Analysis of parameters a↵ecting oxidation level of Cytochrome c
oxidase, during functional activation
Claire Walsh
Supervisors: Jasmina Panovska-Griffiths and Chris Cooper
April 24, 2012
Abstract
The sensitivity analysis of parameters in the Model of Brain NIRS signals [11] has been
undertaken with the specific aim to reproduce experimental data from the functional
activation studies of Tachtsidis et al. [47]. The analysis has shown that the data can be
reproduced by the single alteration of either of two parameters, ck1 and DN ADH . It is also
shown that alteration of these parameters does not a↵ect crucial autoregualtion behaviour.
Other well defined behaviours are also investigated including % CuAr changes in response to
variation in [O2 ], and CM RO2 behaviour in uncoupled mitochondria with changing [O2 ].
1
1
Introduction
a3 . These sites are configured as two binuclear
centres of the CuA , a centre and the CuB ,
a3 centre. The CuA binuclear centre has a
NIR absorption spectra in the 830nm band[16]
and changes in its redox state from copper2+
to copper+ cause a decrease in signal intensity
which can be detected by NIRS[18]. This binuclear centre has been shown to be in close redox equilibrium with CCO[14], and can therefore be used to calculate the oxCCO signal.
This chromophore is present at an abundance
of approximately 10% that of haemoglobin in
the brain. Initial work on NIRS of CCO was
dogged by issues surrounding this low relative
concentration. It was originally thought that
cross-talk of the haemoglobin and oxCCO
signal, caused by the oversimplification of the
modified Beer-Lambert approximation, was
responsible for changes seen. However persistent attempts to show this have proved unsuccessful [50, 19].
Work carried out by Cooper et al.[18] provided
a clinical approach to resolving the cross talk
controversy: Using cyanide to cause maximum
reduction of the CuA site by blocking the a3
acceptor site in bloodless piglets, and following
this with drastic alterations in haemoglobin
levels, the group were able to show that the
redox state of the CCO remained constant in
response to the haemoglobin changes. Hence
they concluded that cross-talk was not a confounding factor. Similar work using rat and
cat subjects showed that cross species and developmental age dependencies were not significant.
Despite this work there remain sceptics as to
the reliability of the oxCCO signal. Yet a
persistence to improve understanding and extend its use are grounded in some significant
clinical findings. Several cases in which the
CCO signal has particular clinical significance
can be found in the literature. These include,
providing the best indicator of long term tissue damage via the measure of actual tissue
dysoxia [16], monitoring of oxCCO signal
during cardiac surgery [40] and monitoring following traumatic brain injury [49]. As well as
Near-infrared spectroscopy is becoming a well
established method for non invasive studies
of cerebral activity, particularly in investigation of functional activation [23, 50, 26, 14].
However doubt over the reliability of the
cytochrome-c oxidase (CCO) signal has lead
to a number of empirical studies on this enzyme [50, 19, 29]. In particular a recent
study by Tachtsidis et. al [47] has shown that
there may be significant variation in CCO redox responses to functional activation amongst
healthy individuals. An alternative approach
to these clinical and empirical studies is a modelling approach. Using BRAINCIRC, a model
of cerebral hemodynamics and metabolism developed by Banaji et al. [4][11], the oxCCO
signal in response to physiological changes has
been modelled. This provides a far more
rigidly controlled environment, not be achievable in clinical studies, which aids and furthers
understanding of the underlying physiology.
This work aims to carry out sensitivity analysis on several parameters of this model during functional activation. One of the key intentions is to see if experimental results showing that a decrease in oxCCO can be replicated by appropriate parameter changes. In
addition, analysis enabling understanding of
whether these parameter changes fall within
physiologically realistic limits is required. The
work consists of four major sections: An
overview of the debate regarding the CCO redox state, a review of the role the BRAINCIRC
model has played, a more complete outline of
the model structure and sensitivity analysis of
parameters relevant to functional activation.
1.1
Origin of cyctochrome-c oxidase
signal and its significance
Cytochrome c oxidase is the terminal enzyme
in the electron transport chain (ETC) and is
responsible for catalysing 95% of oxygen production in mitochondria [47]. The enzyme consists of four metal redox centres; two copper
sites CuA and CuB and two haem sites a and
2
enables non-mathematicians specifically clinicians to have a more intuitive understanding
of the data.
Another central feature of the model is it’s
modularity, this arises from two constraints:
Firstly that models must have artificial boundaries and secondly that it must be possible to
update the model in response to new physiological advances [5, p. 248]. During the
model’s construction these constraints lead to
three main modules: The smooth muscular
component, the vascular biophysics and the
metabolic biochemistry. The vascular biophysics module is taken almost intact from the
Ursino Lodi model [51][52].
