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					Figure 1
Pa functional activation PaCO 2 TOS O 2 concentration CMRO 2 Complex IV redox state ∆ oxCCO

blood flow SaO 2

Complex V (ATPase) turnover

substrate supply

proton motive force

Fig. 1. Summary of the main inputs, variables and processes in the model. Model inputs are enclosed in solid ovals, while outputs are enclosed in dashed ovals. Pa is arterial blood pressure, SaO2 is arterial oxygen saturation level, PaCO2 is arterial CO2 level. TOS and ∆oxCCO are NIRS signals defined in the text.

Figure 2
CuA r f1 R CuA o f2 cyt a3 o cyt a3 r f3 O2

∆p
Fig. 2. Schematic representation of the mitochondrial submodel. The CuA centre is reduced by some reducing substrate, termed R. It in turn passes its electrons on to a terminal substrate, cyt a3 . Finally cyt a3 is oxidised by oxygen. All processes can in general produce proton motive force ∆p, by pumping protons out of the mitochondrial matrix. As a result, they are also inhibited by ∆p. The rates of the three processes – initial reduction of CuA , electron transfer to cyt a3 and final oxidation of cyt a3 , are termed f1 , f2 and f3 respectively.

1

Figure 3
functional activation

O2 CMRO 2

Complex V (ATPase) turnover

uncouplers

Complex IV redox state

∆ oxCCO

proton motive force

Fig. 3. Summary of the main variables and processes in the simplified model. As in Figure 1, inputs are enclosed in solid ovals, while outputs are enclosed in dashed ovals. Components connected with blood flow have been removed from the model. O2 levels are now directly settable.

2

Figure 4
2.5 CBF (normalised) 2 1.5 1 0.5 40 80 120 160 ABP (mmHg) 200 40 80 120 PaCO2 (mmHg) 160

(A) CBF (normalised) 2 1.5 1 0.5

(B)

Fig. 4. The response of model steady state CBF to blood pressure and PaCO2 changes. A) Response to arterial blood pressure changes with data from [1] (red squares) and [2] (green triangles) for comparison. B) Response to PaCO2 changes with data from [3] (with normal blood flow taken as 40 ml/min/100g) for comparison.

To reproduce the model curve in Figure 4 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file pres.dat, run the simulation and output CBF/CBFn.

To reproduce the model curve in Figure 4 B) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file co2.dat, run the simulation and output CBF/CBFn.

3

Figure 5
1.04 CMRO2 (normalised) (A) CBF (normalised) 1.03 1.02 1.01 1 0.99 0 10 time (s) 0.06 71 TOS (%) 70.5 70 69.5 0 10 time (s) 20 30 0 10 time (s) 20 30 (C) ∆oxCCO (µM) 0.04 0.02 0 (D) 20 30 0 10 time (s) 20 30 1.08 (B)

1.04

1

Fig. 5. Model responses to a step up in demand. A) Change in CMRO2 (normalised). B) Change in CBF (normalised). C) Change in TOS (percent). D) Change in ∆oxCCO (µM). All parameters are held at normal values apart from u which is stepped up from 1 to 1.2 for a ten second duration, giving rise to an approximately 3.5 percent increase in CMRO2 and an approximately 6 percent increase in blood flow. TOS increased by a little under 1 percent, and ∆oxCCO also increased by about 0.05 µM corresponding to an oxidation of just under 1 percent.

To reproduce Figure 5 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file activation.dat, run the simulation and output f1/f_n.

To reproduce Figure 5 B) (1) choose the full model descriptor fainvivo.dat (on the model editing interface), 4

(2) (3) (4) (5)

unclick the autoreg button to get a dynamic simulation, choose the input file activation.dat, run the simulation and output CBF/CBFn.

To reproduce Figure 5 C) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file activation.dat, run the simulation and output TOI.

To reproduce Figure 5 D) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file activation.dat, run the simulation and output CCO.

5

Figure 6
∆HbO2, ∆HHb and ∆Hbt (µM) 1.5 1 0.5 0 -0.5 -1 0 10 time (s) 20 30 (A) 1.5 1 0.5 0 -0.5 -1 0 10 time (s) 20 30 ∆HbO2, ∆HHb and ∆Hbt (µM) (B)

Fig. 6. Response of haemoglobin signals to a step up in demand. The response in µM of ∆HbO2 (red), ∆HHb (green) and ∆Hbt (black) to a step up in demand. The stimulus and parameter values are as in Figure 5. In A) τu = 0.5 s (the default value). In B) τu = 1 s. With the slower response time, there is more pronounced transient behaviour including a clear initial decrease in ∆HbO2 before it starts to increase.

To reproduce Figure 6 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file activation.dat, run the simulation and output DHbO2: DHHb: DHbT.

To reproduce Figure 6 B) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file activation_slow.dat, run the simulation and output DHbO2: DHHb: DHbT.

