A Mathematical Model for Trehalose Uptake in the Thermophilic Bacterium Rhodothermus marinus

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A Mathematical Model for Trehalose Uptake in the Thermophilic Bacterium

Rhodothermus marinus

Dissertation zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.),

eingereicht bei der

Mathematisch-Naturwissenschaftlichen Sektion im Fachbereich Biologie der Universit¨at Konstanz

Vorgelegt von Irena Hendekovic

Konstanz 2006

Datum der m¨ undlichen Pr¨ ufung: 12.05.2006 Referent: Prof. Dr. W. Boos

Referent: Prof. Dr. E. Bohl

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Zusammenfassung

Die vorliegende theoretische Arbeit beschäftigt sich mit der Aufklärung des Mech- anismus der Trehaloseaufnahme des thermophilen, halophilen Bakteriums Rho- dothermus marinus. Verschiedene chemische Netzwerke des postulierten Aufnah- memechanismus wurden via mathematischer Modellierung in Systeme gekoppel- ter Differentialgleichungen übersetzt. Sich daraus ergebende analytische Funk- tionen wurden für einen mehrdimensionalen Fehlerquadratsummenfit an experi- mentelle Daten verwendet. Mithilfe dieser Fits kann zwischen verschiedenen Mod- ellen unterschieden werden. Die hier verwendeten Daten wurden freundlicher- weise zur Verfügung gestellt von Carla Jorge vom Labor von Prof. Dr. Helena Santos, Instituto de Tecnologia Quimica e Biologica (ITQB) in Oeiras, Portugal.

Verschiedene Datensätze leisten eine Beitrag zur Klärung obiger Fragestel- lung: Die experimentellen Daten bestehen aus Messungen der Aufnahmerate radioaktiver Glukose, die in Kapitel 2 analysiert werden. Kapitel 3 beschäftigt sich mit Messungen der Glukosefreisetzung ins Medium aufgrund von in vivo Treha- laseaktivität. Aufnahmedaten radioaktiver Trehalose werden in den Kapiteln 4 und 5 modelliert, gefolgt von Simulationen von Kreuzinhibitionsexperimenten in Kapitel 6. Zusätzlich zu Diesen haben in vitro Inhibitionsexperimente mit gere- inigter Trehalase gezeigt, dass Glukose die Trehalaseaktivität mit einer Ki von 12mM inhibiert.

Zwei Modelle für die Trehaloseaufnahme wurden vorgeschlagen: Modell 1 besteht aus einem Diffusionskanal für Glukose und Trehalose in der Außen- membran, einer periplasmatischen Trehalase, die einfließende Trehalose in zwei Glukoseeinheiten spaltet, und einem Glukosetransporter in der Cytoplasmamem- bran. Modell 2 enthält einen zusätzlichen Trehalosetransporter in der Innen- membran. Das Glukosetransportsystem stellt einen funktionell unabhängigen Teil beider Modelle dar. Dadurch können die aus den Glukoseaufnahmedaten gewonnenen Fitparameter für den Fit der Daten der Trehaloseaufnahme sowie der enzymatischen Spaltung verwendet werden. Zur Beschreibung der in vivo Trehalasedaten werden vier weitere Parameter benötigt. Dieselben vier Parame- ter werden für die Trehaloseaufnahme nach Modell 1 benutzt; Modell 2 enthält

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enzymatische Spaltung) beschreiben muss.

Einer der verbleibenden Parameter, z₂, kann aus der Ki der oben erwähnten in vitroGlukoseinhibition von Trehalase, sowie aus der in Kapitel 2.8 berechneten K_m^gt des Glukosetransporters in der Cytoplasmamembran berechnet werden. Die Theorien der enzymatischen Spaltung und der Trehaloseaufnahme liefern Ein- schränkungen, die die verbleibenden Fitparameter an die kinetischen Parameter K_mê,V_maxê der Enzymkurve und K_m^tr,Vmax^tr (Modell 1) oderK_m^su,Vmax^su (Mod- ell 2) der Trehaloseaufnahmekurve koppelt. Für diese Parameter erhalten wir eine Approximation durch Fitten einer Michaelis-Menten Funktion an die Daten.

Diese Beschränkungen reichen aus, um alle Fitparameter von Modell 1 zu berech- nen; für Modell 2 ist ein zusätzlicher Scan überz6 notwendig.

Unter Verwendung von Modell 2 wurde durch Optimierung der vier kinetischen Parameter ein Fit errechnet, der alle Messungen beschreibt. Modell 1 kann entweder die Daten der enzymatischen Spaltung, oder die Trehaloseauf- nahmedaten bei hoher Konzentration beschreiben, nicht jedoch beide Experi- mente gemeinsam. Zudem ist es mit diesem Modell nicht möglich die Messun- gen der Trehaloseaufnahme bei niedrigen Konzentrationen zu beschreiben. Die Unfähigkeit von Modell 1, sämtliche experimentellen Befunde schlüssig erklären zu können, zeigt die Notwendigkeit eines zusätzlichen Trehalosetransporters, der Modell 1 von Modell 2 unterscheidet.

Zur weiteren Best¨atigung des Modells wurden die Parameter des Glukose- fits sowie von Modell 2 benutzt um ein Inhibitionsexperiment zu simulieren.

