132 research outputs found

### The economic basis of periodic enzyme dynamics

Periodic enzyme activities can improve the metabolic performance of cells. As
an adaptation to periodic environments or by driving metabolic cycles that can
shift fluxes and rearrange metabolic processes in time to increase their
efficiency. To study what benefits can ensue from rhythmic gene expression or
posttranslational modification of enzymes, I propose a theory of optimal enzyme
rhythms in periodic or static environments. The theory is based on kinetic
metabolic models with predefined metabolic objectives, scores the effects of
harmonic enzyme oscillations, and determines amplitudes and phase shifts that
maximise cell fitness. In an expansion around optimal steady states, the
optimal enzyme profiles can be computed by solving a quadratic optimality
problem. The formulae show how enzymes can increase their efficiency by
oscillating in phase with their substrates and how cells can benefit from
adapting to external rhythms and from spontaneous, intrinsic enzyme rhythms.
Both types of behaviour may occur different parameter regions of the same
model. Optimal enzyme profiles are not passively adapted to existing substrate
rhythms, but shape them actively to create opportunities for further fitness
advantage: in doing so, they reflect the dynamic effects that enzymes can exert
in the network. The proposed theory combines the dynamics and economics of
metabolic systems and shows how optimal enzyme profiles are shaped by network
structure, dynamics, external rhythms, and metabolic objectives. It covers
static enzyme adaptation as a special case, reveals the conditions for
beneficial metabolic cycles, and predicts optimally combinations of gene
expression and posttranslational modification for creating enzyme rhythms

### How enzyme economy shapes metabolic fluxes

Metabolic fluxes are governed by physical and economic principles.
Stationarity constrains them to a subspace in flux space and thermodynamics
makes them lead from higher to lower chemical potentials. At the same time,
fluxes in cells represent a compromise between metabolic performance and enzyme
cost. To capture this, some flux prediction methods penalise larger fluxes by
heuristic cost terms. Economic flux analysis, in contrast, postulates a balance
between enzyme costs and metabolic benefits as a necessary condition for fluxes
to be realised by kinetic models with optimal enzyme levels. The constraints
are formulated using economic potentials, state variables that capture the
enzyme labour embodied in metabolites. Generally, fluxes must lead from lower
to higher economic potentials. This principle, which resembles thermodynamic
constraints, can complement stationarity and thermodynamic constraints in flux
analysis. Futile modes, which would be incompatible with economic potentials,
are defined algebraically and can be systematically removed from flux
distributions. Enzymes that participate in potential futile modes are likely
targets of regulation. Economic flux analysis can predict high-yield and
low-yield strategies, and captures preemptive expression, multi-objective
optimisation, and flux distributions across several cells living in symbiosis.
Inspired by labour value theories in economics, it justifies and extends the
principle of minimal fluxes and provides an intuitive framework to model the
complex interplay of fluxes, metabolic control, and enzyme costs in cells

### Flux cost functions and the choice of metabolic fluxes

Metabolic fluxes in cells are governed by physical, biochemical,
physiological, and economic principles. Cells may show "economical" behaviour,
trading metabolic performance against the costly side-effects of high enzyme or
metabolite concentrations. Some constraint-based flux prediction methods score
fluxes by heuristic flux costs as proxies of enzyme investments. However,
linear cost functions ignore enzyme kinetics and the tight coupling between
fluxes, metabolite levels and enzyme levels. To derive more realistic cost
functions, I define an apparent "enzymatic flux cost" as the minimal enzyme
cost at which the fluxes can be realised in a given kinetic model, and a
"kinetic flux cost", which includes metabolite cost. I discuss the mathematical
properties of such flux cost functions, their usage for flux prediction, and
their importance for cells' metabolic strategies. The enzymatic flux cost
scales linearly with the fluxes and is a concave function on the flux polytope.
The costs of two flows are usually not additive, due to an additional
"compromise cost". Between flux polytopes, where fluxes change their
directions, the enzymatic cost shows a jump. With strictly concave flux cost
functions, cells can reduce their enzymatic cost by running different fluxes in
different cell compartments or at different moments in time. The enzymactic
flux cost can be translated into an approximated cell growth rate, a convex
function on the flux polytope. Growth-maximising metabolic states can be
predicted by Flux Cost Minimisation (FCM), a variant of FBA based on general
flux cost functions. The solutions are flux distributions in corners of the
flux polytope, i.e. typically elementary flux modes. Enzymatic flux costs can
be linearly or nonlinearly approximated, providing model parameters for linear
FBA based on kinetic parameters and extracellular concentrations, and justified
by a kinetic model

