University of Groningen
Thermodynamic principles governing metabolic operation : inference, analysis, and
prediction
Niebel, Bastian
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Chapter 1
Thermodynamic constraints on
metabolic operations
Bastian Niebel
9
Chapter 1
General overview
Metabolism converts nutrients into energy and biomass precursors. These conversions (or
metabolic processes) are the enzyme-catalyzed chemical transformations and the transports of metabolites across cellular membranes. But metabolic operations are not solely a
consequence of the individual enzyme mechanisms. Instead, concerted action of metabolic
processes are important for the functioning of metabolism. To unravel the complex system
behavior of metabolic processes operating in an orchestrated manner, systems biology approaches have been successfully applied and generated new insights into the operation of
metabolism (1-4). In systems biology, mathematical models of metabolism or other cellular
processes are developed, and the formal analysis of these models typically leads to new insights into the operation of these processes (2).
Given the fact that metabolic networks contain about thousand different metabolic processes,
it is difficult to develop models that describe the mechanism of every metabolic process in
great detail, i.e. with kinetics of the enzymatically catalyzed reactions. Thus, constraint-based
models have been used, circumventing the need for detailed kinetic information on every
metabolic process (4-6). In constraint-based models, metabolism is defined by a set of constraints, where the solution space of these constraints describes the different possible metabolic operations. Generally, the basis for these constraints are steady-state mass balances
for the metabolites, which state that for every metabolite in the metabolic network the consumption rate equals the production rate, and thereby ensuring the conservation of mass.
Adding—besides steady-state mass balances—more constraints to the model reduces the
solution space and thereby can influence the quality of the predictions made with the model
(6). During the last decades, numerous additional constraints have been identified ranging
from constraints based on physical principles, e.g. thermodynamic constraints applied to the
metabolic operations, or heuristically motivated constraints such as enzyme solvent capacity
or transcriptional regulation (5, 6).
Thermodynamic constraints have the advantage of a physical foundation. Also, because thermodynamic principles need to apply irrespective of the cellular growth conditions, these
constraints are independent of the conditions. Therefore, thermodynamic constraints do not
require readjustments when the model is applied to different conditions, unlike for most heuristic constraints, where for every new condition the constraint has to be reevaluated. Because
of the physical foundations and condition-independence of thermodynamic constraints, they
are being thought of as major constraints on the evolution of metabolism (7-13).
Constraint-based models can be analyzed using a multitude of different analysis techniques
(6). The tool of flux balance analysis (FBA) uses a cellular objective functions, e.g. growth
rate, to predict metabolic fluxes (14). This approach has been extended to explore possible
cellular objectives using multi-objective optimization (15). Further, the shape of the solution
space has been explored using flux variability analysis (16) and sampling based approaches
(6, 15, 17, 18), while the topology of the metabolic network has been studied using extreme
pathways (19) and elementary mode analysis (20). All these analyses depend on the quality of
the constraint-based model. Thus, this calls for carefully curated and assembled models supplemented with a maximum number of hard physical constraints, but with a minimal number
of heuristic ad-hoc constraints. Here, thermodynamics could offer such physical constraints.
10
Background
Background
Thermodynamic
principles
Thermodynamic
data
Previous Thermodynamic Analysis of Thermodynamic
work
analysis of
metabolic
analysis of
metabolic
networks with
growth
networks
the loop-law
processes
This work
Connections
between
kinetics and
thermodynamics
Research question and outline
Figure 1. Overview of topics covered in this introduction.
In the following sections (cf. Fig. 1), we briefly review the background of thermodynamic
principles used in biochemistry and methods to determine thermodynamic data. Then, we
discuss how these principles and data were used in previous work to get new insights in metabolic operations. Specifically, we review the topics ‘thermodynamic analysis of metabolic
networks’, ‘thermodynamic analysis of growth processes’, ‘analysis of metabolic networks with
the loop-law’, and ‘connections between kinetics and thermodynamics’. We conclude this introduction with deriving the research question underlying this thesis, and provide an outline
of the following chapters.
Background
Thermodynamic principles
Thermodynamics of metabolic processes can be described by changes in the Gibbs energies,
∆rG (Fig. 2). ∆rG describes the difference in the Gibbs energy stored in the substrates and
products of a metabolic processes (the so-called Gibbs formation energy). For a metabolic
process to obey the second law of thermodynamics, ∆rG has to be negative (21). Strictly spoken, for a metabolic processes to proceed according to the second law of thermodynamics,
entropy has to be produced. However, at the conditions, at which the metabolic process take
place (i.e. constant pressure and temperature), the production of entropy is proportional to
∆rG.