Die Konzentration des Substrats Trehalose ist hierbei konstant, w¨ahrend die Konzentration des Inhibitors Glukose ansteigt. Die Simulation zeigt dass Glukose im Modell 2 die Aufnahme von Trehalose bei hohen Trehalosekonzentrationen wesentlich inhibiert, w¨ahrend bei niedrigen Trehalosekonzentrationen selbst eine sehr hohe Glukosekonzentration nur einen geringen inhibitorischen Effekt hat.

Diese Beobachtung beruht auf der Tatsache, dass bei niedrigen Trehalosekonzen- trationen der dominante Aufnahmeweg ¨uber den direkten Trehalosetransporter f¨uhrt, der in diesem Modell nicht durch Glukose inhibiert wird.

Die Ergebnisse der Simulation stimmen qualitativ mit den experimentellen Daten ¨uberein, wobei letztere schon bei der geringen Konzentration von 1µM eine Inhibition der Trehaloseaufnahme um den Faktor zehn zeigen. Um diese experimentellen Resultate mithilfe der Theorie quantitativ zu erkl¨aren, wird eine Erweiterung des Modellnetzwerks um eine durch Glukose vermittelte Inhibition des Trehalosetransporters in der Cytoplasmamembran vorgeschlagen.

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Acknowledgements

I am indebted to Prof. Winfried Boos, Prof. Erich Bohl and Prof. Helena Santos for the opportunity to work at the juncture between mathematics and biology.

Prof. Bohl and Prof. Boos have been very attentive supervisors and invested a lot of time and work to make this project possible. I especially enjoyed hours and hours of small talk and scientific discussions spent with one or the other or both of them over many liters of coffee. I have certainly learned as much on life, the universe and everything in the past few years as on biomathematics.

Prof. Michael Junk provided financial assistance, administrative help and moral support wherever it was possible and needed, and proved to be a kind, considerate and very witty workmate.

Special thanks go to Carla Jorge who made the measurements my theoretical analysis relies on and patiently answered all the questions on the data I could come up with.

Dr. Eberhard Luik was a great help whenever it came to programming, technical equipment or other pracical details of university life, and a reliable companion for all sorts of social activities, especially if they involved food.

I have shared many enjoyable lunches, coffee breaks and lively discussions with PD Dr. Johannes Schropp, Dr. Vita Rutka and Martin Rheinl¨ander. I particularly want to thank Vita for the hours she invested into improving the appearance of my dissertation.

I am very grateful for the advice, the help and the warm welcome I have always recieved in the Boos Lab, even if I hadn’t shown up for ages. Lab work would have been a lot more frustrating without you all!

Last but not least, this work would never have been finished without the love and support of my friends and family. Xenia Milojevi´c and Marcus Kubitzki encouraged, consoled and pushed me on if necessary, and my parents always provided the comforting security of having a place to go to if everything else fails.

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Chapter 1 Introduction

The focus in microbiological research has broadened in recent years to include a wide variety of organisms beside the classical model systems likeEscherichia coli orBacillus subtilis. Special interest in thermophilic bacteria has arisen due to the biotechnological need for thermostable enzymes. Rhodothermus marinuswith its moderate growth temperatures from 54^◦C to 77^◦C [1] has proven to be a good source for some thermoresistant enzymes, such as a cellulase [7] or a xylanase [10]. Furthermore, halophilic thermophilic bacteria adapt to high osmolarity by accumulating some compatible solutes not usually found in mesophilic organisms such as mannosylglycerate and mannosylglyceramide [12, 16]. These compatible solutes are of biotechnological interest since they have been shown to protect several enzymes against thermal inactivation and other stresses very efficiently [6, 13, 15].

A thorough understanding of all components of the Rhodothermus marinus stress response has therefore become of prime importance. A compatible solute playing a prominent role in the Rhodothermus marinus response to high tem- perature and salinity is trehalose [16], a nonreducing glucose dimer commonly used by bacteria to maintain their turgor and protect enzymes and other proteins against salt-induced stress [2, 11, 17]. The uptake mechanism for trehalose in this bacterium is yet unknown since a gene transfer system for gentic analysis and manipulation in Rhodothermus marinus has been established only very recently [4].

This work presents the mathematical analysis of two models for trehalose transport in Rhodothermus marinus based on experimental data obtained in a collaboration with the research group of Prof. Dr. Helena Santos at the Instituto the Technologia Quimica e Biologica (ITQB) in Oeiras, Portugal. All experimental measurements analyzed in this dissertation have been made by Carla Jorge at the ITQB.

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0 20 40 60 80 100 120 0

500 1000 1500 2000 2500 3000

time [s]

amount of glucose [cpm]

Fig. 1.1: Time course for the uptake of 400µM trehalose. The amount of radioactivity in the cells is given in counts per minute. Clearly a linear fit is not adequate for the measurements and will underestimate the initial uptake rate.

The experimental basis of this study consists of uptake measurements with radioactive glucose and trehalose. Radioactive substrate is offered to a suspen- sion of bacteria, and samples are drawn after defined time spans and washed with medium to remove external traces of radioactive sugar. The amount of radioactivity taken up by each sample is determined by a scintillation counter; the data belonging to one time curve are fitted with a Michaelis-Menten function

v(x) = p1x p₂+x

to obtain the initial uptake rate associated with the known initial substrate concentration. The data could not be described accurately with a linear fit because many sets of data do not form a straight line (see [5], p.170). If they do, the results of the Michaelis-Menten fit are very close to the slope of a linear fit.

A typical time course with a Michaelis-Menten fit curve is shown in figure 1.1.

