We have already obtained w1 = 54.3569 nmol
min·mg, w2 = 0.9137655, w3 = 3.406·10−2 ml nmol in equation (2.54) in chapter 2, p. 21, and
z2 = 2.4467·10−3
in (3.48) in chapter 3.7; to find the remaining parameters z1, z3, z4, z5 and z6, we first have to guess the parameters Vmaxe and Kme from the data for enzy-matic cleavage, and Vmaxsu andKmsu from the trehalose uptake experiment. A first approximation for these values can be obtained by fitting a Michaelis-Menten function through the data. With this information and an assumption for z6 all other parameters can be calculated:
z1 = Vmaxe 2 ¯w1
is given by equation (5.35), then z5 can be obtained from equation (5.39) z5 =
Vmaxsu − w1gpmax
1 +gpmax
z6
gpmax = 1 2
−(1 +w2−2z1) +p
(1 +w2−2z1)2+ 8z1
because all occuring parameters are now known. The only remaining parameters are z3 and z4; to calculate them from equation (5.44) we have to find ¯dp(Kmsu,0) and ¯gp(Kmsu,0), which are implicitly defined in the first two rows of equation (5.41)
Vmaxe
2 = ¯w1g¯p(Kme,0) 2 z1d¯p(Kme,0)
1 + ¯dp(Kme,0) +z2g¯p(Kme,0)−¯gp(Kme,0) = 0
and only depend on parameters calculated before. This means that for a given set of curve parameters
Main parameters: Vmaxe , Kme, Vmaxsu , Kmsu, (5.47)
every value of z6 defines a set of z-parameters to be introduced into equation (5.32) and thus yields a curve vsu(da(0),0) with a certain sum of square errors relative to the measured data points. We then choose z6 so that we obtain a minimum in the sum of square errors for the given set of curve parameters (5.47).
Varying these until we reach a local minimum in the sum of square errors, we find an optimal fit for both data sets. This means that we only have to fit the four variables (5.47) for which the data themselves already provide a good approximation.
The trehalose uptake measurements consist of two data sets, one with a sub-strate range of 0µM to 2µM trehalose taken in May 2005, and another one with 5µM to 350µM trehalose taken in October 2005. To find as good a fit as possible for both data sets it has proven best to fit the data from May alone. The result is shown in fig. 5.3, together with all data points.
We obtained the optimal fit with the following parameters:
Vmaxsu = 18.16minnmol·mg, Kmsu = 79.2µM, Vmaxe = 40.7minnmol·mg, Kme = 128µM,
z1 = 0.34209, z2 = 2.4467·10−3, z3 = 7.83223·10−3nmolml , z4 = 3.377097·10−3, z5 = 4.2552·102minnmol·mg, z6 = 193.8.
(5.48)
The dimensions of the fit parameters are taken from equation (5.33).
CHAPTER 5. TREHALOSE TRANSPORT: MODEL 2 73
0 100 200 300 400 500
0 5 10 15 20
uptake via trehalose transporter uptake via glucose transporter sum of both, comparable to measurements
Uptake rate in nmol/(min*mg)
Trehalose concentration in µM
0 1 2 3 4 5
0 0.5 1 1.5 2 2.5 3
uptake via trehalose transporter
uptake via glucose transporter sum of both, comparable to measurements
Uptake rate in nmol/(min*mg)
Trehalose concentration in µM
Fig. 5.3: Green: Theoretical uptake curve for trehalose with optimal parameters. Red:
Theoretical part of the overall uptake due to trehalase activity and glucose uptake. Blue:
Theoretical part of the overall uptake due to direct uptake by the trehalose transporter.
Squares: Measured data from May. Crosses: Measured data from October. The whole substrate range is shown on top, a blowup of the region around the measurements from May is at the bottom.
0 1000 2000 3000 4000 5000 0
10 20 30 40 50
Release rate of glucose in nmol/(min*mg)
Trehalose concentration in µM
Fig. 5.4: Optimal fit for the in vivo trehalase measurements. Data points marked with squares are the results from the October measurements using the second method if possible.
The data points are multiplied by two to express them in units of glucose. The theoretical curve hasVmaxe = 40.7min·mgnmol and aKme = 128µM.
