dc.contributor.author
Palinkas, Aljoscha
dc.date.accessioned
2018-06-08T01:41:57Z
dc.date.available
2015-07-17T09:35:31.082Z
dc.identifier.uri
https://refubium.fu-berlin.de/handle/fub188/13742
dc.identifier.uri
http://dx.doi.org/10.17169/refubium-17940
dc.description.abstract
Two cellular subsystems are the metabolic network and the gene regulatory
network. In systems biology they have mostly been modelled in isolation with
ordinary differential equations (ODEs) or with tailored formalisms as e.g.
constraint-based methods for metabolism or logical networks for gene
regulation. In reality the two systems are strongly interdependent. For
mathematical modelling the integration is a challenge and a variety of
different approaches has been proposed. Long term alterations in metabolism
result from changes in gene expression, which determines the production of
enzymes. This transcriptional control can adjust the metabolic network to
changes in the environment or the requirements of the cell. In fact, the cell
cycle is connected to cyclic changes in metabolism, so-called metabolic
cycling, but alterations are also observed in non-proliferating cells in a
constant environment. A mathematical model to describe and explain alterations
in metabolism will be proposed here. At first, a resource allocation model for
the enzymes in a metabolic network is developed and integrated into a
constraint-based model of metabolism in Chap. 3. The reaction rates are
bounded depending on the availability of enzymes, which in turn is determined
by the overall distribution of the limited resources. In Chap. 4, this model
is used to test the hypothesis that metabolic alterations are a means of the
cell to achieve the required production of metabolic output most efficiently.
First a toy model is analysed and then the method is applied to a core
metabolic network of the central carbon metabolism. The tasks of this
metabolic network are the production of biomass precursors as well as
constantly providing a minimum of energy and anti-oxidants. The mathematical
model gives a mixed integer linear optimisation problem with a few quadratic
constraints and a quadratic objective function. Instead of searching for a
single flux distribution, a feasible solution corresponds here to a sequence
of several flux distributions together with the time that is spent in each of
them. The consecutive usage of these flux distributions during the associated
time spans yields the required output. The objective is the minimisation of
the total time needed. The computations demonstrate that switching between
several flux distributions allows producing the output in a significantly
shorter time span, compared to an optimal single flux distribution. In a toy
model we could identify the relationship between the model parameters and the
results concerning the efficiency of static versus sequential flux
distributions. Such a comprehensive analysis is not possible for the large
number of parameters in our core metabolic network. To make sure that the
confirmation of the hypothesis is not restricted to a minor region in the
parameter space of the resource allocation model, we perturbed the parameters
randomly and repeated all computations. This empirical analysis showed that
the significant gain in performance is a robust feature of the model. From the
mathematical point of view the proposed resource allocation model defines for
each gene expression state a flux space from which a flux distribution can be
chosen. This flux space is in general not linear and not convex, which turns
out to depend on the space of all possible gene expression states. In our
model the genes regulate the enzyme concentrations in an on-off manner, only
determining the active and inactive parts of metabolism. Furthermore, certain
groups of genes are regulated together as functional units. As a consequence,
the enzyme concentrations cannot be perfectly adjusted to a given flux
distribution in this model and it is for this reason that switching can
increase the efficiency. A simpler model of resource allocation, which is
solely based on molecular crowding, has been proposed before in the
literature. It allows distributing the resources to perfectly match any given
flux distribution and switching is then not necessary to obtain the minimal
production time. In contrast to such a resource allocation model, our
modelling assumptions and computational results suggest a design principle,
where the optimal adjustment to given conditions and requirements is not
achieved by fine-tuning of enzyme concentrations, but by switching between
different flux distributions, which are only roughly determined by
transcriptional control and which do not perfectly match one certain condition
or requirement. In terms of geometry, the difference lies in the convexity of
the flux space. If it is convex, minimal production time can always be
achieved with a single flux distribution. To characterise a set of flux
distributions sufficient to constitute an optimal sequence, the flux space of
the network without the resource allocation model is considered in Chap. 3.
The corresponding polytope allows characterising a finite subset of the flux
space in terms of decomposability, a notion which is closely related to
elementary modes. For any output requirements, an optimal sequence can be
constituted from this finite set of flux distributions. In practice, solving
the optimisation problem that was derived from the modelling approach as well
as computing the sufficient finite subset, is not tractable for large
networks. Also divide and conquer strategies are not promising to obtain
optimal solutions in general, a counterexample is given in Chap. 6.
