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.
6.6 Results of the Simulation
We have used equation (6.49) to simulate inhibition experiments. The results are shown in fig. 6.2: we see the impact of the inhibiting cold glucose on the
uptake rate of radioactive trehalose at a fixed substrate concentration. In the left plot we see that the inhibiting effect is small at low substrate concentrations (1 µM trehalose concentration), whereas the right plot shows substantial inhibition at high substrate concentrations (1 mM trehalose concentration). This result is perfectly understandable if we look at fig. 5.3 in chapter 5: the inhibition reaction (3.3) reduces the cleavage of trehalose into glucose and thus the uptake through the glucose transporter. At very high inhibitor concentrations, the red line in fig. 5.3 showing the uptake via glucose transporter is reduced to almost zero, and the entire transport follows the blue line showing the uptake via trehalose transporter. At low concentrations of trehalose the drop from the green to the blue curve is very small, but at high concentrations the difference is large.
This result is not conform with the experimental findings, where we see a tenfold reduction in the uptake rate at one µM. To model this phenomenon, a further reaction is necessary which inhibits the transport of periplasmatic tre-halose into the cytoplasm.
Chapter 7 Summary
The aim of this work was to elucidate the mechanism of trehalose uptake in the thermophilic halophilic bacterium Rhodothermus marinus. Mathematical mod-eling was employed to translate chemical networks describing subsystems of the postulated trehalose uptake mechanism into coupled differential equations. An-alytical functions resulting from these differential equations were used for a mul-tidimensional least-squares fit to experimental data, thus yielding values for fit parameters and serving as a test to decide between two models. The data used were kindly provided by Carla Jorge from the laboratory of Prof. Dr. Helena Santos at the Instituto de Tecnologia Quimica e Biologica (ITQB) in Oeiras, Portugal.
Several data sets contributed information to resolve the issue: the experi-mental data consist of measurements of the uptake rate for radioactive glucose analyzed in chapter 2, measurements of glucose release into the medium due to in vivo trehalase activity dealt with in chapter 3, uptake data for radioactive trehalose modeled in chapters 4 and 5, and cross-inhibition experiments with glucose and trehalose simulated in chapter 6. In addition to these, in vitro in-hibition experiments with purified trehalase have shown that glucose inhibits trehalase activity with aKi of 12mM.
We have proposed two models for trehalose uptake: model 1 consists of a diffusion channel for glucose and trehalose in the outer membrane, a periplasmic trehalase cleaving incoming trehalose into two units of glucose, and a glucose transporter in the cytoplasmic membrane. Model 2 contains an additional tre-halose transporter in the inner membrane. Since the glucose transport system is a functionally independent part of both trehalose uptake models, the parameters obtained by fitting the glucose uptake data alone can be used for the fit of the trehalose uptake and enzymatic cleavage data. To describe thein vivo trehalase data four additional parameters are needed. The same four parameters are used
99
in model 1 for trehalose uptake; model 2 contains two more. Both experimental curves have to be fitted together, since the same set of parameters has to describe both measurements.
One of the remaining parameters, z2, can be calculated from theKi for thein vitro glucose inhibition of trehalase mentioned above and the Kmgt of the glucose transporter in the cytoplasmic membrane found in chapter 2.8. The theories for enzymatic cleavage and trehalose uptake provide restrictions coupling the remaining fit parameters to the kinetic parametersKme andVmaxe from the enzyme curve and Kmtr andVmaxtr (model 1) orKmsu andVmaxsu (model 2) from the trehalose uptake curve, for which we obtain an approximation by fitting a Michaelis-Menten function through the data. These restrictions together suffice to calculate all fit parameters in model 1, for model two an additional scan over z6 is needed.
By optimizing the four kinetic parameters using model 2 we have obtained a fit that describes all measurements. Model 1 can either fit the cleavage data or the high concentration trehalose uptake data, but not both, and is unable to describe the trehalose uptake measurements at low substrate concentrations.
The inability of model one to coherently describe all experimental findings shows the necessity of the additional threhalose transporter distinguishing model 1 from model 2.
In order to further validate the model we have used the parameters from the glucose fit and from model 2 to simulate an inhibition experiment with a constant concentration of the substrate trehalose and increasing concentrations of the inhibitor glucose. The simulation shows that in model 2 glucose inhibits the uptake of trehalose considerably in the range of high trehalose concentrations, but at low trehalose concentrations even a very high concentration of glucose causes only a slight inhibiting effect. This observation is due to the fact that the dominant uptake pathway at low trehalose concentrations is the direct trehalose transporter which is not inhibited by glucose in our model.
The result of the simulation is in qualitative agreement with experimental findings, but those show a tenfold inhibition of trehalose uptake already at the low concentration of 1µM. To be able to theoretically reproduce this experimental result quantitatively, an extension of the model network by a glucose mediated inhibition of the trehalose transporter in the cytoplasmic membrane should be adressed in further investigations.
Appendix A Tables
Distribution of the Compounds over the Different Volumina:
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
(A.1)
Theoretical variables:
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),
(A.2)
101
Table A.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).