The initial uptake rate is converted into an amount of radioactive sugar taken up per minute and mg bacterial protein and constitutes one data point in the measurements shown on p. 22, 52 and 73.

Glucose was found to be transported through the cytoplasmic membrane by a proton-motive force driven glucose transporter (for a review of secondary transport systems see [8, 14]). Radioactive trehalose is taken up, but no uptake system could be identified as yet. Instead, a periplasmic trehalase was found and pu- rified; it is subject to product inhibition by glucose with an inhibition constant Ki of 12mM. Information onin vivo trehalase activity was obtained by measur- ing the release of glucose at different concentrations of trehalose after destroying

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CHAPTER 1. INTRODUCTION 3

T T

B A D_a

D_p

D_p D_p

D_i

Gp

G_a

Gp

G_i medium

periplasm

cytoplasm

Gp

Fig. 1.2: Schematic depiction of the two trehalose uptake models. Model 1 only consists of the periplasmic trehalaseT cleaving periplasmic trehaloseD_pinto two units of periplasmic glucoseGp and the glucose uptake system A in the inner membrane transporting periplasmic glucoseGpinto the cytoplasm; the sugars can cross the outer membrane without the help of a specific transport protein. Glucose and trehalose in the medium are denoted byGa andDa respectively, in the cytoplasm they are called Gi and Di. The additional trehalose uptake system B for model 2 transporting periplasmic trehaloseDp into the cytoplasm is drawn in a box.

the proton gradient through addition of the protonophore carbonyl cyanide m- chlorophenylhydrazone (CCCP). Trehalose transport as well as glucose uptake are then prevented, the produced glucose can only escape into the medium. Cross- inhibition experiments with glucose and trehalose have been performed to check whether both sugars are transported by the same uptake system; interestingly, the uptake of radioactive trehalose was inhibited by cold glucose but not vice versa.

These findings seem compatible with two different models: in model 1, trehalose is not taken up as such but first cleaved into two units of glucose which is then taken up through the glucose transporter in the inner membrane; model 2 is extended by a specific trehalose transporter in the inner membrane. Figure 1.2 comprises both models: model 1 only consists of a periplasmic trehalase and a glucose transporter in the inner membrane; the sugars can cross the outer membrane without the help of a specific transport protein. The additional trehalose transporter for model 2 is drawn in a box.

The mathematical modeling consists of differential equations derived from the chemical network describing each experiment. The large differences in compartment size are used to introduce different time scales and thus formulate analytical

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expressions for the measured uptake or release rates. These analytical expressions contain fit parameters allowing to adjust the curves to the measured data. If a single set of positive fit parameters can be found that leads all theoretical curves through the respective data points, this is a strong point in favor of the underlying model.

In chapter 2 we will describe the uptake of glucose alone; the data for glucose transport are fitted independently of the trehalose uptake experiments and yield parameters which can be used for the remaining fits. Chapters 3, 4 and 5 formulate restrictions for the fit parameters, coupling them to the kinetic parameters Km andVmaxof the in vivotrehalase data in chapter 3 and trehalose uptake data in chapters 4 (model 1) and 5 (model 2). Chapter 6 finally describes the uptake of radioactive trehalose disturbed by addition of cold glucose. The results of the work are summarized in chapter 7 and the main definitions and equations are put together in tables in the appendix.

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Chapter 2 Glucose Transport

2.1 Experimental Setup

The glucose uptake system inRhodothermus marinus consists of an outer membrane diffusion channel (reaction (2.1) in the network in chapter 2.2) and a proton motive force-driven transporter in the cytoplasmic membrane (reaction (2.2)). It is independent from the trehalose transport and can be analyzed by itself. To that end, we have to introduce the bacteria into a medium with radioctive glucose and measure the amount of radioactivity taken up into the cell after defined time periods. These values can be used to estimate the maximal uptake rate for glucose at the set initial glucose concentration. The substrate concentration is then varied to find the overall maximal uptake rate and the behavior of the transport system at small concentrations.

T T

B A D_a

D_p

D_p D_p

D_i

Gp

Gp a

Gp

G_i G medium

periplasm

cytoplasm

Gp

Fig. 2.1: Schematic depiction of the glucose transport system; active parts are black.

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2.2 Network

The first step in the mathematical analysis of a biological model consists in writing down a chemical network describing the interactions of the different components. In our case we have to describe a sugar transport through four different volumina: the outer medium with the very large volume α, the periplasm with the very small volume β, the inner membrane with the even smaller volume γ, and the cytoplasm with the medium-sized volume δ. The distribution of the different compounds over the four volumes is shown in equation (2.3). The glucose uptake network consists of the following chemical equations:

Glucose in the outer medium Ga can cross the outer membrane to form Gp (the reaction constants are assumed different for the in and out direction to allow for a possible energized transport):

Ga

k3

k⁻3

Gp; (2.1)

the periplasmic glucoseGp is then transported into the cytoplasm by the glucose transporter A, and is thereby transformed intoGi:

Gp+A k₂ k−2

E2

κ₂

→ Gi+A. (glucose transporter) (2.2) Note that the binding of Gp to the periplasmatic trehalase is not modeled here because the effect is negligible in this experiment.

