Flux Eye Strain Download

Computational strain design protocols aim at the system-wide identification of intervention strategies for the enhanced production of biochemicals in microorganisms. Existing approaches relying solely on stoichiometry and rudimentary constraint-based regulation overlook the effects of metabolite concentrations and substrate-level enzyme regulation while identifying metabolic interventions. In this paper, we introduce k-OptForce, which integrates the available kinetic descriptions of metabolic steps with stoichiometric models to sharpen the prediction of intervention strategies for improving the bio-production of a chemical of interest. It enables identification of a minimal set of interventions comprised of both enzymatic parameter changes (for reactions with available kinetics) and reaction flux changes (for reactions with only stoichiometric information). Application of k-OptForce to the overproduction of L-serine in E. coli and triacetic acid lactone (TAL) in S. cerevisiae revealed that the identified interventions tend to cause less dramatic rearrangements of the flux distribution so as not to violate concentration bounds. In some cases the incorporation of kinetic information leads to the need for additional interventions as kinetic expressions render stoichiometry-only derived interventions infeasible by violating concentration bounds, whereas in other cases the kinetic expressions impart flux changes that favor the overproduction of the target product thereby requiring fewer direct interventions. A sensitivity analysis on metabolite concentrations shows that the required number of interventions can be significantly affected by changing the imposed bounds on metabolite concentrations. Furthermore, k-OptForce was capable of finding non-intuitive interventions aiming at alleviating the substrate-level inhibition of key enzymes in order to enhance the flux towards the product of interest, which cannot be captured by stoichiometry-alone analysis. This study paves the way for the integrated analysis of kinetic and stoichiometric models and enables elucidating system-wide metabolic interventions while capturing regulatory and kinetic effects.

A comprehensive approach of metabolite balancing, (13)C tracer studies, gas chromatography-mass spectrometry, matrix-assisted laser desorption ionization-time of flight mass spectrometry, and isotopomer modeling was applied for comparative metabolic network analysis of a genealogy of five successive generations of lysine-producing Corynebacterium glutamicum. The five strains examined (C. glutamicum ATCC 13032, 13287, 21253, 21526, and 21543) were previously obtained by random mutagenesis and selection. Throughout the genealogy, the lysine yield in batch cultures increased markedly from 1.2 to 24.9% relative to the glucose uptake flux. Strain optimization was accompanied by significant changes in intracellular flux distributions. The relative pentose phosphate pathway (PPP) flux successively increased, clearly corresponding to the product yield. Moreover, the anaplerotic net flux increased almost twofold as a consequence of concerted regulation of C(3) carboxylation and C(4) decarboxylation fluxes to cover the increased demand for lysine formation; thus, the overall increase was a consequence of concerted regulation of C(3) carboxylation and C(4) decarboxylation fluxes. The relative flux through isocitrate dehydrogenase dropped from 82.7% in the wild type to 59.9% in the lysine-producing mutants. In contrast to the NADPH demand, which increased from 109 to 172% due to the increasing lysine yield, the overall NADPH supply remained constant between 185 and 196%, resulting in a decrease in the apparent NADPH excess through strain optimization. Extrapolated to industrial lysine producers, the NADPH supply might become a limiting factor. The relative contributions of PPP and the tricarboxylic acid cycle to NADPH generation changed markedly, indicating that C. glutamicum is able to maintain a constant supply of NADPH under completely different flux conditions. Statistical analysis by a Monte Carlo approach revealed high precision for the estimated fluxes, underlining the fact that the observed differences were clearly strain specific.

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Half-integer quantized flux vortices appear in honeycomb lattices when the signs of an odd number of couplings around a plaquette are inverted. We show that states trapped at these vortices can be isolated by applying inhomogeneous strain to the system. A vortex then results in localized midgap states lying between the strain-induced pseudo-Landau levels, with 2n+1 midgap states appearing between the nth and the (n+1)th level. These states are well-defined spectrally isolated and spatially localized excitations that could be realized in electronic and photonic systems based on graphenelike honeycomb lattices. In the context of Kitaev's honeycomb model of interacting spins, the mechanism improves the localization of non-Abelian anyons in the spin-liquid phase, and reduces their mutual interactions. The described states also serve as a testbed for fundamental physics in the emerging low-energy theory, as the correct energies and degeneracies of the excitations are only replicated if one accounts for the effective hyperbolic geometry induced by the strain. We further illuminate this by considering the effects of an additional external magnetic field, resulting in a characteristic spatial dependence that directly maps out the inhomogeneous metric of the emerging hyperbolic space.