The metabolic biochemistry and in particular
the mitochondria sub-model is of particular interest in this work. The necessity for a mitochondria model arose in 2006 when it became clear that at least a caricature of mitochondrial metabolism was necessary as many
metabolic bi-products serve as markers for
CBF regulation[3, p. 501] Additionally there
is interest in modelling the electron transport
chain itself as it has been associated with several neurological conditions including Parkison’s, Huntigdon’s, and Alzheimer’s disease
[21]. Many extensive models of mitochondrial
oxidative metabolism exist and from which
this sub-model has drawn heavily, in particular
Korzeniewski, Zoladz, Beard [31, 32, 7] have
all heavily influenced this sub-model’s design.
Other models which attempt to link oxidative
metabolism into larger models of cerebral function also exist most notably Aubert and Costalat’s previously mentioned model [1].
Features of these other models have been omitted or caricatured by Banaji [3], in order to
make the sub-mode usable and relevant to
NIRS responses. Largely, components involving complexes I-III have been removed and
more detailed analysis given to complex IV
CCO and it’s redox state. The a↵ect of the
proton motive force on Complex V is also considered. The basic sub model can be represented essentially as three coupled reactions
know as f1 , f2 and f3
this, modelling data, which will be discussed
below in more detail, provides further supporting evidence that oxCCO contains significant information independent of cerebral oxygen changes[4]. This combined evidence leads
to interest in better understanding the physiology behind the signal and in how it can best
be used to aid understanding and provide a
measure of cerebral well being.
From the previous sections it is clear that clinical approaches such as that mentioned above
as well as others [48][49], can, and have provided significant results. It is also clear however, that the complexity of the system is
vast, making drawing physiological conclusions
based on NIRS data alone is very difficult. The
BRAINCIRC model provides a powerful tool
that could be used to inform further clinical
work and increase understanding of the physiology.
1.2
The Model
The BRAINCIRC model [4] available open
source[11] and used in this work, was created over a ten year period with the aim to
provide a model of cerebral circulation that
could ultimately be used as a real time interpretive aid for clinicians.The principle motivation, as described in Banaji and Baigent
(2006), was to begin with the physiological
processes as a guide for the dynamical system
of equation. By contrast, other models in the
same field begin with the system of equations
and are designed primarily to fit experimental data e.g.[24][41]. Models built from physiological processes include the Ursino-Lodi
model of vascular biophysics[51] and Aubert
and Costalat’s model of ’Coupling between
Brain Electrical Activity, Metabolism, and
Hemodynamics’[1]. The advantage of this approach is that it is able to provide physiological insight as to the origin of particular
model outputs e.g. variation in CM RO2 can
be described in terms of the mitochondrial enzymatic transport processes. Another advantage that cannot be stressed enough considering the ultimate aim of the model, is that it
3
• f1 is the transfer of electrons from a re- of CuA and the reducing substrate, Z is the
ducing substrate to the CuA centre. The standard physic-chemical constant and p is
reducing substrate can be either NADH or the proton motive force.
FADH/succinate depending on the physif2 = k2 CuAr a3o k 2 CuAo a3r
(3)
cal situation being modelled.
Where k2 and k 2 represent the forward and
backward rate constants and a3o and a3r represent the oxidised and reduced forms of the
a3 centre respectively. The relation between
k2 and k 2 is given by
• f2 is the transfer of electrons from CuA
to a3 (written as cyta3 in [4]). This represents the transfer of the electrons from
one binuclear centre to the second in the
physiological process.
k2
= Keq2 = 10 (p2 p/4
k 2
• f3 is the transfer of the electrons from a3
to oxygen.
E2 )/Z
(4)
A schematic of the process is shown in Fig- Where p2 is the number of protons pumped
across the membrane in association with reure 1:
action 2 and E2 is the di↵erence between the
standard redox potentials of a3 and CuA .
exp( c3 p)(1 + exp(c3 p30 ))
1 + exp( c3( p
p30 ))
(5)
Where k3 is the rate constant, c3 and p30 are
parameters which control the sensitivity of f3
Figure 1: Schematic representation of the reac- to the proton motive force. The proton motive
tion f1 , f2 and f3 , Figure taken from [4, p. 5]. force is described by the equation
f3 = k3,0 [O2 ]a3r
p=
Each of these processes is accompanied by the
pumping of protons across the mitochondrial
inner membrane, which produces an electrochemical gradient enabling the production of
ATP. The amount of proton pumping associated with the process depends on the initial
reducing substrate. The equations that control these rates are as follows:
f1 = k1 CuAo
k
1 CuAr
E1 )/Z
pHo )
(6)
Where
is the mitochondrial inner membrane potential, pHo is the pH in the intermembrane space, and Z = RT
F where F is the
Faraday constant, R is the ideal gas constant
and T the absolute temperature. Protons reenter the matrix via a proton leak channel and
via the ADP/ATP phosphorylation. Equations and further explanation for these reactions can be found in the appendix section 1.