Chapter 6
Nonradioactive Transport
6.1 Experimental Setup
As biological transporters often have more than one substrate, it is a standard part of the characterization of a transport system to perform cross-inhibition experiments. This means that the biologist labels the known substrate X1 and tries to disturb the uptake of the radioactively labeled substrate by adding other nonradioactive substances. If the radioactive transport diminishes with a rising concentration of the nonradioactive substanceX2, the latter is a putative alterna-tive substrate for the transporter. The evidence is strengthened if we now label substance X2 and manage to disturb its uptake by adding the nonradioactive form of X1. In our case, nonradioactive glucose has been found to inhibit the uptake of radioactive trehalose, but not vice versa. Therefore, our model does not consist of one transporter for both glucose and trehalose, but comprises two different transporters for glucose and trehalose and a glucose-inhibited trehalase.
We now have to check whether this model is consistent with the experimental results.
6.2 Network
We assume that the bacterial cells cannot distinguish radioactively labeled sugars from nonradioactive ones. Therefore, the nonradioactive sugars undergo the same reactions with the same reaction constants as the labeled sugars we have examined in the previous chapters. Nevertheless, the reactions have to be written down separately, as only the uptake of radioactive sugars can be measured, but both forms compete for the transporters and the degrading enzyme. Thus, every one of our familiar reactions gets a nonradioactive counterpart:
75
Gi, Ni Gp, Np
T T
B A Da
Dp
Dp
Dp
Di
Np
Na
T Gp
medium
periplasm
cytoplasm
Np
Gp
Fig. 6.1: Schematic depiction of the trehalose transport system containing nonra-dioactive sugars; complete system.
The radioactive and nonradioactive trehalose Da and Ma in the outer medium cross the outer membrane to enter the periplasm and thus become Dp and Mp
(here again, the constants for the in and out direction are different to allow for possible active transport):
Da
k0 k−0
Dp Ma
k0 k−0
Mp; (6.1)
the trehalose in the periplasm is cleaved into two units of periplasmic glucose with the help of the trehalase T; the radioactive trehalose Dp yields the radioactive glucose Gp, and Mp is transformed into cold glucose Np:
Dp+T k1
k−1
E1 κ1
→ 2Gp+T Mp+T k1
k−1
E3 κ1
→ 2Np+T;
trehalase reaction
(6.2)
the periplasmic glucoseGpandNpinhibits the trehalaseT by forming the inactive enzyme complexes Ei and Ej:
Gp+T k4 k−4
Ei Np+T
k4 k−4
Ej; (6.3)
the glucose in the outer medium Ga and Na can cross the outer membrane to form Gp and Np:
Ga
k3
k−3
Gp Na
k3
k−3
Np; (6.4)
CHAPTER 6. NONRADIOACTIVE TRANSPORT 77
the periplasmic glucose Gp and Np is transported into the cytoplasm by the uptake complex A, and is thereby transformed into Gi and Ni:
Gp+A k2 k−2
E2 κ2
→ Gi+A Np+A k2 k−2
E4 κ2
→ Ni+A
glucose transporter
(6.5)
The periplasmic trehalose Dp and Mp is transported into the cytoplasm by the uptake complex B, and is thereby transformed into Di and Mi:
Dp +B k5 k−5
E5
κ5
→ Di+B Mp+B k5 k−5
E6
κ5
→ Mi +B.
trehalose transporter
(6.6)
As the nonradioactive sugars and their complexes reflect the radioactive ones, we can easily see where in the cell the new compounds are located. The overall distribution now takes the form
compartment volume compounds
outer medium α Da, Ga, Ma, Na
periplasm β Dp, Gp, Mp, Np, T, E1, Ei, E3, Ej
inner membrane γ A, E2, E4, B, E5, E6
cytoplasm δ Gi, Di, Mi, Ni
(6.7)
6.3 Differential Equations
In the differential equations for the changes in the amounts of substance, the matrices for the enzyme complexes (2.6), (3.6) and (5.5) are extended by the addition of the equations for the new intermediate products. As we can see in equation (6.5), (2.6) is altered through the addition of the equation forE4:
˙ na(t)
˙ ne2(t)
˙ ne4(t)
= An
a(t) e2(t) e4(t)
An =
−k2(gp(t) +np(t)) k−2+κ2 k−2+κ2
k2gp(t) −(k−2+κ2) 0 k2np(t) 0 −(k−2+κ2)
;
(6.8)
because of (6.6), in (5.5) the new compound is E6:
The (*) elements in the main diagonal are given by the negative sum of the remaining column elements.