The chemical components of the system are distributed over the different volumina as follows:

compartment volume compounds

outer medium α Ga

periplasm β Gp

inner membrane γ A, E2

cytoplasm δ Gi

(2.3)

2.3 Differential equations

A chemical network usually can be translated easily into differential equations for the concentrations of the substances it deals with. Here, the concentration of a

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CHAPTER 2. GLUCOSE TRANSPORT 7

substance is defined as the amount of the substance divided by the volume of the compartment it can be found in. In this text, we will denote the concentration of a substanceXi by xi and the amount of the substance bynxi. The concentration e2 of the enzyme complex E2 for example is defined by the amount ne2 of E2 in the membrane compartment divided by its volume γ:

e₂(t) = ne2(t)

γ . (2.4)

In differential equations for chemical reactions, we describe the rates of change in the amounts of substances due to their reaction with each other. These rates are dependent on the concentrations of the substrates, as the probability for two substrates to meet and form a product is not dependent on the absolute amount of substrate molecules, but on the amount of molecules per volume: the chemical equation

E+F κ

→ G (2.5)

leads to the differential equations

˙

ne(t) = −κe(t)f(t),

˙

nf(t) = −κe(t)f(t),

˙

ng(t) = κe(t)f(t).

To obtain rates of change in concentrations we have to use equation (2.4) and divide every amount of substance by the volume it occurs in. If the whole reaction takes place in a single volume V1 we obtain with _V^κ₁ = −κ⁰

˙

e(t) = ⁿ^˙^e_V^(t)₁ = −_V^κ₁e(t)f(t) = −κ⁰e(t)f(t), f˙(t) = ⁿ^˙^f_V^(t)

1 = −_V^κ

1e(t)f(t) = −κ⁰e(t)f(t),

˙

g(t) = ⁿ^˙^g_V^(t)₁ = _V^κ₁e(t)f(t) = κ⁰e(t)f(t).

This representation is usually used in textbooks, as most experiments are performed in one volume. In this case not only the amount of product formed is the same as the amount of each substrate degraded, but the same holds true for the concentrations. The case is different if the reaction takes place across a membrane separating a large volume V1 from a small volume V2. Let us assume that in equation (2.5) the compounds E and F are located in a compartment with the volumeV₁ and Gis located in a compartment with the volume V₂. The differential equations then read

˙

e(t) = ⁿ^˙^e_V^(t)₁ = −_V^κ₁e(t)f(t) = −κ⁰e(t)f(t), f˙(t) = ⁿ^˙^f_V^(t)₁ = −_V^κ₁e(t)f(t) = −κ⁰e(t)f(t),

˙

g(t) = ⁿ^˙^g_V^(t)₂ = _V^κ₂e(t)f(t) = ^V_V¹₂κ⁰e(t)f(t).

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This means that the concentration of the product G in the small volume V₂ changes more than the concentrations of the substrates E and F in the large volume. This makes sense, as the same absolute change in number of particles is distributed over a much greater volume in V1 than inV2.

We can now set up the differential equations describing the changes in amounts of substance due to the chemical reactions (2.1) and (2.2):

n˙a(t)

˙ ne2(t)

=

−k2gp(t) k−2+κ₂ k₂gp(t) −(k−2+κ₂)

a(t) e₂(t)

(2.6)







˙ ngp(t)

˙ nga(t)

˙ ngi(t)

˙ ne2(t)







= Cg





 gp(t) ga(t) gi(t) e2(t)





 ,

Cg =







−(k2a(t) +k−3) k3 0 k−2

k−3 −k3 0 0

0 0 0 κ2

k₂a(t) 0 0 −(k⁻2+κ₂)





 .

(2.7)

If we want to write both sides of equations (2.6) and (2.7) in concentrations, we have to divide the amounts of substance on the left sides by the volume of the compartment each compound occurs in. Equation (2.3) shows which substance belongs to which compartment. For (2.6), all compounds are localized in the inner membrane with the volume γ, so that we can multiply both sides of the equation by ¹_γ and obtain

a(t)˙

˙ e₂(t)

= 1 γ

n˙a(t)

˙ ne2(t)

= 1 γ

−k2gp(t) k−2+κ2

k₂gp(t) −(k⁻2+κ₂)

a(t) e₂(t)

(2.8) To do the same with the sugar equation (2.7) we have to multiply it by

D^g₃ =







1

β 0 0 0 0 _α¹ 0 0 0 0 ¹_δ 0 0 0 0 _γ¹







and get







˙ gp(t)

˙ ga(t)

˙ gi(t)

˙ e2(t)







= D^g₃







˙ ngp(t)

˙ nga(t)

˙ ngi(t)

˙ ne2(t)







= D^g₃Cg











. (2.9)

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The initial values for gp, gi and e₂ are zero, because they are either intermedi- ate compounds or products. The initial values for external glucose ga and the transport protein a have to be positive for a transport reaction to occur:

a(0) > 0, gp(0) = 0 e2(0) = 0, ga(0) > 0 gi(0) = 0.