The inner membrane potential is a time dependent variable, which depends on the number of
protons pumped across the membrane and the
rates f1 , f2 and f3 , further details of which can
also be found in the appendix section 1 eq. 5.
Using the above model provides scope for identifying physiological causes of empirical results, via a comparison of modelled outputs
and empirically data. As well as this the reverse is also true; using the model to quickly
simulate a number of physiological changes is
(1)
Where k1 is the forward rate constant k 1
is the backward rate constant and CuAo and
CuAr represent the oxidised and reduced CuA
centre. The relation between k1 and k 1 is
given as:
k1
= Keq1 = 10 (p1 p/4
k 1
+ Z(pHm
(2)
Where p1 is the number of protons pumped
across the membrane during f1 , E1 is the difference between the standard redox potentials
4
time efficient way of pinpointing what clinical
trials could provide significant results.
In the former of these two uses the BRAINCIRC model was utilised to investigate the
controversy surrounding cross-talk. The work
[4] comprised a functional activation simulation where the sensitivity of blood flow to activation was abolished by setting a sensitivity parameter to zero1 . When this was done
a reduced (by approx. 40%) but nonetheless
present oxCCO signal was seen[4, p. 9]. This
lead the authors to conclude that the changes
in the oxCCO signal are not merely an artefact of cross-talk, but shows that the ”CCO
redox state primarily associate with changes
in the proton motive force rather than being
slave to changes in oxygen level” [4, p. 10].
This work shows the benefit of this modelling
approach, and provides the basis and framework for the work done here.
2
Parameter
Methods
Choice
Based on these equations the parameters chosen for analysis in this work are ck1 , ck2 and
c3 all of which a↵ect the rate constants k1 , k2
and k3 respectively. ck1 controls the strength
of inhibition the proton motive force has over
f1 .
k1 = k1,0 exp( ck1 ( p
k2 = k2,n exp( ck 2( p
Nrat =
Where V olmit is the mitochondrial volume and
CuAo,n is the initialised value of CuAo . The
rates f1 and f2 a↵ect the CuAo concentration
and CM RO2 via the equation:
CM RO2 = V olmit f3
(9)
(11)
Nrat,n
u2DN ADH
(12)
Further discussion of all parameters and references as the how they are set can be found in
the table on p. 5 of the appendix and section
3.3.
Another significant parameter in this work is
u. This is the demand parameter and can be
considered as analogous to an appropriately
rescaled ADP/ATP ratio. During functional
activation simulation, this parameter is used
to represent neurological e↵ort. In accordance
with [4] this parameter has been increased
from 1 to 1.2 for a 10 second window. This
enables results from this work to be directly
compared to the work in [4], however consideration as to whether this is the best function
to use will be discussed further. The model
used for this work is the Model of Brain NIRS
Signals as obtained from [11]. Descriptions of
how all the figures shown can be created using
the model are found from the same source.
oxCCO = 1000V olmit (CuAo CuAo,n ) (7)
(8)
pn ))
and c3, is the parameter from eq.(5), which
represents the sensitivity of f3 to the proton
motive force p. Also the additional parameter DN ADH was chosen, a parameter which
represents the change glycolytic TCA cycle
flux during functional activation[47].
and
f1 )
(10)
Where k1 is the same as from eq.(1). ck2 has
the same form but for f2 .
Parameters were chosen from the subset of
model parameters which previously have only
been set heuristically and which intuitively
would a↵ect the signals of interest, ( oxCCO
during functional activation). As seen in the
previous section the three equations (1), (3),
(5) govern the rates of three electron transfers
within CCO. These rates are highly significant
in calculating the oxCCO and CM RO2 signals:
dCuAo
= 4(f2
dt
pn ))
3
Results and Discussion
As previously mentioned this work has a
twofold aim:
1
parameter in question is Ru , see appendix section
3.1 for details
5
1. To reproduce key clinical findings of Tach- ulation curve of [4]. This result allows some
sidis et al. in which oxCCO is seen to degree of confidence in the large parameter
decrease upon functional activation.
changes that were suggested by initial investigation. Secondly with regards to the first
2. To analyse how variation in parameters aim of this work, it is important that this bea↵ecting the electron transport chain, af- haviour remains intact regardless of the pafect key modelled behaviours and what in- rameter changes. The clinical study of intersight this gives as to their physiologically est [47] reported on healthy individuals and
realistic values.
hence, the heterogeneity of their oxCCO reTo answer the second of these two aims three sults, must be explained by parameter changes
key experiments were carried out. The first, a which preserve healthy behaviours such as aureproduction of the autoregualtion behaviour toregulation.