### Enzyme economy in metabolic networks

Metabolic systems are governed by a compromise between metabolic benefit and
enzyme cost. This hypothesis and its consequences can be studied by kinetic
models in which enzyme profiles are chosen by optimality principles. In
enzyme-optimal states, active enzymes must provide benefits: a higher enzyme
level must provide a metabolic benefit to justify the additional enzyme cost.
This entails general relations between metabolic fluxes, reaction elasticities,
and enzyme costs, the laws of metabolic economics. The laws can be formulated
using economic potentials and loads, state variables that quantify how
metabolites, reactions, and enzymes affect the metabolic performance in a
steady state. Economic balance equations link them to fluxes, reaction
elasticities, and enzyme levels locally in the network. Economically feasible
fluxes must be free of futile cycles and must lead from lower to higher
economic potentials, just like thermodynamics makes them lead from higher to
lower chemical potentials. Metabolic economics provides algebraic conditions
for economical fluxes, which are independent of the underlying kinetic models.
It justifies and extends the principle of minimal fluxes and shows how to
construct kinetic models in enzyme-optimal states, where all enzymes have a
positive influence on the metabolic performance

### Elasticity sampling links thermodynamics to metabolic control

Metabolic networks can be turned into kinetic models in a predefined steady
state by sampling the reaction elasticities in this state. Elasticities for
many reversible rate laws can be computed from the reaction Gibbs free
energies, which are determined by the state, and from physically unconstrained
saturation values. Starting from a network structure with allosteric regulation
and consistent metabolic fluxes and concentrations, one can sample the
elasticities, compute the control coefficients, and reconstruct a kinetic model
with consistent reversible rate laws. Some of the model variables are manually
chosen, fitted to data, or optimised, while the others are computed from them.
The resulting model ensemble allows for probabilistic predictions, for
instance, about possible dynamic behaviour. By adding more data or tighter
constraints, the predictions can be made more precise. Model variants differing
in network structure, flux distributions, thermodynamic forces, regulation, or
rate laws can be realised by different model ensembles and compared by
significance tests. The thermodynamic forces have specific effects on flux
control, on the synergisms between enzymes, and on the emergence and
propagation of metabolite fluctuations. Large kinetic models could help to
simulate global metabolic dynamics and to predict the effects of enzyme
inhibition, differential expression, genetic modifications, and their
combinations on metabolic fluxes. MATLAB code for elasticity sampling is freely
available

### The protein cost of metabolic fluxes: prediction from enzymatic rate laws and cost minimization

Bacterial growth depends crucially on metabolic fluxes, which are limited by
the cell's capacity to maintain metabolic enzymes. The necessary enzyme amount
per unit flux is a major determinant of metabolic strategies both in evolution
and bioengineering. It depends on enzyme parameters (such as kcat and KM
constants), but also on metabolite concentrations. Moreover, similar amounts of
different enzymes might incur different costs for the cell, depending on
enzyme-specific properties such as protein size and half-life. Here, we
developed enzyme cost minimization (ECM), a scalable method for computing
enzyme amounts that support a given metabolic flux at a minimal protein cost.
The complex interplay of enzyme and metabolite concentrations, e.g. through
thermodynamic driving forces and enzyme saturation, would make it hard to solve
this optimization problem directly. By treating enzyme cost as a function of
metabolite levels, we formulated ECM as a numerically tractable, convex
optimization problem. Its tiered approach allows for building models at
different levels of detail, depending on the amount of available data.
Validating our method with measured metabolite and protein levels in E. coli
central metabolism, we found typical prediction fold errors of 3.8 and 2.7,
respectively, for the two kinds of data. ECM can be used to predict enzyme
levels and protein cost in natural and engineered pathways, establishes a
direct connection between protein cost and thermodynamics, and provides a
physically plausible and computationally tractable way to include enzyme
kinetics into constraint-based metabolic models, where kinetics have usually
been ignored or oversimplified

### A Quantitative Study of the Hog1 MAPK Response to Fluctuating Osmotic Stress in Saccharomyces cerevisiae

Background Yeast cells live in a highly fluctuating environment with respect to temperature, nutrients, and especially osmolarity. The Hog1 mitogen-activated protein kinase (MAPK) pathway is crucial for the adaption of yeast cells to external osmotic changes. Methodology/Principal Findings To better understand the osmo-adaption mechanism in the budding yeast Saccharomyces cerevisiae, we have developed a mathematical model and quantitatively investigated the Hog1 response to osmotic stress. The model agrees well with various experimental data for the Hog1 response to different types of osmotic changes. Kinetic analyses of the model indicate that budding yeast cells have evolved to protect themselves economically: while they show almost no response to fast pulse-like changes of osmolarity, they respond periodically and are well-adapted to osmotic changes with a certain frequency. To quantify the signal transduction efficiency of the osmo-adaption network, we introduced a measure of the signal response gain, which is defined as the ratio of output change integral to input (signal) change integral. Model simulations indicate that the Hog1 response gain shows bell-shaped response curves with respect to the duration of a single osmotic pulse and to the frequency of periodic square osmotic pulses, while for up-staircase (ramp) osmotic changes, the gain depends on the slope. Conclusions/Significance The model analyses suggest that budding yeast cells have selectively evolved to be optimized to some specific types of osmotic changes. In addition, our work implies that the signaling output can be dynamically controlled by fine-tuning the signal input profiles

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