As the reaction equations for the labeled sugars on the left hand side of (6.1) - (6.6) are the same as those in chapter 5, the matrix equation for the radioactive sugars (5.6) does not have to be changed. In addition to that, the nonlabeled sugars, on the right hand side of (6.1) - (6.6), undergo the same reactions as the labeled ones, so that the differential equations can be expressed with the same matrix. By comparing the two sides of (6.1) - (6.6), we see that the matrix has to be applied to the vector
(da(t), dp(t), gp(t), ga(t), gi(t), e1(t), ei(t), e2(t), e5(t), di(t))T
if we want the differential equations for the labeled sugars, and to the vector (ma(t), mp(t), np(t), na(t), ni(t), e3(t), ej(t), e4(t), e6(t), mi(t))T
for the nonradioactive sugars. Therefore we can write down one matrix equation for both the labeled and the nonlabeled sugars and interpret it in two ways:
CHAPTER 6. NONRADIOACTIVE TRANSPORT 79
Again, the stars have to be replaced by the negative sum of the remaining column elements.
Equation (6.11) describes the evolution of the radioactive sugars if (s1(t), s2(t), s3(t), s4(t), s5(t), s6(t), s7(t), s8(t), s9(t), s10(t))
= (da(t), dp(t), gp(t), ga(t), gi(t), e1(t), ei(t), e2(t), e5(t), di(t)) and the nonlabeled sugars if
(s1(t), s2(t), s3(t), s4(t), s5(t), s6(t), s7(t), s8(t), s9(t), s10(t))
= (ma(t), mp(t), np(t), na(t), ni(t), e3(t), ej(t), e4(t), e6(t), mi(t)).
If we want to write both sides of equations (6.8) - (6.11) 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 (6.7) shows which substance
belongs to which compartment. For (6.8) and (6.9), all compounds are localized in the inner membrane with the volume γ, so that we multiplying both sides of the equations by γ1 we obtain
In the beginning of the experiment, only the free enzymes A and B are present;
the initial conditions for the two subsystems are a(0) > 0, b(0) > 0 e2(0) = 0, e5(0) = 0 e4(0) = 0, e6(0) = 0.
(6.14)
From equation (6.7) we see that all components of (6.10) are located in the periplasm β. We can therefore multiply both sides by 1β and get
with the initial conditions
τ(0) > 0, e3(0) = 0 e1(0) = 0, ej(0) = 0.
ei(0) = 0,
(6.16)
In the sugar equation (6.11), the distribution of the compounds over the cell is the same as in (5.8): Da and Ma are located in the medium with the volume α, Dp andMp are localized in the periplasm with the volumeβ, etc. We thus obtain the following distribution:
da(t) dp(t) gp(t) ga(t) gi(t) e1(t) ei(t) e2(t) e5(t) di(t)
α β β α δ β β γ γ δ
ma(t) mp(t) np(t) na(t) ni(t) e3(t) ej(t) e4(t) e6(t) mi(t).
CHAPTER 6. NONRADIOACTIVE TRANSPORT 81
Multiplication of equation (6.11) by
D3n =
The initial concentrations of the two substrates, the radioactive trehaloseda(0) and the nonradioactive glucosena(0) used to disturb the uptake ofDaare positive, all other compounds are either intermediates or products; their concentration is zero at the beginning of the experiment. This means that for the radioactive sugars the initial conditions are
s1(0) = da(0) > 0, s6(0) = e1(0) = 0
and for the nonradioactive ones they are
s1(0) = ma(0) = 0, s6(0) = e3(0) = 0 s2(0) = mp(0) = 0, s7(0) = ej(0) = 0 s3(0) = np(0) = 0, s8(0) = e4(0) = 0 s4(0) = na(0) > 0, s9(0) = e6(0) = 0 s5(0) = ni(0) = 0, s10(0) = mi(0) = 0.