Metabolic processes consist of chemical transformations and metabolite transport. Knowing
Gibbs formation energies for the substrates and products of a process, it is possible to determine the metabolic processes’ changes in the Gibbs energies, ∆rG. For chemical transformations (superscript c), e.g. A → B, ∆rGc is defined by ∆rGcA→B = ∆fGB - ∆fGA, where ∆fG is the
Gibbs energies of formation of the chemical compounds, i.e. substrates and products of the
transformations. ∆fG of a compound X is calculated by, ∆fG = ∆fGo + RT ln aX, where ∆fGo is
the compound’s standard Gibbs energy of formation, a the compound’s chemical activity, R
the Gas constant, and T the temperature. ∆fGo of a compound describes its change in Gibbs
energy with respect to a common reference state, which is indicated by the superscript o.
Note: The standard Gibbs energies of formations for a chemical conversion can be combined
in the standard Gibbs energy of reaction ∆rGo, e.g. ∆rGoA→B = ∆fGoB - ∆fGoA, therefore the ∆rGc
11
Gibbs energy
Chapter 1
A
A
B
∆G
Progress of
metabolic process
A
A
∆G < 0
∆G = 0
∆G > 0
B
B
B
Figure 2. Second law of thermodynamics for metabolic processes. A metabolic process consisting of a chemical transformation of the metabolite A to B can happen when the change
in the Gibbs energy ∆G is negative. ∆G describes the difference in the Gibbs energy between the initial (A) and the final
state (B) of the process.
becomes, ∆rGcA→B = ∆rGoA→B + RT (ln aB - ln aA).
Metabolic networks not only consist of chemical transformations, but they also involve transport of metabolites across membranes. The change of Gibbs energy ∆rGt of a metabolite transport (superscript t), e.g. the transport of a compound X from the outside (superscript out) to
the inside (superscript in) of a cellular compartment, Xout → Xin, is defined by, ∆rGtXout → Xin =
RT (ln aXout - ln aXin) + F zX φ, where z is the charge of X, φ is the electrical membrane potential, and F is the Faraday constant. One feature of many metabolic processes is that different
mechanisms are mechanistically and thermodynamically coupled (22), e.g. the coupling of
a chemical transformation, A → B, to the transport of a compound, Xout → Xin. The change of
Gibbs energy, ∆rG, of this coupled metabolic process is then defined by the sum of the change
in Gibbs energy of the chemical transformation, ∆rGc, and the transport process, ∆rGt, ∆rG =
∆rGcA→B + ∆rGtXout → Xin.
Thermodynamic data
In order to specify the Gibbs energy changes of metabolic processes, we need standard Gibbs
energies of formation (or of reactions). These standard Gibbs energies can be inferred from
the experimental equilibrium constant of chemical conversions (21, 23-27)—also referred as
reactant-contribution method (RC) (27)—or predicted using a group contribution method
(GC) (26-30). Both methods rely on collections of experimentally determined equilibrium
constants for a set of about 400 of enzymatic reactions, which has been collected and curated
from 1000 different articles into a database (31). These two approaches deliver different coverages: With the RC method, one can obtain standard energies Gibbs energies of reactions
for 11 % (~600) of all relevant reactions (full chemical description and chemically balanced)
in the KEGG database, and 88 % (~4800) with GC (27). The median root mean square error
of the estimated standard Gibbs energies of reactions has been determined to 1 KJ mol-1 with
RC and 5.5 KJ mol-1 with GC (27). Another approach has been developed, which infers Gibbs
energy changes based on similarities in the reactions (32), but the high average root mean
square error of 10 KJ mol-1 and the ability to only infer the standard Gibbs energies of 106
reactions, renders this approach not practical.
Recently, different approaches to combine both the RC and GC methods have been developed
(26, 27, 33, 34). When combining both methods, one is faced with the challenge of different
reference states that were used for the estimated (RC) or predicted (GC) Gibbs energies of
formations. If Gibbs energies are used in the thermodynamic constraints within a metabolic
network model, then different reference states would lead to a violation of the first law of thermodynamics, and thus all used standard Gibbs energies of formations must have the same
thermodynamic reference state (27). This problem has been solved by the component contribution method (CC) by ensuring that all standard Gibbs energies of formations are within
12
Background
the null space of the stoichiometric network, i.e. fulfill the loop-law (27). To this end, CC also
allows the exact determination of the estimation errors of the standard Gibbs energies.