Alternative computational methods to obtain optimal or approximative optimal
solutions are then presented. The gene regulatory network behind the metabolic
genes is not fully considered in the resource allocation model of Chap. 3.
Only some constraints are added in the application to the core metabolic
network in order to exclude unrealistic patterns of gene expression.
Incorporating more information about the gene regulation into the
computational model is in fact improving the tractability, because the search
space is reduced. A sufficiently small search space of gene expression
sequences gives the possibility to perform a more precise and extensive
analysis using an alternative computational approach. In Chap. 5, the
perturbations of model parameters, as applied to the core metabolic network to
verify the robustness, are considered in general. From the mathematical point
of view, the linear constraints that bound the flux space are perturbed. The
consequences on the geometry of the flux space and on the objective value of
an optimisation problem over this flux space are analysed and an effect is
discovered, which is surprising at first sight. If the bounds on the reaction
rates are perturbed individually, without a bias for increase or decrease, the
expected objective value of a given linear optimisation problem is decreased
in expectation. This effect emerges from the representation of the flux space.
In particular redundancy of the constraints plays a crucial role. The
modelling and the analysis of the dynamics of gene regulatory networks with
so-called logical networks is a common discrete approach. Logical networks are
often represented by logical functions, which have the advantage of being
mathematical objects that can be given in a natural and easily understandable
format, namely Boolean expressions. In Chap. 7, a method is presented to
obtain a short and well readable representation of a given logical function.
It is based on the minimisation of Boolean expressions, but is designed for
multi-valued logical functions in particular. All possible dynamics of a
logical network can be represented in the so-called state transition graph.
Simply by assigning rates to all edges, which represent the transitions
between different states, this directed graph becomes a continuous time Markov
chain (CTMC) which we call a stochastic logical network. This modelling
approach opens new possibilities for the analysis of quantitative dynamical
properties as shown in Chap. 8. In contrast to this abstract model, detailed
mechanistic and stochastic models of biochemical reaction systems can be
formulated with the chemical master equation, which also defines a CTMC. In
fact, these two formalisms can be combined, so that distinct components of the
biological system are modelled in much detail by the master equation and other
parts on a higher abstraction level as a stochastic logical network. The
combined model can focus on certain aspects, capturing related quantitative
and stochastic effects, while keeping the overall complexity to a minimum.
Finally, Chap. 9 discusses the feedback regulation from metabolism to gene
regulation. In an integrated dynamic model of gene regulation and metabolism,
this aspect should not be missing. Since constraint-based models neglect the
concentrations of metabolites, it is difficult to determine the regulatory
feedback to the genes. This problem can be circumvented by only inferring
metabolic mediated interactions between genes, in the sense that a switch in
gene expression leads to an alteration in the metabolic network, which in turn
gives a new regulatory input to the gene network. To this end, a constraint-
based approach is proposed and compared to a method from the literature, which
is based on metabolic sensitivity analysis. Furthermore, a strategy to derive
concentration changes from changes in flux rates and enzyme activities is
shortly presented.
de
dc.description.abstract
In der vorliegenden Dissertation wird die Regulation des zellulären
Stoffwechsels durch Gene mit mathematischen Modellen beschrieben.
Veränderungen in der Expression von Genen, die für die Produktion von Enzymen
verantwortlich sind, bewirken längerfristige Umstellungen im Stoffwechsel. Auf
diese Weise wird auf der Transkriptionsebene z.B. die Anpassung des
Stoffwechsels an äußere Bedingungen oder sich ändernde Bedürfnisse der Zelle
gesteuert. Ein Modell der Verteilung von Ressourcen für die Produktion von
Enzymen wird in Kap. 3 entwickelt und in ein constraintbasiertes
Stoffwechselmodell integriert. Die Flussraten der einzelnen Reaktionen werden
dabei von der Menge an verfügbaren Enzymen beschränkt, welche wiederum von der
Verteilung der Ressourcen auf das ganze Stoffwechselnetzwerk abhängt. In Kap.
4 wird dieses Modell dann verwendet, um der Hypothese nachzugehen, dass
Umstellungen des Stoffwechsels der Zelle dazu dienen könnten, verschiedene
benötigte Metaboliten mit höchstmöglicher Effizienz zu produzieren. Zuerst
analysieren wir die Effizienz an einem Spielmodell, bevor dann ein Netzwerk
des zentralen Kohlenstoffwechsels untersucht wird. In diesem Modell betrachten
wir die Produktion von einigen Bausteinen der Biomasse. Zusätzlich wird die
permanente Bereitstellung von genügend Energie und Antioxidantien gefordert.
