decision trees.
First, a large number of SK-models is created based on randomly sampled parameter sets. They allow the detection of those parameter sets that lead to a stable or unstable steady state, respectively. Using the model parameters as feature vectors and the stability information as class labels, a classifier can learn those patterns in the parameter space with highest discriminatory power between both classes. These patterns then describe quantitative criteria for the degree of saturation of individual enzymes in the pathway that ensure stability or instability.
Figure 2.
The principles of structural kinetic modeling.
Normalization of the pathway-specific stoichiometric matrix with respect to steady state concentrations
and fluxes
produces the normalized matrix
. Together with the model parameters in the matrix
, it uniquely defines the Jacobian matrix of the system in the steady state. Evaluation of the eigenvalues of the Jacobian matrix then indicates whether the steady state is stable.
Figure 3.
Network underlying the SK-model of the CBC and related pathways.
Compounds written in italics represent external substances the concentrations of which are kept constant in the model. Dotted lines indicate the reactions of cofactors. Dashed lines connect metabolites that are assumed to be in equilibrium so that their concentration changes are directly proportional to each other. The proportions of the individual concentrations of these metabolites then depend solely on their equilibrium constants.
Figure 4.
Stable steady states for increasing values of regulatory TPT parameters.
Effect of increasing SK-model parameters for the triose-phosphate translocator (TPT) under different assumptions regarding regulatory mechanisms. Transporter-associated model parameters were sampled from consecutive intervals of length 0.1. For each interval, SK-models were generated.
Table 1.
The impact of regulation on plant energy metabolism.
Figure 5.
Isolated subnetwork after restriction to metabolites which are exclusively used by the CBC. Compounds written in italics represent external substances the concentrations of which are kept constant in the model. Dotted lines indicate the reactions of cofactors. Metabolites connected by dashed lines are assumed to be in equilibrium.
Table 2.
Classifier performance.
Figure 6.
Relative frequencies of metabolites in the patterns.
The boxplots show the occurrence frequencies of a) the metabolites and b) the reactions in the detected stability patterns with Laplace ratios in different cross-validation runs. Cytosolic reactions begin with a lowercase ‘c’.
Table 3.
CBC enzyme occurrences in the derived patterns.
Table 4.
Enzyme occurrences in the derived patterns (ATP, starch and sucrose metabolism).
Figure 7.
Example patterns with a) two or b) three stability conditions each.
For each enzyme-metabolite pair, a threshold for the saturation is given. Enzymes are marked in red, their reactants are marked in green. Pattern 1 exhibited an average Laplace ratio of
(
,
). Pattern 2 affected less training samples because of its less strict threshold on the parameter associated with PGK and GAPDH but also produced more training errors and a lower Laplace value (
,
). Pattern 3 affected an even larger number of hits but nevertheless, it produced fewer training errors than pattern 2. As a consequence, it exhibited the highest Laplace value of all the depicted patterns (
,
,
).