From fig. 3.2 we see that ¯ve(da(0)) = ¯w1x(d¯ a(0)) is a strictly increasing saturating function with a starting point ¯ve(0) = 0; we can therefore indicate a saturation value Vmaxe
Vmaxe := ¯w1·x¯m, x¯m = lim
da(0)→∞
¯
x(da(0)), (3.49)
and a substrate concentration Kme at which 12Vmaxe is reached:
1
2Vmaxe = ¯w1·x(K¯ me). (3.50)
To find Vmaxe , we first have to find the saturation value ¯xm for ¯x(da(0)); to this end the two implicit equations from (3.35) are used:
2· z1y(d¯ a(0))
1 + ¯y(da(0)) +z2x(d¯ a(0)) = ¯x(da(0)),
1 + z4
1 + ¯y(da(0)) +z2x(d¯ a(0))
¯
y(da(0)) = z3da(0).
(3.51)
Since all constants are positive and
¯
x(da(0)) ≥ 0, y(d¯ a(0)) ≥ 0 (3.52)
being concentrations of periplasmic sugars, we see that
0 ≤ z4
1 + ¯y(da(0)) +z2x(d¯ a(0)) ≤z4, 0< z3da(0) =
1 + z4
1 + ¯y(da(0)) +z2x(d¯ a(0))
¯
y(da(0))≤y(d¯ a(0))·(1 +z4),
CHAPTER 3. IN VIVO TREHALASE ACTIVITY 39
and find
¯
y(da(0))−→ ∞ for da(0)−→ ∞. (3.53) We now see why ¯y(da(0)) in fig. 3.2 does not saturate. Defining the function
F(da(0)) := 2· z1y(d¯ a(0))
1 + ¯y(da(0)) +z2x(d¯ a(0)), (3.54) we see from the second equation in (3.49) that forda(0) −→ ∞, ¯x(da(0)) converges to ¯xm, and with (3.53) we obtain
F(da(0))−→2·z1 for da(0)−→ ∞. (3.55) But the upper equation of (3.51) shows that
F(da(0)) = ¯x(da(0)), so that with (3.49) we get
2·z1 = lim
da(0)→∞
F(da(0)) = lim
da(0)→∞
¯
x(da(0)) = ¯xm, Vmaxe = ¯w1·x¯m = ¯w1·2·z1.
(3.56)
To find Kme (see (3.50)), we must use equation (3.56):
¯
w1x(K¯ me) = 1
2 ·Vmaxe = ¯w1·z1, (3.57)
that is
¯
x(Kme) = z1. (3.58)
Then, the upper equation of (3.51) leads us to 2· z1y(K¯ me)
1 + ¯y(Kme) +z2z1 = z1, 2·z1y(K¯ me) = z1(1 + ¯y(Kme) +z2z1),
2·y(K¯ me) = 1 + ¯y(Kme) +z2z1, and
¯
y(Kme) = 1 +z2z1. (3.59)
Inserting ¯x(Kme) and ¯y(Kme) from equations (3.58) and (3.59) into the lower equa-tion of (3.51) yields
1 + z4
1 + 1 +z2z1+z2z1
(1 +z2z1) = z3Kme
Kme =
1 + z4
2(1 +z2z1)
(1 +z2z1)· 1
z3 = 1 z3 ·
1 +z2z1+z4
2
.
(3.60)
In this way restrictions for thez-parameters in equation (3.35) can be formulated by introducing the two parameters from equations (3.60) and (3.56)
Kme = 1 z3
·
1 +z2z1+ z4 2
, Vmaxe = ¯w1·2·z1. (3.61) These restrictions, together with those developed in the next two chapters, are necessary for the fits shown in figure 4.3, 5.3 and 5.4. The data described by equation (3.35) are at the bottom of figure 4.3 and in figure 5.4.
Chapter 4
Trehalose Transport: Model 1
4.1 Experimental Setup
In this experiment, radioactive trehalose in varying initial concentrations is taken up and measured over time. As in the glucose transport experiment, the raw data comprising measurements of radioactivity over time is used to determine the maximal uptake rate at a given initial concentration of substrate. These sets of concentrations and respective uptake rates then constitute the actually fitted data.
T T
B A Da
Dp
Dp Dp
Di
Gp Gp
Ga
Gp
Gi cytoplasm
periplasm medium
Gp
Fig. 4.1: Schematic depiction of the trehalose transport system according to model 1 (without cytoplasmic trehalose transporter); active parts are black.
41
4.2 Network
We have two different hypotheses to test: model 1 only uses the components of glucose uptake and trehalase activity we have encountered in chapters 2 and 3, whereas model 2 features an additional specific trehalose transporter in the cytoplasmic membrane. In this chapter we will deal with model 1.