Because the chemical conditions—at which metabolic processes take place within the cellular
compartments—can be approximated by a dilute mixture of compounds in an electrolyte
solution with constant ionic strength, I, and constant pH, we can transform the standard
Gibbs energies to take into account theses chemical conditions. The assumption of a dilute
electrolyte solution allows to approximate the chemical activity, a, by the molar concentration, C, using the extended Debye-Hückel theory (21), i.e. a = γ(I)C , where γ is the activity
coefficient determined as a function of I. Often, the activity coefficients are included in the
standard Gibbs energy of formations, ∆fGo(I) = ∆fGo + RT ln γ(I), and thereby ∆fGo becomes
a function of the ionic strength I.
In aqueous solutions, in which biochemical processes take place, metabolites (i.e. the reactants) are typically present as different chemical species (i.e. differently protonated). The
distribution between the abundance of these species is often pH-dependent. As handling of
individual species would be cumbersome, we only consider reactants in biochemical thermodynamics. In order to determine the pH-dependent Gibbs energy of a reactant, we first make
the standard Gibbs energy of formation of the species ι, ∆fGoι(I), pH-dependent using the
Legendre transformation (21), ∆fG’oι = ∆fGoι(I) – NH (∆fGoH+(I)- RT pH ln 10), where NH is the
number of hydrogen atoms of the species, and ∆fGoH+(I) the standard Gibbs energy of formation of hydrogen ions (protons, H+). This Legendre transformation transforms the standard
Gibbs energy of the species to the biochemical reference state (indicated by the apostrophe ‘),
and thereby making ∆fGoι(I) a function of the pH and the ionic strength I. The standard Gibbs
energy of formation of a reactant is then determined from all its chemical species ι, using the
relationship (21) ∆fG’o = -RT ln[∑ιexp(-∆fG’oι/RT)], where we assume an equilibrium between
the differently protonated species. Note, that while this concept is very practical, it complicates things once it comes to the thermodynamic description of transport processes, where
we still also need to consider, for instance, the charge of individual species. For a detailed
treatment of thermodynamics of transport processes, however, the reader is referred to the
work of Jol et al. (35).
Gibbs energy changes of metabolic processes are also dependent on the processes’ reactant
concentrations. In microorganisms and mammalian cells, metabolite concentrations typically ranges between 1 uM to 10 mM (36-39). When using thermodynamic constraints on
metabolic network operation, one can constrain metabolite concentrations to such generic
physiological bounds. Notably, in some studies wider concentration ranges were used (34,
40, 41). Likely, metabolite concentration bounds had to be relaxed, because of inconsistencies in the reference state of the Gibbs energies of formation or missing adjustments of the
thermodynamic data to pH or ionic strength (42, 43). Such inconsistencies in the thermodynamic data are regretful in the first place, because they might lead to wrong conclusions.
But, furthermore, these inconsistencies then typically required researchers to apply large metabolite concentration ranges to get the network feasible at all. Consequently the constraints
imposed by the Gibbs energies are relaxed significantly and might not be active after all.
Therefore, it is key to use standard Gibbs energies estimated by methods which ensure the
same thermodynamic reference state, such as component contribution (27). Then, narrow
13
Chapter 1
concentration ranges between 1 uM and 10 mM can be used and consequently thermodynamic constraints are more active.
Previous work
Thermodynamic analysis of metabolic networks
The second law of thermodynamics together with detailed information of the Gibbs energies
of reactions and concentration ranges have been used to determine the feasibility of metabolic
pathways (9-13, 42-46) and metabolic networks (33, 34, 38, 40, 41, 47-54). Early studies focused on the thermodynamic feasibilities of metabolic pathways, where especially reactions
were identified that serve as thermodynamic bottlenecks in glycolysis for different concentration ranges of the substrates and products of the glycolytic pathway (44, 45). Also the sensitivity of the thermodynamic bottlenecks with respect to pH, ionic strength, and magnesium has
been studied (42, 43). More recent thermodynamic feasibility studies have been carried out
to unravel the biochemical logic behind the glycolytic pathway (11, 13) and other pathways
in central metabolism (13). Thermodynamic feasibility statements have been also applied for
industrial applications. Here, feasibility of thermodynamic pathways have been used to optimize the penicillin production of Penicillium chrysogenum (46), and for the design of new
carbon fixation pathways (9, 10, 12).