(6.19)
6.4 Separating Different Time Scales
The set of variables (5.3) is now completed by nonradioactive sugars and sugar complexes to form
slow fast
v1(t) = da( ¯αt), u1(t) = a( ¯αt), u9(t) = e5( ¯αt) v2(t) = ga( ¯αt), u2(t) = e2( ¯αt), u10(t) = e4( ¯αt) v3(t) = gi( ¯αt), u3(t) = τ( ¯αt), u11(t) = e3( ¯αt) v4(t) = di( ¯αt), u4(t) = e1( ¯αt), u12(t) = ej( ¯αt) v5(t) = ma( ¯αt), u5(t) = ei( ¯αt), u13(t) = mp( ¯αt) v6(t) = na( ¯αt), u6(t) = dp( ¯αt), u14(t) = np( ¯αt) v7(t) = ni( ¯αt), u7(t) = gp( ¯αt), u15(t) = e6( ¯αt).
v8(t) = mi( ¯αt), u8(t) = b( ¯αt),
(6.20)
With these variables, the procedure for time scale separation from the previous chapters leads to the following equations:
˙ u1(t)
˙ u2(t)
˙ u10(t)
= 1 η · α
γA00n
u1(t) u2(t) u10(t)
A00n =
−k2(u7(t) +u14(t)) k−2+κ2 k−2+κ2
k2u7(t) −(k−2+κ2) 0 k2u14(t) 0 −(k−2 +κ2)
,
u1(t) +u2(t) +u10(t) = Cun1, Cun1 := u1(0) +u2(0) +u10(0);
(6.21)
CHAPTER 6. NONRADIOACTIVE TRANSPORT 83
Cn00=
∗ k−0 0 0 0 0 0 0 0 0
k0 ∗ 0 0 0 k−1 0 0 k−5 0
0 0 ∗ k3 0 2κ1 k−4 k−2 0 0
0 0 k−3 ∗ 0 0 0 0 0 0
0 0 0 0 ∗ 0 0 αδκ2 0 0
0 k1u3(t) 0 0 0 −(k−1+κ1) 0 0 0 0
0 0 k4u3(t) 0 0 0 ∗ 0 0 0
0 0 k2u1(t) 0 0 0 0 −(k−2+κ2) 0 0 0 k5u8(t) 0 0 0 0 0 0 −(k−5+κ5) 0
0 0 0 0 0 0 0 0 αδκ5 ∗
2s1(t) + 2βαs2(t) + βαs3(t) +s4(t) + αδs5(t) + 2βαs6(t) + βαs7(t) +γαs8(t) + 2αγs9(t) + 2αδs10(t) = Csn1
Csn1 := 2s1(0) + 2βαs2(0) + βαs3(0) +s4(0) + αδs5(0) + 2βαs6(0) + βαs7(0) +γαs8(0) + 2γαs9(0) + 2αδs10(0).
(6.25)
The stars in the matrix Cn00 stand for the negative sums of the remaining column elements. Note that the fractions αδ in the equations for the products s5 and s10, which also have to stay dynamic, have been drawn into the matrix to create the factor one in the diagonal matrix. Equations (6.14) and (6.16) show that all initial conditions for the enzyme matrices (6.21), (6.22) and (6.23) are zero except for a(0), b(0) and τ(0). Using (6.20) this means that
Cun1 = u1(0) = a(0) > 0, Cun8 = u8(0) = b(0) > 0, Cun3 = u3(0) = τ(0) > 0.
(6.26)
In equation (6.25) we now have to differentiate: for the labeled sugars, all initial values except for v1(0) =da(0) are zero, and we obtain
Csn1 = 2v1(0) = 2da(0) > 0; (6.27)
for the nonlabeled sugars, v6(0) =na(0) is positive, and we get
Csn1 = v6(0) = na(0) > 0. (6.28)
Looking at (6.21), (6.22), (6.23) and (6.24), we see that some of the differential equations carry a large factor αβ or αγ, and some do not. This means the system
CHAPTER 6. NONRADIOACTIVE TRANSPORT 85
Table 6.1: Conversion table for variables
% = u6(t) +u13(t), σ = u7(t) +u14(t)
labeled unlabeled
slow v1(t) = da( ¯αt) s1(t) v5(t) = ma( ¯αt)
u6(t) = dp( ¯αt) s2(t) u13(t) = mp( ¯αt) = %−u6(t) fast
u7(t) = gp( ¯αt) s3(t) u14(t) = np( ¯αt) = σ−u7(t)
v2(t) = ga( ¯αt) s4(t) v6(t) = na( ¯αt) slow
v3(t) = gi( ¯αt) s5(t) v7(t) = ni( ¯αt)
u4(t) = e1( ¯αt) s6(t) u11(t) = e3( ¯αt) u5(t) = ei( ¯αt) s7(t) u12(t) = ej( ¯αt) fast
u2(t) = e2( ¯αt) s8(t) u10(t) = e4( ¯αt) u9(t) = e5( ¯αt) s9(t) u15(t) = e6( ¯αt)
slow v4(t) = di( ¯αt) s10(t) v8(t) = mi( ¯αt) remaining fast variables:
u1(t) = a( ¯αt), u3(t) = τ( ¯αt), u8(t) = b( ¯αt).