(2.10)

2.4 Separating Different Time Scales

The different volumina in which the reactions take place are of widely different size. We have calculated them from the experimental conditions (l = 1 liter):

α = 0.3·10⁻³l, β = 2.3·10⁻⁹l, γ = 4·10⁻⁹l, δ = 38.87·10⁻⁹l; (2.11) they lead to the volume fractions

α

β = 0.13·10⁶, α

γ = 0.075·10⁶, α

δ = 0.07718·10⁵. (2.12) We would now like to use the differences in size of the different compartments to motivate a pseudo steady state analysis. To that end, we have to multiply the equations (2.8) - (2.9) by the large volume α, so that the input variable ga(t) in the outer medium, which has to stay dynamic, carries a one in front of the differential equation, whereas all others obtain one of the large factors from (2.12):

α·a(t)˙ α·e˙₂(t)

= α γ

−k2gp(t) k⁻2+κ2

k₂gp(t) −(k⁻2 +κ₂)

a(t) e₂(t)

(2.13) and







α·g˙p(t) α·g˙a(t) α·g˙i(t) α·e˙2(t)







=







α

β 0 0 0

0 1 0 0 0 0 ^α_δ 0 0 0 0 ^α_γ





 Cg











. (2.14)

To get rid ofα on the left hand side of equations (2.13) and (2.14), we would like to scale the timet withαwithout changing the dimension of the time. Therefore we use the dimensionless variable

¯ α = α

η, η = 1l, (2.15)

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and consider the equations (2.13) and (2.14) as well as (2.8) and (2.9) at the time

¯

αt. This leads us to α·a⁰( ¯αt)

α·e⁰₂( ¯αt)

= ^α_γ

−k₂gp( ¯αt) k−2+κ₂ k2gp( ¯αt) −(k−2+κ2)

a( ¯αt) e2( ¯αt)

(2.16) and







α·g_p⁰( ¯αt) α·g⁰_a( ¯αt) α·g⁰_i( ¯αt) α·e⁰₂( ¯αt)







=







α

β 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ^α_γ





 C_g⁰





 gp( ¯αt) ga( ¯αt) gi( ¯αt) e₂( ¯αt)





 ,

C_g⁰ =







−(k2a( ¯αt) +k−3) k₃ 0 k−2

k−3 −k3 0 0

0 0 0 ^α_δκ2

k2a( ¯αt) 0 0 −(k⁻2+κ2)







(2.17)

on the one side and, using f⁰(s) := df(s)

ds , f⁰( ¯αt) = f⁰(s) with s = ¯αt, (2.18) to

η· _dt^da( ¯αt) η· _dt^de2( ¯αt)

=

η·α¯·a⁰( ¯αt) η·α¯·e⁰₂( ¯αt)

=

α·a⁰( ¯αt) α·e⁰₂( ¯αt)

(2.19) and analogously







η·_dt^dgp( ¯αt) η· _dt^dga( ¯αt) η· _dt^dgi( ¯αt) η· _dt^de₂( ¯αt)







=







α·g_p⁰( ¯αt) α·g_a⁰( ¯αt) α·g_i⁰( ¯αt) α·e⁰₂( ¯αt)







(2.20)

on the other. We can now combine equations (2.16) and (2.19) to η·_dt^da( ¯αt)

η·_dt^de2( ¯αt)

= α γ

−k₂gp( ¯αt) k−2+κ₂ k2gp( ¯αt) −(k−2+κ2)

a( ¯αt) e2( ¯αt)

(2.21) and equations (2.17) and (2.20) to







η· _dt^dgp( ¯αt) η· _dt^dga( ¯αt) η· _dt^dgi( ¯αt) η· _dt^de2( ¯αt)







=







α

β 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ^α_γ





 C_g⁰





 gp( ¯αt) ga( ¯αt) gi( ¯αt) e2( ¯αt)







. (2.22)

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Note that the fraction ^α_δ in the equation for the productgi, which also has to stay dynamic, has been drawn into the matrix to create the factor one in the diagonal matrix.

In order to return to our experimental time t we introduce the variables v₂(t) = ga( ¯αt), u₁(t) = a( ¯αt)

v₃(t) = gi( ¯αt), u₂(t) = e₂( ¯αt) u7(t) = gp( ¯αt)

(2.23)

and rewrite equation (2.21) u˙₁(t)

˙ u2(t)

= ¹_η ·^α_γ

−k₂u₇(t) k−2+κ₂ k2u7(t) −(k−2+κ2)

u₁(t) u2(t)

(2.24) with the conservation equation

u1(t) +u2(t) = C_u^g1, C_u^g1 := u1(0) +u2(0); (2.25) according to (2.23), u2(0) =e2(0) and u1(0) =a(0) because

u2(0) = e2( ¯α·0) = e2(0), u1(0) = a( ¯α·0) = a(0), but e2(0) = 0 as we can see from (2.10), and we are left with

C_u^g₁ = u₁(0) = a(0) > 0; (2.26)

equation (2.22) leads to







˙ u7(t)

˙ v2(t)

˙ v₃(t)

˙ u₂(t)







= _η¹







α

β 0 0 0 0 1 0 0 0 0 1 0 0 0 0 ^α_γ





 C_g⁰⁰





 u7(t) v2(t) v₃(t) u₂(t)





 ,

C_g⁰⁰ =







−(k2u1(t) +k−3) k3 0 k−2

k−3 −k3 0 0

0 0 0 ^α_δκ2

k2u1(t) 0 0 −(k⁻2+κ2)







(2.27)

and the sugar conservation equation β

αu₇(t) +v₂(t) + δ

αv₃(t) + γ

αu₂(t) = C_v^g₂ C_v^g₂ := β

αu₇(0) +v₂(0) + δ

αv₃(0) + γ αu₂(0).

(2.28)

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Again, we can see from equation (2.23) and the initial values (2.10) that u₇(0) = gp(0) = 0, v₃(0) = gi(0) = 0, u₂(0) = e₂(0) = 0, and

C_v^g₂ = v2(0) = ga(0) > 0.