The network describing the uptake of radioactive trehalose according to model 1 is generated by joining the networks for in vivocleavage of trehalose in chapter 3, p. 23 and glucose uptake from chapter 2, p. 6: The trehalose from the outer medium Da crosses the outer membrane (see (3.1):
Da
k0 k−0
Dp;
it is then cleaved by the trehalase T into two units of glucose (see (3.2)):
Dp+T k1 k−1
E1 κ1
→ 2Gp+T; (trehalase reaction) the periplasmic glucose Gp inhibits the trehalase (see (3.3)):
Gp+T k4 k−4
Ei;
glucose can leave the periplasm into the outer medium to become Ga (see (2.1)):
Ga
k3 k−3
Gp;
and the periplasmic glucose Gp is taken up into the cytoplasm by the uptake complex A (see (2.2)):
Gp+A k2 k−2
E2 κ2
→ Gi+A. (glucose transporter)
Since we are working with the same bacteria, the distibution of the compounds over the different volumina must remain the same as in (2.3) and (3.5):
compartment volume compounds outer medium α Da, Ga
periplasm β Dp, Gp, T, E1, Ei
inner membrane γ A, E2
cytoplasm δ Gi
(4.1)
CHAPTER 4. TREHALOSE TRANSPORT: MODEL 1 43
4.3 Differential Equations
Since the network consists of parts that are already discussed in chapter 2 and 3, some parts of the system of differential equations are known already: the differential equations describing the glucose transporter (2.2) are already dealt with in equation (2.6)
n˙a(t)
˙ ne2(t)
=
−k2gp(t) k−2+κ2 k2gp(t) −(k−2+κ2)
a(t) e2(t)
,
and those for the trehalase reaction (3.2) are put together in equation (3.6)
˙ nτ(t)
˙ ne1(t)
˙ nei(t)
=
−(k1dp(t) +k4gp(t)) k−1+κ1 k−4
k1dp(t) −(k−1+κ1) 0 k4gp(t) 0 −k−4
τ(t) e1(t) ei(t)
. Using the variables from (2.23) and (3.13)
v1(t) = da( ¯αt), u1(t) = a( ¯αt), u4(t) = e1( ¯αt) v2(t) = ga( ¯αt), u2(t) = e2( ¯αt), u5(t) = ei( ¯αt) v3(t) = gi( ¯αt), u3(t) = τ( ¯αt), u6(t) = dp( ¯αt) u7(t) = gp( ¯αt)
(4.2)
we see in equation (2.29) and (3.18) that the left sides can be set to zero because of the large differences between the volume of the outer medium αand the inner membrane γ or the periplasm β, respectively (see (2.12)). With equations (2.26) and (3.17)
Cug1 = a(0) > 0, Cue3 = τ(0) > 0
equations (2.6) and (3.6) yield the explicit expressions (2.32) and (3.21) u1(t) = Cug1
1 + ¯K2u7(t), u2(t) = Cug1K¯2u7(t) 1 + ¯K2u7(t), u3(t) = Cue3
1 +K4u7(t) + ¯K1u6(t), u4(t) = Cue3K¯1u6(t)
1 +K4u7(t) + ¯K1u6(t), u5(t) = Cue3K4u7(t) 1 +K4u7(t) + ¯K1u6(t).
(4.3)
In contrast to equations (2.6) and (3.6), we cannot use the differential equations for the radioactive sugars from the former chapters because some of the com-pounds, although they have occured before, here take part in new reactions. For
this set of equations we have to repeat our analysis analogously to chapters 2 and 3: the differential equations
with the diagonal matrix
Dt3 =
are changed into differential equations for the changes in concentrations
CHAPTER 4. TREHALOSE TRANSPORT: MODEL 1 45
The missing main diagonal elements in the matrix Ct are given by the negative sums of the remaining column elements.
The initial concentrations of the substrate da(0) and the enzymes a(0) and τ(0) are positive, all other compounds are either intermediates or products; their concentration is zero at the beginning of the experiment:
a(0) > 0, τ(0) > 0, da(0) > 0
4.4 Separating Different Time Scales
To make the different time scales in the remaining equations visible, we reuse the method from chapter 2.4 with equation (2.15)
¯ α = α
η, η = 1l, and obtain from equation (4.5)
or,using the new variables from (4.2),
The stars in the main diagonals of the matricesCt0 and Ct00have to be substituted by the negative sums of the remaining column elements.
In equation (4.6) we see that with (4.2), all of the above initial values are zero except for v1(0) = da(0) > 0, and we obtain
Cvt1 = 2v1(0) > 0. (4.9)
CHAPTER 4. TREHALOSE TRANSPORT: MODEL 1 47
4.5 Pseudosteady State Approximation
From equation (2.12) on page 9 we see that both αβ and αγ are very large. This means that we can solve the equations
with Ct00 from equation (4.8) to obtain a good approximation for the solution of (4.8).