CBF Hmlê100g-minL
shown in Figure 4 in [4], and well described
experimentally by [24] and [34]. The second 3.2 Real data performance
being a reproduction of Figure 9 from [4] in
which real experimental data for cytochrome
c reduction from [54], was compared against
the modelled outcome. Finally the e↵ect of
oxygen concentration changes in the CM RO2
from which an apparent Km can be derived.
Initial investigation into changing parameter
values lead to maximum changes of between
5% - 1000% in all parameter values, these were
then tested as mentioned above for a physiological basis.
Figure 3: Plot of the a↵ects of oxygen concenc3=0.0
tration variation on the % cytochrome c reduc0.020
c3=0.117
c =0.0
tion. Figure taken from [4], a trace of original
c = 0.1
c =0.0
results from [54]
0.015
c =0.04
k1
k1
k2
k2
DNADH =0.0
DNADH =0.1
Reproduction of real data is an important
test of parameter range. Figure 9 from [4],
taken originally from the work of Wilson on in
vitro mitochondria, [54] is reproduced in Figures 3 and 4.
As can be seen ck2 , (Figure 4(b)) induces small
output changes in response to large variation
in its values, DN ADH (4(c)) changes have no
impact whatsoever on the modelled outcome,
ck1 (4(a)) and c3 (4(d)) changes, both significantly alter the modelled output.
The lack of a↵ect of DN ADH has on the output
is not surprising as in the simple model used to
create the data the reducing substrate is set to
succinate and changes to eq.(10) give a new expression of f1 as eq.(17) of the appendix. The
direction of the ck1 and ck2 responses are as
expected from the equations in section 2. A
0.010
0.005
50
100
150
200
ABPHmmHgL
Figure 2: Showing the atuoregulation responses to changes in arterial blood pressure
changes. The figure shows 8 curves, in each
curve one of the parameters of interest was
taken to the maximum or minimum value used
in the rest of the work.
3.1
Autoregualtion
As can be seen in Figure 2, changing each of
the parameters to both the maximum and minimum values, shows no impact on the autoreg6
50
35
30
40
25
35
30
25
20
20
15
15
0
10
20
30
40
50
ck2 #0.00
ck2 #0.01
ck2 #0.016
ck2 # 0.018
ck2 #0.02
ck2 #0.022
ck2 #0.024
ck2 #0.3
ck2 #0.04
ck2 #0.2
45
" CuA reduction
40
" CuA reduction
50
ck1 #0
ck1 #0.005
ck1 #0.008
ck1 #0.009
ck1 #0.01
ck1 #0.011
ck1 #0.012
ck1 #0.015
ck1 #0.05
ck1 #0.1
45
0
60
10
20
O2 ΜM
30
(a)
60
50
DNADH #0
DNADH #0.005
DNADH #0.008
DNADH #0.009
DNADH #0.01
DNADH #0.011
DNADH #0.012
DNADH #0.015
DNADH #0.05
DNADH #0.1
40
35
30
25
c3#0
45
c3#0.055
40
" CuA reduction
45
" CuA reduction
50
(b)
50
20
c3#0.088
35
c3#0.099
c3#0.11
30
25
20
15
0
40
O2 ΜM
15
10
20
30
40
50
60
0
O2 ΜM
10
20
30
40
50
60
O2 ΜM
(c)
(d)
Figure 4: Showing the a↵ect of changing parameters on model output using real data input
taken from [54]. (a) Shows a change in %CuA reduction with varying ck1 from 0 to 0.1, (b)
shows change caused when ck2 is varied through the range of 0-0.2. (c) Shows that no change
is caused when DN ADH is varied from 0-0.1 and (d) shows a↵ect of varying c3 from 0-0.117. In
all situations the plots were created based on the simple mitochondrial model with the reducing
substrate set to be succinate and the demand parameter to be low at u = 0.4
decrease in f1 leads to a decrease in the rate of
electrons being transferred to CuA and hence
a decrease in the % of reduced CuA as seen
in 4(a). The converse is true for ck2 , increasing this parameter decreases the rate at which
electrons move from the CuA to the a3 site.