We see that both factors ^α_β and ^α_γ in equations (2.24) and (2.27) are very large.

This means that the solutions of the differential equations foru₇, u₁ andu₂ reach a point of equilibrium a lot faster than the solutions of the differential equations forv₂ andv₃. This allows us to approximate u₇, u₁ and u₂ by their pseudosteady states.

2.5 Pseudosteady State Approximation

Using the different time scales in equations (2.24) and (2.27) we therefore have to solve

0 0

= ¹_η

−k₂u₇(t) k−2+κ₂ k2u7(t) −(k−2+κ2)

u₁(t) u2(t)

u₁(t) +u₂(t) = C_u^g₁ = a(0) > 0

(2.29)

and





 0

˙ v₂(t)

˙ v₃(t)

0







= 1 η







−(k2u₁(t) +k−3) k₃ 0 k−2

k−3 −k3 0 0

0 0 0 ^α_δκ₂

k2u1(t) 0 0 −(k−2+κ2)











 u₇(t) v₂(t) v₃(t) u2(t)







. (2.30)

Equation (2.29) leads to analytical expressions for u1 and u2: with K¯₂ := k₂

k−2+κ₂, (2.31)

we get

u2(t) = ¯K2u7(t)u1(t),

and with the conservation equation (2.25) u₁(t) = C_u^g₁

1 + ¯K₂u₇(t), u₂(t) = C_u^g₁K¯₂u₇(t)

1 + ¯K₂u₇(t). (2.32)

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In equation (2.30) the lowest row does not provide any new information since we have already found an expression for u₂; the only remaining fast variable in the system isu7, which is implicitly defined by the uppermost equation in (2.7). We can simplify that equation by adding the lowest row of equation (2.30), which sums up to zero but eliminates some of the terms in the implicit equation: adding

−(k2u1(t) +k⁻3)u7(t) +k3v2(t) +k⁻2u2(t) = 0 and

k₂u₁(t)u7(t)−(k⁻2+κ₂)u2(t) = 0 leads to

−k−3u₇(t) +k₃v₂(t)−κ₂u₂(t) = 0 or, using u2(t) from equation (2.32),

k3v2(t)−

k⁻3+κ2

C_u^g₁K¯₂ 1 + ¯K₂u₇(t)

u7(t) = 0. (2.33)

2.6 Constructing a Time-Independent Velocity Function

Next we turn to the definition of the transport velocity in this system. We need the velocity as a function of the glucose concentration in the medium since the data we actually want to fit are measurements of velocity in dependence of the sugar concentration offered to the bacteria. The velocity always describes a change in the measurable product; what we can measure in this system is the change in radioactive glucose inside ˙ngi, which, according to (2.23), is obtained from ˙v₃(t) multiplied by the cytoplasmic volume δ in order to convert the concentration into an amount of substance:

δv˙₃(t) = 1

η ·α·κ₂u₂(t) = 1

η ·α·κ₂C_u^g₁K¯₂u₇(t)

1 + ¯K2u7(t) (2.34) with the implicit equation (2.33) foru₇(t)

k3v2(t)−

k−3+κ2

C_u^g₁K¯2

1 + ¯K2u7(t)

u7(t) = 0, or, using

K₃ := k₃ k−3

(2.35)

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and after division by k−3, K₃v₂(t)−

1 + κ₂ k⁻3

· C_u^g₁K¯₂ 1 + ¯K₂u₇(t)

u₇(t) = 0. (2.36)

The biologically important variables are the glucose on the outside and the inside of the cell, as the outside concentration is the experimental input, and the amount of radioactive glucose inside is the only experimentally accessible output. These variables have to be expressed in the original dimensions and cannot be transformed into scaled variables. On the other hand, the variable for the periplasmatic sugar cannot be measured anyway; it is a mere operand and we can only profit by making it dimensionless by scaling. This reduces the number of parameters and we can operate with dimensionless numbers. To make u7 dimensionless we scale it with

¯

u₇(t) = ¯K₂u₇(t). (2.37)

From equation (2.34) we obtain δv˙₃(t) = 1

η ·α·κ₂C_u^g₁u¯₇(t) 1 + ¯u7(t) K3v2(t)−

1 K¯2

+ κ2

k−3

· C_u^g₁ 1 + ¯u₇(t)

¯

u7(t) = 0,

(2.38)

or after multiplication of the lower row of equation (2.38) by ¯K2, δv˙₃(t) = 1

η ·α·κ₂C_u^g₁u¯₇(t) 1 + ¯u₇(t) K¯₂K₃v₂(t)−

1 + κ₂ k−3

· K¯₂C_u^g₁ 1 + ¯u7(t)

¯

u₇(t) = 0.

(2.39)

We have now described the rate of change for the radioactive glucose inside using the transformed variables v₂ and v₃, not the experimental variables nga

and ngi. To compare theoretical results with the experimental data we have to retransform equation (2.39) into the original variables; since ¯u7 is only an operand, we substitute it by the placeholder x. With the definitions

v₂(t) = ga( ¯αt), v₃(t) = gi( ¯αt) from equation (2.23) and choosing

w₁ = κ₂C_u^g₁ w₂ = κ₂C_u^g₁ · _k^K^¯²

−3

w₃ = K¯₂K₃,

(2.40)

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we arrive at δ d

dtgi( ¯αt) = 1

η ·α· w1x(ga( ¯αt)) 1 +x(ga( ¯αt)), w3ga( ¯αt)−

1 + w₂

1 +x(ga( ¯αt))

x(ga( ¯αt)) = 0

(2.41)

or, using the definition (2.18),

ηαδg¯ _i⁰( ¯αt) = α·n⁰_g_i( ¯αt) = α· w1x(ga( ¯αt)) 1 +x(ga( ¯αt)), w3ga( ¯αt)−

1 + w2

1 +x(ga( ¯αt))

x(ga( ¯αt)) = 0.