The second law of thermodynamics, applied to the reactions of a metabolic network model,
has been used in thermodynamic analysis of metabolic networks (38) to predict ranges for
the Gibbs energies of reactions and concentrations in Escherichia coli (33, 38, 49-51, 54),
Geobacter sulfurreducens (52), Saccharomyces cerevisiae (34, 38, 40, 41) and mammalian cells
(34, 48, 51, 53) and to further check the thermodynamic consistency of metabolome data for
E. coli and S. cerevisiae (38). Based on the predicted ranges of the Gibbs energies of reactions,
potential regulatory reactions in metabolic networks have been identified (38, 41, 50, 52). By
varying concentrations of toxic compounds, thermodynamic analysis of metabolic networks
has been used to predict the responses of intracellular metabolite concentrations and Gibbs
energies of reactions to different dosages of this toxic compounds (48). Also, the second law
has been directly integrated as a mixed integer constraint allowing the computationally tractable integration of network thermodynamics and the second law of thermodynamics (49,
51). This mixed integer constraint has been used together with flux balance analysis to predict
flux distributions based on measured intracellular and extracellular metabolic concentrations
(51), and to study the effect of thermodynamic constraints on the feasible solution space of
metabolic networks (54).
Thermodynamic analysis of metabolic networks has further been used to define the directionality of reactions within the metabolic network. In constraint-based models reactions
are either classified as irreversible or reversible, where typically the majority of the reactions are classified as irreversible. This irreversibility classifications have been incorrectly
referred to as thermodynamic constraints (55, 56). Thermodynamic analysis of metabolic
networks allowed to determine which reactions can be correctly classified as irreversible on
the basis of thermodynamics. Therefore, we collected from different studies that classified
the irreversibility of reactions using thermodynamic analysis, the fraction of reactions that
were correctly classified as irreversible. With this literature summary, we found that for dif14
Previous work
ferent genome scale metabolic reconstructions of E. coli, 28% (47), 29% (57), 34% (33), and
mammalian reconstructions 39% (53), 29% (34), were correctly classified as irreversible on
the basis of thermodynamics. In summary, on average 30% of the reactions classified as irreversible in this genome scale metabolic reconstructions are correctly classified on thermodynamic basis.
Thermodynamic analysis of growth processes
Thermodynamic principles not only apply to intracellular metabolic processes, but also to
the overall growth process, i.e the conversion of substrates, S, e.g. glucose, phosphate, oxygen, into biomass, B, and by-products, P, e.g. carbon-dioxide, ethanol, acetate, lactate, by the
growth process, S → B + P. Similar to every metabolic process, the growth process has to fulfill
the second law of thermodynamics, and thus, the change in Gibbs energy associated with the
growth process needs to be negative, ∆rGS→B+P. To analyze these Gibbs energy changes of the
growth process, black box models have been developed that determine ∆rGS→B+P on the basis
of measured extracellular rates and Gibbs energies of formation of the different substrates,
products and the biomass as input (8, 58-67). The Gibbs energy of formation of the biomass
has been determined using low temperature calorimetry, statistical mechanics, or empirical
relationships (67-69). Using these black box models that describe ∆rGS→B+P, the thermodynamic efficiency of different growth condition has been analyzed (59, 60, 62, 65, 67), and a
number of different empirical relationships have been developed to predict the biomass yield
based on measured ∆rGS→B+P (63, 65-67).
Analysis of metabolic networks with the loop-law
For metabolic networks, the first law of thermodynamics, i.e. conservation of energy, ensures
that no energy can be produced or destroyed. Therefore, the first law is the energetic analogue
to the mass conversation. For the operation of a metabolic network, this means that the Gibbs
energy changes of a cyclic series of metabolic processes, e.g. loops of chemical conversions,
A → B → C → A, of the metabolites, A, B, and C, must be zero, e.g. ∆rGA→B + ∆rGB→C + ∆rGC→A =
0. Combining the first law with the second law of thermodynamics forbids a metabolic flux
through a loop of metabolic processes (in the following, we refer to this as the loop-law (70)).
