Figure 1.
Feasibility Analysis (FA) explained.
Panel fig:Feasibility-FBA illustrates constraint-based modeling often used within Flux Balance Analysis, starting from the unconstrained solution space and ending in the optimal solution (adapted from [8]). Feasibility Analysis is inspired from this constraint-based approach and combines it with the molecular rigor of a detailed kinetic model. The regulatory and physiological spaces are connected to each other with available kinetic rate equations for each reaction (usually a non-linear function of enzyme levels , metabolite levels
and kinetic parameter set
). Under a number of constraints (e.g. thermodynamic, kinetic etc), only a subspace of both the enzyme and physiological space in panel fig:Feasibility-Spaces is viable, i.e. fulfills the constraints, as represented in panel fig:Feasibility-Allowed. Considering the list of feasibility criteria, only a subspace of this viable space is also feasible (panel fig:Feasibility-Feasible). The feasible enzyme space is constructed by evaluating the list of feasibility criteria for each physiological state in the viable space. The final feasible enzyme space can further be inspected within the scope of regulation analysis.
Figure 2.
The small synthetic pathway used for illustration of the feasibility analysis.
fig:ToyModel: The metabolic reaction network used. The solid arrows represent the base network and dashed lines indicate the additional kinetic interactions considered. fig:ElasticityMatrix: the reference steady state and the kinetic parameters for the small model.
Figure 3.
Thermodynamic constraints as limits to the physiological space.
For the synthetic small problem, these constraints can be implemented before sampling. is presented in logarithmic scale.
Figure 4.
Feasibility analysis for the toy problem.
The first two columns are the physiological space (first colum: flux vs. , second column:
vs
), and the last two columns are the enzyme space (third column: viable enzyme space with all enzymes, fourth column: selected 2D slices from the third column at
). fig:FluxFeasToy: Optimal flux as feasibility criterion for function. fig:HomeostasisFeasToy: Homeostasis of the both metabolites as feasibility criterion. fig:PromptFeasToy: Turn-over time as feasibility criterion for temporal responsiveness. The red-blue color gradient indicates continuous values for the feasibility criteria in consideration, red indicates feasible states while blue indicates infeasible states The feasibility criteria are
, for function, homeostasis and temporal responsiveness respectively. The quantitative measures for homeostasis and temporal responsiveness have been changed sign and added offset for visualization purposes. The gray points in the physiological states are those for which the corresponding enzyme levels are outside the viable range, after applying the constraints. All axes in all plots are presented relative to their reference state, and
is presented in logarithmic scale.
Figure 5.
Combining feasibility criteria.
fig:ObjSpace represents the objective space, where each feasibility criterion is taken along (no cutoff is used in this plot). fig:CombinedObj presents a 3D plot of the feasible enzyme space fig:CombinedObjLayers presents decompositions of the feasible enzyme space into a series of 2D slices, each differing by the value of (indicated on the plot). Blue points describe the functionally feasible enzyme levels (
i.e. fluxes with top 25% are considered as feasible), red points are homeostatically feasible enzyme levels (
), green points are the feasible enzyme levels considering the temporal responsiveness and black points are the states that are feasible for all three criteria.
Figure 6.
The effect of additional feedback inhibition of
on
on the feasible enzyme space with respect to homeostasis (
) and function (
). fig:HomeostasisEv1x2: The feasible enzyme space for
(left),
(center), and
(right). In every subplot, red points: solely homeostatically feasible enzyme levels; blue points: solely functionally feasible enzyme levels; black points: feasible enzyme levels on both criteria. The axes for all 3 plots are the same, enzyme levels relative to the reference state. fig:AdditionalFeedbackEv1x2-MetLev: The maximum achievable metabolite levels as function of the inhibition strength.
Figure 7.
Feasibility analysis of the changes in enzyme levels during long term chemostat cultivation.
In each plot, the dots (•) describe the sampled enzyme levels relative to reference state, colored according to the feasibility criteria specified in each plot above the color legend bar; the squares (▪) are the experimental data either from [41] or [43], white being the “wild-type” (10th generation) and black being the “evolved strain” (200th generation) and the arrow indicates the direction of the number of generations during the experiment (time). Enzymes not shown change only 10% from their reference state. fig:LongChemostatPGI-ALD: The function feasibility in terms of PGI and ALD, the color corresponds the total cost of the enzymes (). The experimental data from [43] shows that the cells evolved to an economized state.fig:LongChemostatGLT-HK: Function feasibility inspected for glucose transporter (GLT) and hexokinase (HK). The colouring is similar to fig:LongChemostatPGI-ALD, the sum of enzyme levels. The hypothesis on enzyme economy fails to predict the levels of these two enzymes for the evolved strain. fig:LongChemostatPGI-GLT: The evolution of glucose transporter and PGI enzymes inspected via homeostasis feasibility, as the co-response of extracellular glucose and uptake rate (
). Cells evolve to a state where they are more apt to use extracellular resources. fig:LongChemostatPGI-ALD-ATP: The evolved state is predicted to allow yeast to respond quicker to external perturbations, as indicated by ATP temporal responsiveness feasibility as the color code for PGI and ALD. The experimental verification of this prediction is presented in fig:LongChemostatATPpulse where the response of ATP to a glucose perturbation (taken from [41]) is presented. The y-axis is the ATP level relative to the state before perturbation and x-axis represent the time in seconds. Evolved cells (•) respond quicker to glucose perturbation, when compared to wild-type cells (♦
). fig:LongChemostatPGI-PFK: the function feasibility inspected for PFK and PGI. PFK levels, being already at the edge of the feasible space, can not further be decreased.
Figure 8.
The competition experiment, to illustrate the optimal enzyme distribution considering the trade-off between enzyme economy and competitive ability for extracellular glucose.
The radar plot on the left represents the enzyme levels, relative to wild-type, and the plot on the left represents the competition of each subpopulation with a specific enzyme setting as described in the radar plot. The color for each subpopulation is the same in both plots and is described in the legend.
Figure 9.
The danger of Turbo design and a potential solution investigated using FA.
fig:TurboDesignAnalysisA: The original model considered in [24]. fig:TurboDesignAnalysisB: The feasible regulatory space of relative enzyme activities of HK and GLT. Increasing the enzyme levels leads to infeasible states for hexoses (red points on the upper right corner on the plot). fig:TurboDesignAnalysisD: The new design of the system with added metabolite T6P and its inhibition on HK. The new model parameters for storage branch are: . fig:TurboDesignAnalysisE: the same regulatory space as in fig:TurboDesignAnalysisB after addition of the feedback inhibition of T6P on HK. In fig:TurboDesignAnalysisA and fig:TurboDesignAnalysisD, only interactions within the focus are shown for simplicity where blue arrows indicate the kinetic activation and red arrow indicates inhibition. In fig:TurboDesignAnalysisB and fig:TurboDesignAnalysisE, only HK and GLT are monitored, remaining enzymes are held at their reference levels. The color code used in plots fig:TurboDesignAnalysisB and fig:TurboDesignAnalysisE is the co-response coefficient
.
Table 1.
The reference conditions for the yeast problem.