Figure 1.
A pathway map of L. lactis central metabolism.
The pathway involves 21 metabolic interconversions between 24 metabolites and includes three conserved moieties and two internal metabolites whose concentrations are assumed to be constant (FMT, CoA). External metabolites are indicated by light blue boxes. The inset provides an overview of the regulatory interactions. Abbreviations are defined in Text S2.
Figure 2.
Probabilistic Modeling of L. lactis.
The topological properties of the pathway, including the stoichiometry and all known regulatory interactions, are assembled. The analysis is then based on knowledge of a metabolic state of the system, as defined by a steady-state flux distribution and a set of associated thermodynamically consistent metabolite concentrations. Based on this information, the state-specific dynamic properties of the corresponding pathway model are evaluated. Of particular interest are control coefficients, the role of regulatory interactions, as well as the dynamic response to periods of starvation.
Figure 3.
Probabilistic distribution of flux control coefficients.
Shown is the distribution of the scaled flux control coefficients corresponding to the pathway model of L. lactis central metabolism given in Figure 1. Each plot corresponds to the interval [−1,1] on the abscissa. The diagram in the th column and on the
th row gives the distribution of the control coefficient quantifying the extent to which enzyme
controls the flux through the reaction j. Each distribution provides information about the magnitude and uncertainty of one control coefficient. Narrow distributions indicate control coefficients that do not change appreciably due to parameter sampling, whereas broad distributions indicate that the precise value of the coefficient is more strongly dependent on parameter values. The corresponding sign distribution is shown in Figure 4.
Figure 4.
Probabilistic sign distribution of flux control coefficients.
Grey-scale representation of the sign distribution of the flux control coefficient shown in Figure 3. The shade of the entry represents the percentage of the calculated control coefficients that are positive. Dark colors correspond to a distribution of flux control coefficients that lies predominantly on the negative semiaxis, whereas light colors indicate that the sampled control coefficients are predominantely positive. For example, for this metabolic phenotype, an increase of the enzyme PYK will for almost all sampled parameter values result in decreased flux through the LDH reaction as indicated by the dark circle in the row for LDH and the column for PFK.
Figure 5.
The width of the distribution of control coefficients correlates with distance from equilibrium.
Shown is the average standard deviation of the sampled flux control distribution as a function of displacement from equilibrium of the respective enzyme. Reactions close to equilibrium (
close to unity) typically have narrow distributions of flux control coefficients, centered at zero, indicating they can only exert little control over the flux through the system. Contrary, reactions far from equilibrium (
) exhibit broad distributions, indicating a potential, but no necessity, for high control coefficients. For definitions see Materials and Methods.
Figure 6.
Dependency of the glycolytic flux on the maximal activity of the glucose transporter (PTS).
Shown is the glucose uptake as a function of , evaluated using a kinetic model with a representative set of the sampled parameters. The black dot indicates the reference state.
Figure 7.
Probabilistic distribution of flux control coefficients in the absence of metabolic regulation.
Same as Figure 3 except for the absence of metabolic regulation. Any diagram refers to the control of one flux (i.e. through the step indicated to the left of the row) by one enzyme (i.e. the enzyme indicated above each diagram corresponds to the interval [-1,1] on the abscissa.
Figure 8.
Probabilistic sign distribution of flux control coefficients in the absence of regulation.
The shade of the entry represents the percentage of the calculated control coefficients that are positive. Dark colors correspond to a distribution of flux control coefficients that lies predominantly on the negative semiaxis, whereas light colors indicate that the sampled control coefficients are predominantly positive.
Figure 9.
Metabolic recovery after periods of starvation.
Starting at the defined metabolic state, external glucose is lowered from 20 mM to 0.1 mM at time t = 1 min, mimicking a brief period of starvation. At time t = 10 min external glucose is restored to its orginal value. The upper panel shows a histogram of intracellular ATP after a recovery period at time t = 100 min. For the regulated system (A), approximately 54% of all models recover to the initial metabolic state (542 of 1000 instances tested), whereas in the absence of regulation the probability of recovery is below 5% (39 of 1000 instances tested). The lower panel shows the median of the time-course, with .25 and 0.75 quantiles included as dashed lines. For the regulated system, the time-course is split between recovering and non-recovering instances.
Figure 10.
Recovery as a function of starvation time.
Shown is the percentage of model instances that recovered to the original metabolic state after a starvation time and a recovery time
in the presence (left panel) and absence (right panel) of regulation. We emphasise the different scales on the z-axis on both panels. In the presence of regulation, the probability to recover does not appreciably depend on length of starvation and recovery time. In the absence of regulation a longer starvation time decreases the probability to recover.
Table 1.
Percentage of recovering systems in the presence of individual regulation mechanisms.
Figure 11.
Time-courses of intermediate metabolites.
The median of the concentration of FBP, ATP, and PEP following a withdrawal of external glucose at in models that include activation of PYK by FBP is shown. After withdrawal of glucose, the concentration of PEP quickly rises and attains a new steady state. With the restoration of external glucose at
, PEP undergoes a quick drop, fuelling glucose uptake and subsequent production of ATP. The corresponding figure for systems that lack regulatory interactions but are nonetheless able to recover from periods of starvation is discussed in the Text S2.
Figure 12.
Bistability and hysteresis with respect to external glucose.
Shown is a non-recovering system in the absence of regulation (A,B) and a recovering system in the presence of regulation (C,D). In the upper panels (A,C), the concentration of external glucose was varied between the initial level of to a lower value of
and back. The lower panels (B,D) show a corresponding time-course of the rate of glucose uptake. The original level of external glucose,
, was lowered to
within the time interval t = 1 min to 10 min.