splits up into two time scales, where the components carrying a factor reach an equilibrium a lot faster than the others. This allows us to approximate the fast variables by a pseudo steady state.
6.5 Pseudosteady State Approximation
Instead of solving the differential equations with the large factors, we can approx-imate them by setting the left hand sides to zero. Again, we use s1(t)−s10(t), the time scale information, % and σ from table 6.1:
CHAPTER 6. NONRADIOACTIVE TRANSPORT 87
The stars (*) symbolize the negative sum of the remaining column elements.
In (6.29) - (6.32) we can give analytical expressions for all the pseudosteady state variables. We summarized them in table 6.2, and will now explain how they are constructed:
The expressions foru1(t), u2(t) and u10(t) are derived from equation (6.29) using K¯2 = k2
k−2+κ2 (see (2.31)) : The lower two rows lead to
u10(t) = ¯K2(σ−u7(t))u1(t) and u2(t) = ¯K2u7(t)u1(t), and with the conservation equation
u1(t) +u2(t) +u10(t) = Cun1
we obtain the respective equations in table 6.2.
Table 6.2: Analytical expressions for some fast variables
% = u6(t) +u13(t), σ = u7(t) +u14(t),
N1 = 1 + ¯K2σ, N2 = 1 + ¯K1%+K4σ, N3 = 1 + ¯K5%,
u1(t) = Cun1 N1
u2(t) = Cun1K¯2u7(t) N1
, u3(t) = Cun3 N2
,
u4(t) = Cun3K¯1u6(t)
N2 , u5(t) = Cun3K4u7(t)
N2 , u8(t) = Cun8 N3 , u9(t) = Cun8K¯5u6(t)
N3 , u10(t) =Cun1K¯2(σ−u7(t))
N1 , u11(t) =Cun3K¯1(%−u6(t))
N2 ,
u12(t) =Cun3K4(σ−u7(t)) N2
, u15(t) =Cun8K¯5(%−u6(t)) N3
.
CHAPTER 6. NONRADIOACTIVE TRANSPORT 89
Equation (6.30) and K¯5 = k5
k−5+κ5 from (5.17) lead to
u15(t) = ¯K5(%−u6(t))u8(t) and u9(t) = ¯K5u6(t)u8(t), and using the conservation equation
u8(t) +u9(t) +u15(t) = Cun8
to the equations for u8(t), u9(t) and u15(t).
Equation (6.31) yields
u12(t) = K4(σ−u7(t))u3(t), u11(t) = ¯K1(%−u6(t))u3(t), u5(t) = K4u7(t)u3(t), u4(t) = ¯K1u6(t)u3(t),
and, with the conservation equation
u3(t) +u4(t) +u5(t) +u11(t) +u12(t) = Cun3,
the expressions foru3(t), u4(t), u5(t), u11(t) and u12(t). Thus, all equations from table 6.2 are accounted for.
We see in table 6.2 that all fast variables have an explicit equation except for u6(t), u7(t), u13(t) = %−u6(t) and u14(t) = σ−u7(t). These can be obtained from equation (6.32), although not explicitly, as this subsystem contains pseudo steady state equations as well as dynamical ones. The missing variables are defined in the equations fors2(t) ands3(t); thus, all fast variables are accounted for, and the remaining pseudo steady state equations do not contribute any new information since they are already dealt with in table 6.2.