These loops in the metabolic network are defined by the null space of the stoichiometric
matrix, i.e. the mathematical representation of the metabolic network (70). Using the looplaw does not require any information about Gibbs energies of formation. It only requires the
calculation of the null space of the stoichiometric matrix. The loop-law was used in a series
of constraint-based models as constraints and was used to exclude loops from the flux distributions (18, 19, 71-74), to predict Gibbs energies of formations (73), and was included in
flux variability analysis (75). It has been mathematically proven that using the loop-law only
constrains thermodynamic infeasible loops and does not remove thermodynamically possible flux distributions (76).
Connections between kinetics and thermodynamics
For elemental reaction steps, the kinetics of these reactions are described by the law of mass
action and therefore the Gibbs energy is proportional to the natural logarithm of the ratio
of the forward, v+, and the backward rate, v-, of this elementary reaction step, i.e. ∆rG = RT
ln(v+/v-) (77, 78). This relationship between ∆rG and ln(v+/v-) has been integrated in con15
Chapter 1
straint-based models (72, 79-81) and been used to identify kinetically limited reactions in
metabolic pathways (13). But since this relationship assumes mass action kinetics, the generality of this relationship is questionable, since enzymatic reactions are not described by a
single elementary reactions but consist of a series of elementary reactions, which make up the
enzymatic mechanism. To this end, several attempts have been made to extend this relationship and develop new kinetic relationships between the Gibbs energy and reaction and the
enzymatic rates (82, 83).
This work
Research question
Thermodynamic principles apply to every aspect of metabolic operations, although from previous work it remains unclear to what degree metabolic operations are constrained by thermodynamics beyond pure feasibility statements. Metabolic operations that generally occur
in most organisms, e.g. bacteria (84), fungi (85), mammalian cells (86, 87), and even plants
(88), are respiration and aerobic fermentation (89). While with respiration, ATP is generated
at high yields, with aerobic fermentation ATP is only generated at low yields through substrate-level phosphorylation (90). Different research fields postulated numerous explanations
why cells under aerobic conditions choose an ATP-inefficient fermentative metabolism over
an ATP-efficient respiratory metabolism, amongst which are economics of enzyme production (89), ‘make-accumulate-consume’ strategy (85), intracellular crowding (84), limited nutrient transport capacity (91), and adjustments to growth-dependent requirements (86, 87).
These explanations have all the short coming that they do not give a detailed mechanistic
reason, therefore they could not be quantified and validated by experimental data.
We asked whether—in contrast to the previously proposed explanations—rather a common
inevitable principle would underlie the specific choice of metabolic operation, with fermentation being seemingly connected with high, and respiration with low rates of glucose uptake and glycolysis (92). Specifically we investigated, using constraint-based modeling (4, 6),
whether thermodynamic constraints could be the cause for the ubiquitously observed and
obviously sugar uptake rate-dependent choice between respiration and aerobic fermentation.
Here, we aimed to identify these constraints by integrating previous work of thermodynamic
analysis of metabolic networks, thermodynamic analysis of cellular growth, and the use of the
loop-law to formulate a new thermodynamic metabolic network model.
Outline of this thesis
In Chapter 2, we develop a constraint-based model for Saccharomyces cerevisiae, which combines mass and charge balances with comprehensive description of the biochemical thermodynamics governing metabolic operations. As the model does not use any heuristic irreversibility assignments of the intracellular rates, therefore all conclusions drawn from the model
are based on thermodynamic principles. We use this model together with experimental physiological and metabolome data of S. cerevisiae and identify a global thermodynamic constraint, i.e. a limit in the cellular rate of entropy production. Using this constraint in a flux
balance analysis together with a cellular objective of maximizing the growth rate, we correctly
predict the intracellular and extracellular rates of S. cerevisiae for a wide range of different
16
This work
glucose uptake rates.
In Chapter 3, we present a computational workflow to develop thermodynamic metabolic network models similar to the one developed in Chapter 2. Here, we especially focus on
the potential pitfalls, when gathering the necessary biochemical information from different
sources. Further, we give advice on potential model reductions and how to gather the necessary data to train such a model.
In Chapter 4, we use the thermodynamic metabolic model of S. cerevisiae and develop a statistical workflow to accurately quantify the metabolic operation based on experimental data,
where we combine the thermodynamic metabolic model with isotopomer balancing. We use
this workflow to infer intracellular metabolic fluxes and also backward fluxes from measured
extracellular rates, metabolomics data, standard Gibbs energies of reactions, and measured
isotopomer patterns.
Lastly, we conclude this work by a short discussion of the impact of the here identified principles on metabolic operations and suggest potential following studies.
17