Since ¯α >0, this equation holds true for all t≥0 and we can write

˙

ngi(t) = w₁x(ga(t))

1 +x(ga(t)), t≥0, w3ga(t)−

1 + w2

1 +x(ga(t))

x(ga(t)) = 0.

(2.42)

The velocity of the process is hence described by the function vg(ga) := w₁x(ga)

1 +x(ga), w₃ga−

1 + w2

1 +x(ga)

x(ga) = 0

(2.43)

with constants w1, w2, w3 from equation (2.40) . Although the concentration ga

changes as time evolves, it is well defined at each single point in time. The velocity vg of the process depends only on this concentration, not on the time it took for the system to reach it.

We have introducedxin equation (2.41) as a function of the outer glucosega; to find the biological meaning ofxwe have to return to its definition. If we insert the parametersw₁, w₂ and w₃ from equation (2.40) into (2.41) and compare with equation (2.39), we see that it actually stands for ¯u₇(t):

x(ga( ¯αt)) = ¯u7(t), t ≥ 0. (2.44)

From equation (2.37) we see that

¯

u₇(t) = ¯K₂u₇(t),

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and from (2.23) we get u7(t) = gp( ¯αt);

thus,

x(ga( ¯αt)) = ¯u₇(t) = ¯K₂u₇(t) = ¯K₂gp( ¯αt), t ≥ 0, and hence

x(ga(t)) = ¯K2gp(t), t ≥ 0.

Introducing the scaled variable

¯

gp(t) := ¯K₂gp(t), (2.45)

we finally obtain

x(ga(t)) = ¯gp(t), t ≥ 0. (2.46)

Note that the scaled variable ¯gp(t) is dimensionless. We now see that x(ga) is not only an implicitly defined function helping to calculate the reaction velocity vg(ga) but also gives us the periplasmic concentration of glucose belonging to a given outer glucose concentration.

x(ga) is implicitly defined by the lower row of equation (2.43); if we multiply it by 1 +x(ga), we obtain a simple quadratic equation in x(ga):

w₃ga(1 +x(ga))−(1 +x(ga) +w₂)x(ga) = 0, or

x(ga)²+ (1 +w₂−w₃ga)x(ga)−w₃ga = 0. (2.47) The explicit representation of equation (2.47) reads

x(ga) = −1

2·(1 +w2−w3ga) + r1

4 ·(1 +w2−w3ga)²+w3ga. (2.48) Equation (2.48) shows thatx(ga)≥0 for allga≥0: x(0) = 0, and with w₃ >0 and ga>0,

r1

4·(1 +w2−w3ga)² +w3ga > 1

2 · |1 +w2−w3ga| so that

x(ga) > 0 for all ga > 0.

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0 200 400 600 800 1000

0 10 20 30 40 50

ga [µM]

x(ga)

0 200 400 600 800 1000

0 0.005 0.01 0.015 0.02 0.025

ga [µM]

vg(ga)[nmol min·mg]

Fig. 2.2: Left: Periplasmic glucose x(ga) (dimensionless) plotted against ga. Right:

vg(ga) plotted againstga. We can see thatx(ga) continues to increase withgaandvg(ga) increases strictly but saturates.

To understand equation (2.43) a little better, we can plot the functionsx(ga) and vg(ga) with positive parameters w₁, w₂, w₃ against the positive variable ga. ga is the concentration of glucose in the medium and equation (2.46) states that x(ga(t)) = ¯gp(t) is the normalized concentration of periplasmic glucose. This means that the left plot of fig. 2.2 shows the concentration ¯gp(ga) of glucose in the periplasm rising continuously with the concentration ga of glucose in the medium. From the biological point of view this makes sense since the transfer of glucose from the medium into the periplasm is not enzymatic and thus not limited. On the other hand, the uptake of glucose into the cytoplasm vg(ga) in the right plot of fig. 2.2 is limited by the amount of transport protein na(0).

Therefore, there is a maximal rate at which glucose can be transported from the medium into the cytoplasm, depending on the transport rate from the periplasm to the cytoplasm when all transport proteins are active.

2.7 Fit Parameters and Dimensions

We now have a look at the dimensions of our variables and constants; in the first place, all concentrations are expressed inµM, and the rates of change in amounts of substance are of the dimension ^nmol_min. With this knowledge we can compute the dimensions of the constants from equations (2.6) and (2.7):

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[α] = [β] = [γ] = [δ] = l, [k2] = l²

mmol·min,

[k−2] = [κ₂] = [k₃] = [k−3] = ml min.

(2.49)

The unit for the rates of change in concentrations from equation (2.8) and (2.9) is then _min^nM.

The biological dimensions are defined as follows:

1mol = 6·10²³molecules, 1M = 1mol

l , 1mM = 10⁻³mol l , 1µM = 1·10⁻⁶mol

l , 1nM = 1·10⁻⁹mol l .