We can simplify the implicit equations substantially by adding them to other pseudo steady state equations we know to be zero; we get an implicit equation fors2(t) by adding the rows fors6(t) ands9(t) to the second row:
k0s1(t)−k−0s2(t)−κ1s6(t)−κ5s9(t) = 0;
and one fors3(t) by adding the rows fors7(t) ands8(t) in equation (6.32) to the third row:
−k−3s3(t) +k3s4(t) + 2κ1s6(t)−κ2s8(t) = 0.
Inserting the variables from table 6.1 we obtain k0v1(t)−k−0u6(t)−κ1u4(t)−κ5u9(t) = 0
−k−3u7(t) +k3v2(t) + 2κ1u4(t)−κ2u2(t) = 0,
(6.33)
for the radioactive set and
k0v5(t)−k−0u13(t)−κ1u11(t)−κ5u15(t) = 0
−k−3u14(t) +k3v6(t) + 2κ1u11(t)−κ2u10(t) = 0
(6.34)
for the nonradioactive one. Using equation (6.33) andu2(t), u4(t) and u9(t) from table 6.2 we construct the implicit equations for the labeled sugars in table 6.3; the implicit equations for the unlabeled sugars are obtained by inserting u10(t), u11(t) and u15(t) from table 6.2 into (6.34).
The four equations in table 6.3 define the missing four fast variablesu6(t), u7(t), u13(t) = %−u6(t) and u14(t) = σ −u7(t) implicitly. We see here why the fast variables still change with time although they are in a pseudo steady state: all fast variables from table 6.2 are dependent on the four variables defined by the implicit equations in table 6.3, and these are dependent on the slow variables v1(t), v2(t), v5(t) and v6(t).
As we see in (6.32), the remaining differential equations now read
˙
s1(t) = 1
η ·(−k0s1(t) +k−0s2(t)), s˙5(t) = 1 η ·α
δκ2s8(t),
˙
s4(t) = 1
η ·(−k3s4(t) +k−3s3(t)), s˙10(t) = 1 η · α
δκ5s9(t);
(6.35)
using the explicit expressions from table 6.2 and s1(t)−s10(t) from table 6.1, we end up with the equations in table 6.3.
In table 6.3 we see that the differential equations are only dependent on dynamical variables and the four variables%, σ, u6(t) andu7(t) which are defined by the four implicit equations in table 6.3.
Next we turn to the definition of the transport velocity in this system. We need the velocity as a function of the sugar concentration in the medium, as 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 radioactivity inside, without having the means of distinguishing
CHAPTER 6. NONRADIOACTIVE TRANSPORT 91
Table 6.3: Remaining differential and implicit equations
% = u6(t) +u13(t), σ = u7(t) +u14(t),
N1 = 1 + ¯K2σ, N2 = 1 + ¯K1%+K4σ, N3 = 1 + ¯K5%,
labeled unlabeled
˙
v1(t) = 1
η ·(−k0v1(t) +k−0u6(t)), v˙5(t) = 1
η ·(−k0v5(t) +k−0(%−u6(t))),
˙
v2(t) = 1
η ·(−k3v2(t) +k−3u7(t)), v˙6(t) = 1
η ·(−k3v6(t) +k−3(σ−u7(t))),
˙
v3(t) = 1 η ·α
δκ2Cun1K¯2u7(t) N1
, v˙7(t) = 1 η ·α
δκ2Cun1K¯2(σ−u7(t)) N1
,
˙
v4(t) = 1 η ·α
δκ5Cun8K¯5u6(t)
N3 , v˙8(t) = 1 η ·α
δκ5Cun8K¯5(%−u6(t))
N3 ,
k0v1(t)−
k−0+κ1Cun3K¯1
N2 +κ5K¯5Cun8 N3
u6(t) = 0, labeled
k3v2(t) + 2κ1
Cun3K¯1u6(t)
N2 −
k−3+κ2
Cun1K¯2
N1
u7(t) = 0,
k0v5(t)−
k−0+κ1
Cun3K¯1 N2
+κ5
K¯5Cun8 N3
(%−u6(t)) = 0, unlabeled
k3v6(t) + 2κ1
Cun3K¯1(%−u6(t))
N2 −
k−3+κ2Cun1K¯2 N1
(σ−u7(t)) = 0.