6

Figure 7
100 81.2 % CuA oxidation 80.8 80.4 80 70 0 10 time (s) 20 30 0.8 1.2 1.6 CMRO2 (normalised) (A) % CuA oxidation 90 (B)

80

Fig. 7. Response of CuA redox state in the simplified model to changes in u. A) The time course of oxidised CuA in response to functional activation. As in the in vivo simulations, u was changed from 1 to 1.2 for a ten second duration, resulting in an approximately 1 percent increase in CuA oxidation. B) The steady state level of CuA oxidation in response to varying levels of activation. u was varied from 0.2 to 100 resulting in variation in CMRO2 from 80 to 170 percent of baseline. CuA oxidation increased steadily.

To reproduce Figure 7 A (1) (2) (3) (4) (5) choose the simplified model descriptor o2param.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file funcact_isolated.dat, run the simulation and output a/cytox_tot*100.

To reproduce Figure 7 B (1) (2) (3) (4) (5) choose the simplified model descriptor o2param.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file funcact_u.dat, run the simulation and output xvar: f1/f_n a/cytox_tot*100.

7

Figure 8
mitochondrial O2 (normalised) 1.2

0.8

0.4

0 1 1.2 1.4 CMRO2 (normalised)

Fig. 8. Relationship between CMRO2 amd mitochondrial oxygen levels during activation. The full model was run with parameter Ru set to zero so that an increase in demand had no effect on blood flow. Increasing u allowed increases in CMRO2 up to approximately 145 percent of baseline. The three data points shown are calculated from Figure 2 of [4] in which predictions on how tissue oxygen levels in the “lethal corner” should vary with activation level during normoxia are presented.

To reproduce the model curve in Figure 8 (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file funcact_u_CBFfix.dat, run the simulation and output xvar: f1/f_n O2/O2_n.

8

Figure 9

55 % CuA reduction 45 35 25 15

(B)

6

18

30 O2 (µ M)

42

54

Fig. 9. Comparison of experimentally measured and modelled CCO redox states. A) Figure 5A from [5] is redrawn showing how the level of reduction of cytochrome c varies with oxygen concentration. B) The equivalent data for CuA from model simulations is presented. For the simulation, the reducing substrate is set to be succinate, and the demand parameter u is set to be low (u = 0.4) to represent a high phosphorylation potential.

To reproduce Figure 9 B) (1) (2) (3) (4) (5) choose the simplified model descriptor o2param.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file fwilson.dat, run the simulation and output xvar: 1000*O2 100*ared/cytox_tot.

9

Figure 10
5 0.8 CMRO2 (a.u.) 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 16 O2 (µM) (A) 4 CMRO2 (a.u.) 3 2 1 0 0 2 4 6 8 10 O2 (µM) 12 14 16 (B)

Fig. 10. The response of steady state CMRO2 to a drop in mitochondrial O2 level. CMRO2 is in arbitrary units. A) In coupled mitochondria. B) Uncoupled mitochondria. As above, for both simulations, the reducing substrate is set to be succinate, so that input to the system is by electron transfer to ubiquinone, and the demand parameter u is set to be low (u = 0.4 in both simulations). For the uncoupled mitochondria, the parameter kunc is raised from its normal value of 1 to a value of 1000. During uncoupling there is an approximately four-fold increase in maximum CMRO2 . The behaviour of CuA was similar to Figure 6 in [5] with baseline oxidation now at approximately 99 percent.

To reproduce Figure 10 A) (1) (2) (3) (4) (5) choose the simplified model descriptor o2param.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file funcact_O2wilson, run the simulation and output xvar: O2*1000 f1/f_n.

To reproduce Figure 10 B) (1) (2) (3) (4) (5) choose the simplified model descriptor o2param.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file funcact_O2wilsuncoup, run the simulation and output xvar: O2*1000 f1/f_n.

10

Figure 11
75 70 TOS (%) 65 60 55 0 60 time (s) 120 180 0.1 (A) ∆oxCCO (µM) 0 -0.1 -0.2 -0.3 -0.4 0 60 time (s) 120 180 (B)

Fig. 11. Model response of TOS and ∆oxCCO to a step down in arterial oxygen saturation. A) Response of TOS (percent). B) Response of ∆oxCCO (µM). A hyperaemic effect is seen in both signals.

To reproduce Figure 11 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file hypoxia2.dat, run the simulation and output TOI.

To reproduce Figure 11 B) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file hypoxia2.dat, run the simulation and output CCO.

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Figure 12

0 ∆oxCCO (µM) -1 -2 -3 -40 0

(A) ∆oxCCO (µM)

0 -1 -2 -3 -30 -20 -10 ∆HbO2 (µM) 0 0

(B)

0.5

0.6 0.7 0.8 0.9 CMRO2 (normalised)

1

(C) ∆oxCCO (µM) ∆oxCCO (µM) -1 -2 -3 -4 -30 -1 -2 -3 -4 -20 -10 ∆HbO2 (µM) 0 0.1

(D)

0.2 0.3 0.4 0.5 CMRO2 (normalised)

0.6

Fig. 12. Relationship between ∆HbO2, ∆oxCCO and CMRO2 during changes in arterial oxygen saturation A) The model was run with normal parameter values and an approximately linear relationship between ∆HbO2 and ∆oxCCO held. B) At these same normal parameter values CMRO2 showed an approximately linear realationship with ∆oxCCO. C) Baseline CMRO2 was lowered to about 60 percent of the normal model baseline, by setting u = 0.1 , while normal CBF was also lowered by about the same amount by setting CBFn = 0.007 ml blood per ml brain tissue per second. A more clearly biphasic relationship between ∆HbO2 and ∆oxCCO was obtained. D) Again, at the changed parameter values, CMRO2 had an approximately linear relationship with ∆oxCCO.