Strain dependence of the energy levels of (a) a triangular honeycomb flake, (b) in the presence of an additional background magnetic field B=0.05max, and (c) with a half-integer flux vortex instead placed into the center of the flake. The flake measures 90 hexagons across and is terminated by zigzag edges, conforming to the geometry in Fig. 2. The strain is given in terms of the strength of the dimensionless pseudomagnetic field [see Eq. (2)], while the eigenvalues are given in units of the coupling strength t of the pristine system. As we show in Sec. 3, the precise energies and degeneracies of the levels in (a), marked by the horizontal and vertical lines, reveal the effects of an emergent curvature in the continuum description of the model, including the Wen-Zee shift (vertical tick marks show the results without these shifts). The underlying hyperbolic geometry leads to the broadening of levels in (b), which is in striking contrast to the splitting of the levels expected in conventional low-energy theory, as we describe in Sec. 4. Flux vortices induce a characteristic sequence of midgap states (c), which we describe in Sec. 5, while Sec. 6 contains the application to the Kitaev honeycomb model of interacting spins.

System-size dependence of the pLL energies (6) at maximal strain max in the continuum theory with curvature (thick blue curves), which exactly recovers the result (4) of the microscopic model in the optimally strained geometry. The red lines show the conventional estimate (3) of these levels when the curvature is ignored. For clarity, we only show the levels with index n=1,...,5.

Division of an optimally strained triangle into zigzag chains labeled by l, as used for the construction of exact pLL states in Sec. 5a. The amplitudes specify a state in the first pLL for system size N=5. This state is supported by the blue trapezoidal region, and hence is unaffected by the indicated flux vortex, which is generated by a phase shift of the deep red coupling.

Construction principle of exact pLL states in an optimally strained triangle with a flux vortex, marked in red, which is obtained by a Peierls substitution along the line of the shaded plaquettes. The desired states are obtained by adopting their construction principle in absence of the vortex, which can be used to produce a basis of zero modes with trapezoidal support. We use this to construct three trapezoidal systems whose pLLs are exact solutions of the triangle with the vortex, but are not affected by its existence (the remaining shaded plaquettes do not carry any flux, so this also holds true for subsystem 1). The combined Wen-Zee shift from the three systems exactly accounts for the observed number of midgap states produced by the vortex.

Formation of flux-induced midgap states in an optimally strained triangle (N=90), as a function of the flux in a vortex placed at its center. (a) The analytically predicted midgap energies from Eqs. (47) (orange) and (48) (green) along with the pLL energies in Eq. (6) (blue) and (b) the numerical result from the microscopic model.

Single-particle energy levels in an optimally strained triangle of size N=90, as a function of vortex positions. (a) A single half-integer vortex is placed at a position along a line from the edge to the center. (b) The system contains an additional half-integer vortex fixed at the center.

Vortex energetics in the Kitaev honeycomb model. (a) Strain dependence of the ground-state energies in the flux-free sector and in the sector with a flux vortex at the center of the system. (b) Excess energy from the centered vortex. (c) Position dependence of excess energies due to a single vortex or a vortex pair in the optimally strained system (solid symbols) and in the unstrained system (open symbols). The results for the vortex pair represent an effective interaction energy, as defined in Eq. (51).

Magnetic field dependence of the energies in a triangular system of size N=30, without any vortices (left), and with a half-integer vortex placed into the center (right). In the top panels, the system is pristine; in the bottom panels it is optimally strained.

(a) Determination of the effective vortex interaction energy in Fig. 9 from the ground states of sectors in different vortex configurations, amounting to the difference between the shaded and green curves in accordance with Eq. (51). This interaction energy is consistent with that obtained by displacing both vortices symmetrically from the center, giving the ground-state energies of (b). These data are for optimally strained triangles of size N=90. 2351a5e196