between radioactive glucose and trehalose, which consists of two labeled glucose subunits. This means that we can only look at the sum of all radioactive glucose subunits, and the change of this sum can be written as
vtot(t) = ˙ngi(t) + 2 ˙ndi(t). (6.36) from table 6.1 we see that in order to find the relevant rates ˙ngi(t) and ˙ndi(t), we first have to multiply the differential equations forv3(t) and v4(t), found in table 6.3, by the cytoplasmic volume δ (see (6.7)):
δv˙3(t) = 1
η ·α·κ2
Cun1K¯2u7(t) 1 + ¯K2σ δv˙4(t) = 1
η ·α·κ5Cun8K¯5u6(t) 1 + ¯K5%
(6.37)
with the implicit equations from table 6.3.
Division of the first equations of the labeled and unlabeled implicit equations in table 6.3 by k−0 and of the second ones by k−3 yields, with K0 = kk0
−0 and K3 = kk3
−3 from (3.25), the labeled case K0v1(t)−
1 + κ1
k−0
· Cun3K¯1 N2 + κ5
k−0
·K¯5Cun8 N3
u6(t) = 0,
K3v2(t) + 2 κ1
k−3
·Cun3K¯1u6(t)
N2 −
1 + κ2
k−3
·Cun1K¯2
N1
u7(t) = 0
(6.38)
and the unlabeled case K0v5(t)−
1 + κ1 k−0
· Cun3K¯1 N2
+ κ5 k−0
·K¯5Cun8 N3
(%−u6(t)) = 0,
K3v6(t) + 2 κ1
k−3
·Cun3K¯1(%−u6(t))
N2 −
1 + κ2
k−3
· Cun1K¯2
N1
(σ−u7(t)) = 0.
The biologically important variables are the sugars on the outside and the in-side of the cell, as the outin-side concentrations are the experimental input, and the amount of radioactive sugar inside is the only experimentally accessible out-put. These variables have to be expressed in the original dimensions and cannot
CHAPTER 6. NONRADIOACTIVE TRANSPORT 93
be transformed into scaled variables. On the other hand, the variables for the periplasmatic sugars cannot be measured anyway; they are mere operands and we can only profit by making them dimensionless by scaling. This reduces the number of parameters, and we can operate with plain numbers. Hence, we can also suppress their dependence on time, which is mediated by the slow variables only. The definitions of the scaled variables can be found in table 6.4.
With the scaled variables from table 6.4 we write equation (6.37) as
δv˙3(t) = 1
η ·α·κ2Cun1u¯7 N1 , δv˙4(t) = 1
η ·α·κ5Cun8KK¯¯5
1u¯6
N3
(6.39)
with
K0v1(t)− 1 K¯1
+ κ1 k−0
·Cun3 N2 + κ5
k−0
·
K¯5
K¯1Cun8 N3
!
¯
u6 = 0,
K3v2(t) + 2 κ1 k−3
· Cun3u¯6 N2
− 1
K¯2 + κ2 k−3
· Cun1 N1
¯ u7 = 0
(6.40)
for the labeled sugars and K0v5(t)− 1
K¯1 + κ1 k−0
· Cun3 N2
+ κ5 k−0
·
K¯5
K¯1Cun8 N3
!
(¯%−u¯6) = 0,
K3v6(t) + 2 κ1 k−3
·Cun3(¯%−u¯6)
N2 −
1 K¯2
+ κ2 k−3
· Cun1 N1
(¯σ−u¯7) = 0
(6.41)
for the unlabeled ones. After multiplication of the first equations in (6.40) and (6.41) with ¯K1 and the second equations with ¯K2 we obtain the implicit equations in table 6.4.