The di↵erence between the magnitude of the
reactions of ck1 and ck2 however seems more
difficult to explain, being that it does not reflect the symmetry of the eqs. 1 and 3 and the
relation of eqs. 8 and 7. The smaller response
to ck2 changes implies that, either changes in
sensitivity to p do not have a great impact
on f2 , or, that changes in f2 have a smaller
e↵ect on the level of CuAo . The first of these
implies that p has less of an influence on the
transfer of electrons from CuA to a3 than it
does on succinate to CuA ; a conclusion not
born physiologically or mathematically. The
second implies that it is the transfer of succinate electrons which is the rate limiting step
in the reaction chain of Complex IV. This can
also be seen in the model equations where the
initial value of k1 (k1n ), is 8.92 whereas k2n is
3912
A comparison with 4(d) shows a qualitatively
di↵erent response to any of the other parameters. Changing c3 a↵ects the rate at which
the %CuA reduction decreases but does not
a↵ect the initial or final values unlike ck1 or
ck2 . Qualitatively the direction of the change
can be understood as previously, whereby, decrease in c3 leads to an increase in f3 , this
in turn causes electron transfer from the a3
site to O2 to increase, causing increase transfer
from CuA to a3. All of this leads to a decrease
2
these values were found from the model outdat
files.
7
25
0.8
ck1 "0
ck1 "0.005
ck1 "0.008
ck1 "0.009
ck1 "0.01
ck1 "0.011
ck1 "0.012
ck1 "0.015
ck1 "0.05
20
ck1 "0
ck1 "0.005
ck1 "0.008
ck1 "0.009
ck1 "0.01
ck1 "0.011
ck1 "0.012
ck1 "0.015
ck1 "0.05
ck1 "0.1
0.4
0.2
0.0
0
CMRO2 !a.u."
CMRO2 !a.u."
0.6
5
10
15
10
5
0
0
15
5
(a)
15
(b)
1.0
5
0.8
4
CMRO2 !a.u."
CMRO2 !a.u."
10
O2 ΜM
O2 ΜM
0.6
c3"0
0.4
c3"0.055
3
c3"0
c3"0.055
2
c3"0.088
c3"0.088
c3"0.099
0.2
1
c3"0.099
c3"0.11
c3"0.11
0.0
0
5
10
0
0
15
5
10
15
O2 ΜM
O2 ΜM
(c)
(d)
Figure 5: Showing the a↵ect of parameter changes on the CM RO2 response to [O2 ] conc. in
both coupled and uncoupled mitochondria. (a) and (b) show changes in coupled and uncoupled
mitochondria respectively with ck1 variation and (c) and (d) show changes for coupled and
uncoupled mitochondria with c3 respectively. As previous the reducing substrate is set as
succinate, and u is set to be 0.4. For uncoupled the parameter kunc is raised to 1000 from 1
giving a four fold increase in max CM RO2 .
in the % of reduced CuA . The lack of change
in the initial and final values can also be understood from the fact that while f1 and f2 are
reversible reactions f3 is not; hence changes in
this rate will not a↵ect the equilibrium concentrates of the end product. Additionally it
is interesting to note that increases in c3 from
the set value of 0.11 lead to a total loss of
the qualitative behaviour and are not shown
in 4(d). This results from a numerical feature
of the model whereby a singularity is reached
in f3 at a c3 values greater than 0.117. In addition, it is important to note that all parameter
changes excluding the two highest values of ck1
are consistent with the real data as shown in
Figure 3.
The variation in CM RO2 with varying O2 concentration is also a key response which is affected by the parameters under investigation.
A plot of this behaviour as reproduced from
Figure 10 in [4] is shown in Figure 5. The
plots of DN ADH and ck2 are not shown as the
reducing substrate was set as succinate hence
DN ADH as expected showed no a↵ect and ck2
as previously, showed almost no a↵ect.3 Figure
5(a) shows that in the coupled case, even large
changes in ck1 cause only a small shift in the
qualitative and quantitative behaviour. For c3
variation, a slightly greater a↵ect is seen where
CM RO2 reaches maximum value at a faster
rate, a result expected from eqs. 9 and 5. The
apparent half maximal CM RO2 in these cases,
is reduced for an increase ck1 and a decrease
in c3. In the uncoupled case this behaviour is
reversed, increasing ck1 to 0.05 changes cause
an increase of approximately %600 in the maximum value of CM RO2 . In contrast changes
3
8
The plots can be seen in the appendix section 5.
tary neural e↵ort mechanistically causes the
physiological hallmarks of functional activation. Therefore the main references for setting the value of u and its dynamic relations,
are that qualitatively these hallmarks can be
produced as model outputs. As seen from Fig-
in c3 cause minimal changes to the max values
of CM RO2 . Notably ck1 and c3 are opposite
in the direction of change they cause in max.