Das Ziel ist dabei, die geforderte Produktion von Metaboliten in möglichst
kurzer Zeit zu erfüllen. Das mathematische Modell ist ein Optimierungsproblem
mit gemischt-ganzzahligen Variablen, linearen und wenigen quadratischen
Nebenbedingungen sowie einer quadratischen Zielfunktion. Eine Lösung
entspricht einer Abfolge von Flussverteilungen und deren Dauer. Die
Berechnungen in Kap. 4 zeigen, dass das Umschalten zwischen verschiedenen
Flussverteilungen des Stoffwechsels es ermöglicht, die Biomasse in einer
signifikant kürzeren Zeit zu produzieren als es eine einzelne Flussverteilung
erlauben würde. Die Robustheit dieser Ergebnisse bezgl. der Parameterwahl
wurde empirisch bestätigt. Die mathematischen Eigenschaften des
Ressourcenverteilungsmodells werden in Kap. 3 analysiert. Unser Modell geht
von einer groben Steuerung der Enzymkonzentrationen auf der
Transkriptionsebene aus, was durch binäre Genexpression modelliert wird, die
nur festlegt, welche Stoffwechselpfade aktiviert sind und welche nicht.
Weiterhin gibt Kap. 3 eine Charakterisierung von bestimmten besonders
effizienten Flussverteilungen. Aus dieser endlichen Menge kann immer eine
Abfolge zusammengestellt werden kann, die optimal die gegebenen Anforderungen
erfüllt, d.h. eine optimale Lösung unseres Optimierungsproblems ist. Für große
Netzwerke sind die Optimierungsprobleme, die das Ressourcenverteilungsmodell
formuliert, numerisch nicht lösbar. Daher werden in Kap. 6 verschiedene
alternative Berechnungsmethoden vorgestellt. In Kap. 5 werden stochastische
Störungen der Nebenbedingungen, die den Flussraum beschränken, untersucht. Es
zeigt sich, dass hier ein unerwarteter Effekt auftritt. Er wird durch die im
mathematischen Sinne nicht eindeutige Darstellung des Stoffwechselmodells
durch lineare Nebenbedingungen bestimmt. Zur Modellierung und Untersuchung der
Dynamik von genregulatorischen Netzwerken wird oft der Formalismus der
sogenannten logischen Netzwerke verwendet. In Kap. 7 wird ein Algorithmus
vorgeschlagen, der eine kurze und gut lesbare Darstellung der benötigten
logischen Funktionen liefert. Eine Erweiterung von logischen Netzwerken zu
einem Markov-Prozess wird in Kap. 8 vorgeschlagen, um stochastische und
quantitative Aspekte darzustellen. Exemplarisch wird gezeigt, wie sich dieser
Formalismus direkt mit einer Mastergleichung für bestimmte Reaktionen oder
regulatorische Interaktionen kombinieren lässt. In einem Ausblick wird in Kap.
9 das Problem der Feedback-Regulation des genregulatorischen Netzwerkes durch
den Stoffwechsel behandelt.
en
dc.rights.uri
http://www.fu-berlin.de/sites/refubium/rechtliches/Nutzungsbedingungen
dc.subject
mathematical optimization
dc.subject
systems biology
dc.subject
metabolic networks
dc.subject.ddc
500 Naturwissenschaften und Mathematik::510 Mathematik
dc.title
Integrated modelling of metabolic and regulatory networks
dc.contributor.contact
aljoscha.palinkas@fu-berlin.de
dc.contributor.firstReferee
Prof. Dr. Alexander Bockmayr
dc.contributor.furtherReferee
Prof. Dr. Hermann-Georg Holzhütter
dc.date.accepted
2015-05-11
dc.identifier.urn
urn:nbn:de:kobv:188-fudissthesis000000099801-6
dc.title.translated
Integrierte Modelle metabolischer und regulatorischer Netzwerke
de
refubium.affiliation
Mathematik und Informatik
de
refubium.mycore.fudocsId
FUDISS_thesis_000000099801
refubium.mycore.derivateId
FUDISS_derivate_000000017437
dcterms.accessRights.dnb
free
dcterms.accessRights.openaire
open access