To reproduce Figure 12 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file springett0.dat, run the simulation and output xvar: DHbO2 CCO.

To reproduce Figure 12 B) 12

(1) (2) (3) (4) (5)

choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file springett.dat, (this chooses the smooth approximation to J_O2) run the simulation and output xvar: DHbO2 CCO.

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Figure 13
modelled and measured TOS (%) 80 70 0 60 50 0 500 1000 time (s) 1500 2000 0 500 1000 time (s) 1500 2000 -0.4 (A) 0.4 modelled and measured ∆oxCCO (µM) (B)

Fig. 13. Responses of measured and modelled TOS and ∆oxCCO during a hypoxia challenge. Measured (red) and modelled (black) responses of A) TOS (%) and B) ∆oxCCO (µM) are shown. Details are given in the text.

To reproduce Figure 13 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file Hypoxia_cyt20ana1.dat, run the simulation and output TOI: TOIsup.

To reproduce Figure 13 B) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file Hypoxia_cyt20ana1.dat, run the simulation and output CCOsup1-0.35: CCO.

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Figure 14

(A) 80 80 70

(B)

60 0 400 time (s) (C) 800

70 0 400 time (s) 800

80

80

(D)

70

70

60 0 400 time (s) 800

60 0 400 time (s) 800

Fig. 14. Responses of measured and modelled TOS during a hypercapnia challenge. Measured (red) and modelled (black) responses of TOS: A) For subject 1 without optimisation. B) For subject 1 following optimisation of AVRn and RC , which gave values of AVRn = 0.78 and RC = 1.31. C) For subject 2 without optimisation. D) For subject 2 following optimisation of AVRn and RC , which gave values of AVRn = 3.5 and RC = 1.62.

To reproduce Figure 14 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file Hypdata/study17.dat run the simulation and output TOI: TOIsup.

To reproduce Figure 14 B) (1) choose the full model descriptor fainvivo.dat (on the model editing interface), (2) unclick the autoreg button to get a dynamic simulation, 15

(3) choose the input file Hypdata/study17opt.dat (4) run the simulation and (5) output TOI: TOIsup. To reproduce Figure 14 C) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file Hypdata/study09.dat run the simulation and output TOI: TOIsup.

To reproduce Figure 14 D) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), unclick the autoreg button to get a dynamic simulation, choose the input file Hypdata/study09opt.dat run the simulation and output TOI: TOIsup.

16

Figure 4 in supplementary material
2.5 CBF (normalised) 2 1.5 1 0.5 40 80 120 160 ABP (mmHg) 200

(A) CBF (normalised)

2 1.5 1 0.5

(B)

40

80 120 160 ABP (mmHg)

200

Fig. 15. [Fig. 4 in supplementary material]The response of model steady state CBF to blood pressure changes. A) Model autoregulation curve fitted to data from [2]. The following parameters were reset: Pa,n = 91.6, RP = 3.05 and r0 = 0.015. B) Model autoregulation curve fitted to data from [1]. Pa,n = 112.0, RP = 3.28 and r0 = 0.0133.

To reproduce the model output in Figure E.1 A) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file presharper.dat run the simulation and output CBF/CBFn.

To reproduce the model output in Figure E.1 B) (1) (2) (3) (4) (5) choose the full model descriptor fainvivo.dat (on the model editing interface), click the autoreg button to get a static simulation, choose the input file presgao.dat run the simulation and output CBF/CBFn.

References
[1] E. Gao, W. L. Young, J. Pile-Spellman, E. Ornstein, Q. Ma, Mathematical considerations for modelling cerebral blood flow autoregulation to systemic arterial pressure, Am J Physiol Heart Circ Physiol 274 (3) (1998) H1023–H1031. [2] S. L. Harper, H. G. Bohlen, M. J. Rubin, Arterial and microvascular contributions to cerebral cortical autoregulation in rats, Am J Physiol Heart Circ Physiol 246 (1) (1984) H17–24. 17

[3] M. Reivich, Arterial PCO2 and cerebral hemodynamics, Am J Physiol 206 (1) (1964) 25–35. [4] M. A. Mintun, B. N. Lundstrom, A. Z. Snyder, A. G. Vlassenko, G. L. Schulman, M. E. Raichle, Blood flow and oxygen delivery to human brain during functional activity: Theoretical modeling and experimental data, Proc Natl Acad Sci USA 98 (12) (2001) 6859–64. [5] D. F. Wilson, W. L. Rumsey, T. J. Green, J. M. Vanderkooi, The oxygen dependence of mitochondrial oxidative phosphorylation measured by a new optical method for measuring oxygen concentration, J Biol Chem 263 (6) (1988) 2712–2718.

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