CM RO2 (as expected by similar arguments to
previous section) and c3 has an opposite e↵ect
on max CM RO2 in coupled (c) vs uncoupled
(d) mitochondria.
#HbO2
1.0
Functional
sponses
Activation
Re-
#HHb
0.5
ΜM
3.3
Due to its non-invasive and good spaciotemporal features, NIRS responses to functional activation have been of particular interest in recent years. Several groups have
adopted various protocols to particularly study
the changes in CCO signal. Protocols to
date include passive visual stimuli[26] passive
blob and interblob [50] and anagram solving
[47]. The di↵erences in protocols poses a difficulty in comparing results of these groups,
however there are some notable similarities as
drawn out by [47]. All the groups have shown
that redox changes in CCO are heterogeneous
amongst individuals, with instances of both increases and decreases in the signal in response
to functional activation. Tachtsidis et al.[47] in
particular have proposed a physiological basis
for this heterogeneity. Their conclusions are
based on the variety of physiological mechanisms which can a↵ect the redox state of CCO
and are in turn a↵ected by functional activation. A schematic diagram of these factors is
shown in Figure 6.
The protocol used for modelling functional activation in [4] has been to increase the demand parameter u stepwise from 1 to 1.2 for
10 seconds then return to 1. In the clinical setting of [47] functional activation was induced
via anagram solving of 4 and 7 letter words
for 1 minute each. Data from each individual
was included in the final results based on reporting ”a statistically significant increase in
HbO2 and corresponding decrease in HHb
signals.” [47, p. 8]. The use of the parameter u
to model this behaviour attempts to simulate
the physiological e↵ects of functional activation. Little is understood about how volun-
0.0
!0.5
0
5
10
15
Time !s"
20
25
30
Figure 7: Showing that the protocol for functional activation vis a vis the stepwise increase in u and then decrease, produces model
behaviour consistent with criteria for clinical
functional activation.
ure 6 increasing metabolic demand a↵ects the
mitochondrial control network through several
pathways. When trying to follow the e↵ects of
altering u through the equations of the model,
this complexity becomes apparent. For instance it is clear that u directly a↵ects the
NAD/NADH ratio via eq. 12, but how it
causes the increase in HbO2 is less apparent.
It is therefore important to check that this protocol for modelling functional activation does
indeed reproduce the important clinical test
of functional activation. Figure 7 shows the
modelled output for HbO2 and HHb with
normal parameter settings and confirms that
the protocol produces the required output.
Having shown that the modelled protocol
passes clinical criteria for functional activation, it can be used to investigate the behaviours of the four NIRS outputs oxCCO,
CM RO2 , TOI and CBF. This work is mainly
focused on the oxCCO signal shown in Figure 8 Figures showing TOI and CBF and
CM RO2 responses are in shown in the appendix section 4.3. It is interesting to note
that Figures 8(d) and (a) show the reduction
9
Figure 6: Schematic representation of the relations of metabolic control, image taken from [11]
in CCO oxidation level during functional activation as described by [47].
Being able to reproduce this observed behaviour by parameter modification is a successful result for the model. It suggests that,
as per its conception, the model may be able
to capture the di↵erences between individuals
and eventually produce a personalised output.
Whether these specific parameter changes are
responsible for the heterogeneity of the results
in [47], is unclear from the modelled outputs.
oxCCO showed the largest response to
changes in ck1 . The response of the maximum
value of the CCO signal during the functional
activation4 is plotted in Figure 9. This shows
a non linear change of over 150% with variation of ck1 through the full range of values.
Additionally the gradient is largest in the region of the current value for the parameter,
leading to a small change of 5% in the current
value of ck1 giving rise to approx. 1.5% change
in the observed oxCCO max. By comparison to the data of [47], the values of CCO
are in the range of what was observed in experimentation. c3, as seen in Figure8(c) also
produces a large, non-linear response in the
max value of oxCCO. It shows the largest
%change in oxCCO for parameter changes
within 5% of the value currently used in the
model,(up to a 10% in oxCCO). While this
may be interesting in terms of getting a range
of positive values; changes in c3 cannot induce
the reduction in oxCCO upon functional ac-
tivation seen in [47]. The direction of the responses in both ck1 and c3 are explained in the
same way as in the previous section. DN ADH
is the only other parameter which has the ability to cause a reduction in the oxCCO signal,
this occurs for a large increase of the parameter (DN ADH > 0.05). As can be seen from
eq.12, this parameter a↵ects the NAD/NADH
ratio, a change from 0.01, the normal value
of DN ADH , to 0.1 (at which point the reduction in oxCCO is seen) corresponds to
a fairly small change in the NAD/NADH ratio from 8.96 to 8.67. A change of this magnitude is easily plausible as variations in reported NAD/NADH ratios range from 5:1 to
20:1[29]. It could therefore be argued that the
DN ADH is the most plausible of the four candidate parameters to be the cause the decrease
in oxCCO. However, although this parameter a↵ects the ratio it does this via during
functional activation only and hence in concluding that it is the parameter responsible
for the physiological changes it assumes that
there is no variation amongst individuals in
the normal or simply pre-functional activation
NAD/NADH ratio. This assumption would
appear to be a more fundamental misrepresentation of the physiology as it is clear that many
factors a↵ect this ratio including glucose load
[29] which cannot be assumed the same for all
individuals.