In equation (4.10), the three lowest rows are already dealt with, and v1, v2 and v3 remain dynamic. Adding the two bottom rows to the third one and the third last to the second one, we are left with
−k−3u7(t) +k3v2(t) + 2κ1u4(t)−κ2u2(t) = 0
The velocity of the transport process is described by the rate of change in the amount of the transported product; in this case the relevant rate is ˙ngi. Since gi( ¯αt) = v3(t) (see (4.2)), we have to multiply the differential equation for v3(t)
Division of the upper equation of (4.13) by k−3 and of the lower equation by k−0
yields, using equation (3.25), K0 = k0 To be able to find parameters fitting all experiments at the same time, the scaling in all subnetworks should be compatible. We therefore use the scaling from equation (3.27) to find the dimensionless variables
¯
u6(t) = ¯K1u6(t), u¯7(t) = ¯K2u7(t). (4.15) From (4.12) we then get
δv˙3(t) = 1 or, after multiplication of the upper equation in (4.17) with ¯K2 and the lower equation with ¯K1,
CHAPTER 4. TREHALOSE TRANSPORT: MODEL 1 49
The experimental data we have to fit are expressed in concentrations da and ga and in the amount of glucose inside ngi, not in v1, v2 and v3. We therefore have to retransform equation (4.16) and the implicit equations (4.18) accordingly:
with equations (2.45) and (3.38)
¯
gp = ¯K2gp, d¯p = ¯K1dp
and equation (4.2) we obtain from (4.16)
ακ2Cug1u¯7(t)
1 + ¯u7(t) =δηd
dtv3(t) = δηd
dtgi( ¯αt) =δηα·¯ g0i( ¯αt) =δα·g0i( ¯αt) =α·n0gi( ¯αt) (4.19) and
K¯2K3ga( ¯αt) + 2 κ1 k−3
· K¯2Cue3d¯p( ¯αt) 1 + ¯dp( ¯αt) + KK¯4
2¯gp( ¯αt)−
1+ κ2 k−3
· K¯2Cug1
1 + ¯gp( ¯αt)
¯
gp( ¯αt) = 0
K0K¯1da( ¯αt)− 1 + κ1 k−0
· K¯1Cue3 1 + ¯dp( ¯αt) + KK¯4
2g¯p( ¯αt)
!
d¯p( ¯αt) = 0.
(4.20) The function describing the data is then defined by
vtr(da(t), ga(t)) := κ2
Cug1¯gp(t) 1 + ¯gp(t) K¯2K3ga(t) + 2 κ1
k−3
· K¯2Cue3d¯p(t) 1 + ¯dp(t) + KK¯42¯gp(t) −
1 + κ2
k−3
· K¯2Cug1 1 + ¯gp(t)
¯
gp(t) = 0
K0K¯1da(t)− 1 + κ1
k−0
· K¯1Cue3
1 + ¯dp(t) + KK¯42¯gp(t)
!
d¯p(t) = 0.
(4.21) vtr describes the change in the amount of glucose in the cytoplasm in dependence on the concentrations of trehalose and glucose outside. The system of differential equations from equation (4.10) now takes the form
˙
ngi(t) = vtr(da(t), ga(t))
˙
nda(t) = −k0da(t) +k−0
K¯1u¯6(t)
˙
nga(t) = −k3ga(t) + k−3
K¯2u¯7(t)
(4.22)
with vtr(da(t), ga(t)) as in equation (4.21).
Comparing the clusters of individual velocity constants with equations (2.40) and (3.30) shows that all fit parameters have occured already:
w1 = κ2Cug1, z1 = kκ1
−3 ·K¯2Cue3 w2 = κ2K¯2C
g u1
k−3 , z2 = KK¯42 w3 = K¯2K3, z3 = K0K¯1
z4 = kκ1
−0 ·K¯1Cue3.
(4.23)
w1 = 54.3569minnmol·mg, w2 = 0.9137655 andw3 = 3.406·10−2nmolml are already known from the fit of the glucose transport data, and z2 = 2.4467 · 10−3 has been calculated in (3.48). The other z-variables are the same as in chapter 3 and follow the restrictions (3.61)
Kme = 1 z3 ·
1 +z2z1+z4
2
, Vmaxe = ¯w1·2·z1.
Introducing the fit variables we arrive at
vtr(da(t), ga(t)) = w1g¯p(t) 1 + ¯gp(t) w3ga(t) + 2 z1d¯p(t)
1 + ¯dp(t) +z2g¯p(t)−
1 + w2
1 + ¯gp(t)
¯
gp(t) = 0
z3da(t)−
1 + z4
1 + ¯dp(t) +z2¯gp(t)
d¯p(t) = 0
CHAPTER 4. TREHALOSE TRANSPORT: MODEL 1 51
or, ifga(0) = 0,
vtr(da(0),0) = w1¯gp
1 + ¯gp
2 z1d¯p
1 + ¯dp+z2¯gp
−
1 + w2 1 + ¯gp
¯ gp = 0
z3da(0)−
1 + z4
1 + ¯dp+z2g¯p
d¯p = 0.
(4.24)
0 100 200 300 400 500
0 0.5 1 1.5 2 2.5 3 3.5