APPENDIX A. TABLES 103
Table A.2: Fit Results
Vmaxsu = 18.16minnmol·mg, Kmsu = 79.2µM, Vmaxe = 40.7minnmol·mg, Kme = 128µM, w1 = κ2Cug1 = 54.3569minnmol·mg, w2 = κ2Cug1·kK¯2
−3 = 0.9137655, w3 = K¯2K3 = 3.406·10−2nmolml ,
¯
w1 = ww12 = 59.4867minnmol·mg, w¯2 = ww12·w3 = 2.026117minml·mg, z1 = kκ1
−3
K¯2Cue3 = 0.34209, z2 = KK¯42 = 2.4467·10−3, z3 = K0K¯1 = 7.83223·10−3nmolml , z4 = kκ1
−0
K¯1Cue3 = 3.377097·10−3, z5 = 2·κ5·KK¯¯51Cuz8 = 4.2552·102minnmol·mg, z6 = KK¯¯51 = 193.8.
Table A.3: Main Results I
Glucose transport: vg(v2) = w1x(v2) 1 +x(v2), w3v2−
1 + w2
1 +x(v2)
x(v2) = 0
Trehalase activity: ¯ve(da(0)) = ¯w1x(d¯ a(0)), 2· z1y(d¯ a(0))
1 + ¯y(da(0)) +z2x(d¯ a(0)) −x(d¯ a(0)) = 0, z3da(0)−
1 + z4
1 + ¯y(da(0)) +z2x(d¯ a(0))
¯
y(da(0)) = 0
K4 = 12mM1 , K¯1
2 = 29.36µM,
Kme = z13 · 1 +z2z1+z24
, Vmaxe = w¯1·2·z1 Trehalose transport model 1: 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+z2¯gp
d¯p = 0
APPENDIX A. TABLES 105
Table A.4: Main Results II
Trehalose transport model 2: vsu(da(0),0) = w1g¯p
1 + ¯gp
+1+zz5d¯p
6d¯p, 2 z1d¯p
1 + ¯dp+z2g¯p
−
1 + w2 1 + ¯gp
¯ gp = 0
z3da(0)−
1 + z4
1 + ¯dp+z2g¯p
+
z5z4w2
2z1w1
1 +z6d¯p
d¯p = 0
Cross-Inhibition: vsum(da(0),0,0, na(0)) = w1¯gp
1 + ¯σ + z5d¯p
1 +z6%¯, 2 z1d¯p
1 + ¯%+z2σ¯ −
1 + w2
1 + ¯σ
¯ gp = 0
z3da(0)−
1 + z4
1 + ¯%+z2σ¯ +
z5z4w2
2z1w1
1 +z6%¯
d¯p = 0
w3na(0) + 2 z1(¯%−d¯p) 1 + ¯%+z2σ¯ −
1 + w2
1 + ¯σ
(¯σ−g¯p) = 0
−
1 + z4
1 + ¯%+z2σ¯ +
z5z4w2
2z1w1
1 +z6%¯
(¯%−d¯p) = 0.
Table A.5: Analytical Expressions for Some Fast Variables
% = u6(t) +u13(t), σ = u7(t) +u14(t),
u1(t) = Cun1
1 + ¯K2σ, u2(t) = Cun1K¯2u7(t)
1 + ¯K2σ , u3(t) = Cun3
1 + ¯K1%+K4σ, u4(t) = Cun3K¯1u6(t)
1 + ¯K1%+K4σ, u5(t) = Cun3K4u7(t)
1 + ¯K1%+K4σ, u8(t) = Cun8 1 + ¯K5%, u9(t) = Cun8K¯5u6(t)
1 + ¯K5% , u10(t) = Cun1K¯2(σ−u7(t))
1 + ¯K2σ , u11(t) = Cun3K¯1(%−u6(t)) 1 + ¯K1%+K4σ , u12(t) = Cun3K4(σ−u7(t))
1 + ¯K1%+K4σ , u15(t) = Cun8K¯5(%−u6(t)) 1 + ¯K5% .
APPENDIX A. TABLES 107
Table A.6: Remaining Differential and Implicit Equations and Scaled Variables
% = u6(t) +u13(t), σ = u7(t) +u14(t),
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
δκ2
Cun1K¯2u7(t)
1 + ¯K2σ , v˙7(t) = 1 η ·α
δκ2
Cun1K¯2(σ−u7(t)) 1 + ¯K2σ ,
˙
v4(t) = 1 η ·α
δκ5Cun8K¯5u6(t)
1 + ¯K5% , v˙8(t) = 1 η ·α
δκ5Cun8K¯5(%−u6(t)) 1 + ¯K5% ,
¯
u6 = ¯K1u6, u¯13 = ¯K1u13, ¯s2 = ¯K1s2, d¯p = ¯K1dp, m¯p = ¯K1mp, %¯ = ¯K1%,
¯
u7 = ¯K2u7, u¯14 = ¯K2u14, ¯s3 = ¯K2s3,
¯
gp = ¯K2gp, n¯p = ¯K2np, σ¯ = ¯K2σ
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