4
The maximum value was take from time period of
between 11-16secs to exclude the sudden spike in the
signal at 10 seconds which is a numerical feature of the
stepwise change in u.
10
ck1 $0
ck1 $0.005
ck1 $0.008
ck1 $0.009
ck1 $0.01
ck1 $0.011
ck1 $0.012
ck1 $0.015
ck1 $0.05
ck1 $0.02
ck1 $0.1
"oxCCO ΜM
0.04
0.02
ck2 #0.01
0.05
ck2 #0.016
ck2 # 0.018
0.04
!oxCCO ΜM
0.06
0.00
ck2 #0.02
ck2 #0.022
ck2 #0.024
0.03
ck2 #0.3
ck2 #0.04
0.02
ck2 #0.2
!0.02
0.01
!0.04
0.00
0
5
10
15
20
25
30
0
5
10
15
Time!s"
20
25
30
Time!s"
(a)
(b)
c3#0
0.05
c3#0.044
DNADH !0.00
DNADH !0.005
DNADH ! 0.008
DNADH !0.009
DNADH !0.01
DNADH !0.011
DNADH !0.012
DNADH !0.015
DNADH !0.02
DNADH !0.05
DNADH !0.1
c3#0.055
c3#0.066
0.04
0.04
c3#0.077
c3#0.099
"oxCCO ΜM
!oxCCO ΜM
c3#0.088
0.03
c3#0.11
c3#0.117
0.02
0.02
0.00
0.01
0.00
!0.02
0
5
10
15
20
25
30
0
Time!s"
5
10
15
20
25
30
Time!s"
(c)
(d)
Figure 8: Showing the e↵ect of changing (a) ck1 , (b) ck2 , (c) c3 and (d) DN ADH on oxCCO
during functional activation. In all graphs functional activation was modelled b varying the
parameter u from 1 to 1.2 for a 10 second time period. In each Graph the black line represents
the value currently used in the model.
4
Conclusions
Work
and
Further
Sensitivity analysis of the model parameters
ck1 ,ck2 , DN ADH and c3 has shown that alteration of parameters does not a↵ect crucial
autoregualtion behaviour. % CuAr changes
in response to O2 variation are in line with
behaviour expected from the equations of the
model. CM RO2 behaviour in uncoupled mitochondria is drastically a↵ect by increases in
ck1 and in coupled mitochondria half maximal
CM RO2 concentration are decreased by an increase in ck1 and a decrease in c3. Changes
to ck1 and DN ADH are able to simulate the
reduction in oxCCO signal on functional activation as noted in [47]. c3 can only a↵ect
the level of positive oxCCO reaction to functional activation and ck2 shows minimal a↵ect
on all modelled outputs.
Continuatuion of sensitivity analysis of all
BRAINCIRC model parameters it essential for
the models progression towards a clinical aid.
Use of real data and optimisation of parameters to individuals will also make headway towards achieveing personalised model. In addition it would be of interest to further explore
the e↵ects of functional activation via a more
extensive sensitivity analysis. The use of real
functional activation data within the model,
would enable the exploration of the CM RO2
signal and might provide interesting insight
into the physiological basis of u and to understand whether the experimental protocol four
functional activation could be improved.
11
! Change in Max "oxCCO
" Change in #oxCCO Max
10
0
!50
!100
!150
0.00
8
6
4
2
0
0.02
0.04
0.06
ck1 Value
0.08
0.10
0.05
(a)
0.10
ck2 Value
0.15
0.20
(b)
0
" change in max #oxCCO
" change in max #oxCCO
0
!20
!40
!60
!80
!100
0.00
0.02
0.04
0.06
c3 Value
0.08
0.10
!20
!40
!60
!80
!100
!120
0.00
0.12
(c)
0.02
0.04
0.06
DNADH Value
0.08
0.10
(d)
Figure 9: Showing how the CCO value reached at plateau changes with various parameter
changes (a)The %change in CCO with varying ck1 .(b) The % change in max CCO with varying
ck2 . (c) The % change in max CCO with changes in c3. (d) The % change in max CCO with
varying DN ADH . All figure show marked in red the normal value of the parameter.