We have now described the rates of change for the radioactive sugars inside using transformed variables and not the measured variablesngi andndi. To compare our theoretical results with the experimental data we have to retransform equation (6.39) and the implicit equations in table 6.4 into the original variables: with the definitions in table 6.1 we get
Table 6.4: Scaled variables
¯
u6 = ¯K1u6, u¯13 = ¯K1u13, s¯2 = ¯K1s2, d¯p = ¯K1dp, m¯p = ¯K1mp, %¯ = ¯K1%
¯
u7 = ¯K2u7, u¯14 = ¯K2u14, s¯3 = ¯K2s3,
¯
gp = ¯K2gp, n¯p = ¯K2np, σ¯ = ¯K2σ
¯
% = ¯u6+ ¯u13, σ¯ = ¯u7+ ¯u14, N1 = 1 + ¯σ, N2 = 1 + ¯%+K4
K¯2
¯
σ, N3 = 1 +K¯5 K¯1
¯
%,
K0K¯1v1(t)−
1 + κ1
k−0
· K¯1Cun3 N2 + κ5
k−0
·K¯5Cun8 N3
¯
u6 = 0, labeled
K¯2K3v2(t) + 2 κ1 k−3
· K¯2Cun3u¯6
N2 −
1 + κ2 k−3
·K¯2Cun1
N1
¯
u7 = 0,
K0K¯1v5(t)−
1 + κ1 k−0
· K¯1Cun3 N2
+ κ5 k−0
·K¯5Cun8 N3
(¯%−u¯6) = 0, unlabeled
K¯2K3v6(t) + 2 κ1
k−3
·K¯2Cun3(¯%−u¯6)
N2 −
1 + κ2
k−3
·K¯2Cun1 N1
(¯σ−u¯7) = 0.
CHAPTER 6. NONRADIOACTIVE TRANSPORT 95
with the scaled variables as in table 6.4 and K0K¯1da( ¯αt)−
for the labeled sugars and K0K¯1ma( ¯αt)− for the nonlabeled ones.
All variables are now considered at the same time ¯αt, so that we can return to the time t, or even leave the time dependence of the sugars away completely.
In that case, vsum describes the uptake rate in dependence of the actual sugar concentrations in the medium:
vsum(da, ga, ma, na) = κ2
in the labeled case and
in the nonlabeled one. Note that the fast variables are not time dependent any more, as we only look at a single point in time.
In order to fit the constants in our system we have to group them to fit variables:
w1 = κ2Cun1, z1 = kκ1 We can see that these are the same variables we already encountered in the previous chapter on radioactive trehalose transport, except for Cun1, Cun3 and Cun8; but if we compare equation (6.26) with equations (2.26), (3.17) and (5.11) we see that
Cun1 = Cug1, Cun3 = Cue3, Cun8 = Cuz8,
hence, both (6.48) and (5.31) define the same parameters. Inserting them into (6.45) and using the definitions
¯
% = ¯dp+ ¯mp, σ¯ = ¯gp+ ¯np
from table 6.4 we obtain
vsum(da, ga, ma, na) = w1¯gp
from the radioactive subsystem and z3ma−
CHAPTER 6. NONRADIOACTIVE TRANSPORT 97
from the nonradioactive one.
Experimentally known are only the concentrationsda(0), ga(0), ma(0) and na(0), wherega(0) =ma(0) = 0. This leads to
vsum(da(0),0,0, na(0)) = w1¯gp
1 + ¯σ + z5d¯p
1 +z6%¯, z3da(0)−
1 + z4
1 + ¯%+z2σ¯ +
z5z4w2
2z1w1
1 +z6%¯
d¯p = 0
2 z1d¯p
1 + ¯%+z2σ¯ −
1 + w2 1 + ¯σ
¯ gp = 0
−
1 + z4
1 + ¯%+z2σ¯ +
z5z4w2
2z1w1
1 +z6%¯
(¯%−d¯p) = 0
w3na(0) + 2 z1(¯%−d¯p) 1 + ¯%+z2¯σ −
1 + w2 1 + ¯σ
(¯σ−¯gp) = 0.
(6.52)
0 100 200 300 400 500
0 0.5 1 1.5 2
cold glucose n a(0)[µM]
uptakeratevsum(1µM,0,0,na(0))[nmol min·mg]
0 500 1000 1500
0 5 10 15 20
cold glucose n a(0)[µM]
uptakeratevsum(1mM,0,0,na(0))[nmol min·mg]
Fig. 6.2: Left: Uptake ratevsum(da(0),0,0, na(0)) withda(0) = 1µM plotted against the concentration of the inhibiting cold glucose na(0). At low trehalose concentrations, the inhibition effect is minimal. Right: Uptake ratevsum(da(0),0,0, na(0)) withda(0) = 1000µM = 1mM plotted against the inhibitor concentration na(0). At high substrate concentrations, the inhibition effect becomes considerable.