(2.50)

From this reasoning we see that ga has the dimension µM and vg(ga) is of the dimension ^nmol_min. This conforms with the experimental setup where all concentrations are measured inµM. The experimental data forvg are expressed in _min^nmol_·_mg; the scaling to mgprotein is done by taking into account the amount of cells used in the experiment.

The dimension of w₁ can be seen from the first equation in (2.43): x is dimensionless, so thatw1 carries the same dimension asvg(ga), in this case _min^nmol_·_mg. To find the dimensions of w2 and w3, we have to study the second equation in (2.43): all summands have to be dimensionless in order to fit to the 1·x, so that w₂ is dimensionless and w₃ compensates for [ga] = µM = ^µmol_l . This leads us to

[w1] = nmol

min·mg, [w2] = 1, [w3] = 1

µM = l

µmol = ml nmol.

(2.51)

To check the consistency of these dimensions with those in equation (2.49) we use the definitions of w1, w2 and w3 from (2.40): w1 has the dimension _min^nmol_·_mg, because

[C_u^g₁] = µM = µmol

l and [κ₂] = ml min.

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This leads to

[w1] = µmol

1000·ml · ml

min = µmol

1000·min = nmol min ; the normalization tomg protein has to be kept in mind.

w₂ has to be dimensionless, because [κ₂] = [k−3], [ ¯K₂] =

l² mmol·min

1 1000l min

= l

µmol = 1

µM, and [C_u^g₁] = µM.

w3 carries the dimension _µM¹ = _nmol^ml asK3 is dimensionless and [ ¯K2] = 1

µM = l

µmol = ml nmol.

2.8 Numerical Fitting of the Parameters to Glu- cose Uptake Data

Our experimental data consist of seven data points (g⁽ⁱ⁾, v⁽ⁱ⁾), i = 1. . .7

where g⁽ⁱ⁾ is the initial condition ga(0) in the i-th uptake experiment and v⁽ⁱ⁾ is the corresponding maximal uptake rate. Under the assumption that the maximal uptake rate is close to the beginning of the uptake experiment, they can be described by the function (2.43) with (2.48)

vg(g⁽ⁱ⁾) = w₁x(g⁽ⁱ⁾) 1 +x(g⁽ⁱ⁾) x(g⁽ⁱ⁾) = −1

2·(1 +w₂−w₃g⁽ⁱ⁾) + r1

4·(1 +w₂−w₃g⁽ⁱ⁾)²+w₃g⁽ⁱ⁾.

(2.52)

To find the optimal fit ( ˜w₁,w˜₂,w˜₃), we have to look upon w₁, w₂ and w₃ as variables instead of parameters; we can define the function

vvSSE(g, w) = w1xSSE(g, w)

1 +xSSE(g, w), w = (w₁, w₂, w₃), xSSE(g, w) = −1

2 ·(1 +w₂−w₃g) + r1

4·(1 +w₂−w₃g)²+w₃g,

(2.53)

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and minimize the sum of square errors SSE(w) = 1

2·

7

X

i=1

(v⁽ⁱ⁾−vvSSE(g⁽ⁱ⁾, w))².

We have chosen to scan over w₃, searching for each fixed ¯w₃ for the optimal ¯w₁ and ¯w₂. To this end, the equation

grad(SSE^∗(w1, w₂)) = 0 SSE^∗(w1, w2) = SSE(w1, w2,w¯3).

needs to be solved. Explicitly, 0 = ∂SSE^∗

∂w₁ =

7

X

i=1

(v⁽ⁱ⁾−vvSSE(g⁽ⁱ⁾, w1, w2,w¯3))· ∂vvSSE(g⁽ⁱ⁾, w1, w2,w¯3)

∂w₁ ,

0 = ∂SSE^∗

∂w₂ =

7

X

i=1

(v⁽ⁱ⁾−vvSSE(g⁽ⁱ⁾, w1, w2,w¯3))·∂vvSSE(g⁽ⁱ⁾, w1, w2,w¯3)

∂w₂ with

∂vvSSE(g⁽ⁱ⁾, w₁, w₂,w¯₃)

∂w1

= xSSE(g⁽ⁱ⁾, w₁, w₂,w¯₃) 1 +xSSE(g⁽ⁱ⁾, w1, w2,w¯3) and

∂vvSSE(g⁽ⁱ⁾, w1, w2,w¯3)

∂w₂ = w1

(1 +xSSE(g⁽ⁱ⁾, w₁, w₂,w¯₃))²·∂xSSE(g⁽ⁱ⁾, w1, w2,w¯3)

∂w₂ ,

∂xSSE(g⁽ⁱ⁾, w₁, w₂,w¯₃)

∂w2

= 1 2 ·



−1 + 1 +w₂−w¯₃g 2q

1

4 ·(1 +w₂−w¯₃g)²+ ¯w₃g



. To find the optimal solution ¯w= ( ¯w1,w¯2) for each ¯w3, we have numerically solved the differential equation

w˙1(t)

˙ w₂(t)

= F(w1(t), w2(t))

F(w₁(t), w₂(t)) = −grad(SSE^∗(w₁(t), w₂(t))).

The solution (w1(t), w2(t)) saturates for t 1 at a value (w1s, w2s). This satu- ration point is a zero of F(w1, w₂) =grad(SSE^∗(w1, w₂)).

A Mathematical Model for Trehalose Uptake in the Thermophilic Bacterium Rhodothermus marinus