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15
Appendix and Supplementary Material from [4]
Claire Walsh
January 18, 2012
1
Model Equations
dvx
dt
d CuAo
dt
d a3r
dt
d [H]
dt
d
dt
d [O2 ]
dt
=
1
(x
⌧x
vx ),
=
4(f2
f1 )
(2)
=
4(f2
f3 )
(3)
=
( p1 f 1
=
p 1 f 1 + p 2 f 2 + p3 f 3
Cim
=
JO2 /Volmit
L = LCV,max
x = Pa , O2 , PaCO2, u
p2 f 2
✓
(1)
p3 f3 + L)/VolHi
(4)
L
(5)
f3
(6)
◆
1 exp[ kCV ( p
1 + rCV exp[ kCV ( p
pCV 0 + Z ln(u))]
pCV 0 + Z ln(u))]
+ kunc Llk0 (exp(klk2 p)
1)
(7)
1000 Cbuffi Volmit dpH
[H] (1 10 dpH )
= min{DO2 ([O2,c ] [O2 ]), CBF[HbO2,a ]} with smooth approximation
VolHi =
JO2
(8)
(9)
p
(x + c)2 + ✏2 (x + c)
where
2
c = CBF[HbO2,a ], x = DO2 ([O2,c ] [O2 ]), ✏ = CBFn [HbO2,a,n ]/10
JO2 = c
CBF = KG (Pa
✓
[O2,c ] =
r =
h =
(
q
Pv )r
(11)
2SaO2 JO2 /(CBF [Hbtot])
2 + JO2 /(CBF [Hbtot])
e0 (exp(K
(r
r0 )/r0 )
1)
◆
1
nh
coll )h
(12)
+ Tmax0 (1 + kaut µ) exp
(Pa + Pv )/2
r2 + (r0 + h0 )2
r02
Pic
⇣
r rm
r t rm
nm ⌘
r
(13)
(14)
e⌘
µmin + µmax
1 + e⌘
!
v Pa
⌘ = RP
1 + RO
vPa ,n
µ =
(10)
4
(15)
!
v O2
vO2 ,n
1 + RC 1
1
vPaCO2
vPaCO2,n
!
+ Ru 1
vu
vu,n
!
. (16)
In the case of the simplified model Equations 1, 6 and 9-16 are omitted, mitochondrial oxygen [O2 ]
is a controllable parameter, and f1 (Equation 7) takes the form
f1 = k1,n
UQH2
exp( ck1 ( p
UQH2,n

pn )) CuAo
10
CuAr
(p1 p/4 E1 (UQH2 ))/Z
.
(17)
Note that several of the equations are implicit or need to be solved simultaneously. Apart from CBF
above, important model output variables (or observables), which can potentially be used to compare
model behaviour to measured quantities in vivo are
SvO2 = SaO2
JO2
CBF[Hbtot]
(18)
CMRO2 = Volmit f3
AVRn(r/rn )2 SaO2 + SvO2
TOS =
(AVRn(r/rn )2 + 1)
Hbt =
1000
4
HbO2 =
1000
4
HHb = Hbt
Volart,n
Volart,n
HbO2
✓
✓
r
rn
r
rn
◆2
(20)
!
+ Volven [Hbtot]Volblood,n
!
Hbtn
SaO2 + Volven SvO2 [Hbtot]Volblood,n
(21)
HbO2n
HHbn
oxCCO = 1000 Volmit (CuAo
2
◆2
(19)
(22)
(23)
CuAo,n ) .
(24)
Glossary of model variables
Where concentrations are given units of mM without further characterisation, the reference volume is
that of the compartment in which the quantity resides. Where there is ambiguity the volume in question
is made explicit. In the case of NIRS quantities, unit conversions are carried out to give concentrations
in micromoles per unit tissue volume. Initialisation is only needed for di↵erential variables.
Where concentrations are given units of mM without further characterisation, the reference volume
is that of the compartment in which the quantity resides. Where there is ambiguity the volume
in question is made explicit. In the case of NIRS quantities, unit conversions are carried out to
give concentrations in micromoles per unit tissue volume. Initialisation is only needed for di↵erential
variables.
variable
brief description
Di↵erential variables
CuAo
oxidised CuA concentration
a3r
reduced cyt a3 concentration
[H]
mitochondrial proton concentration
mitochondrial inner membrane potential
[O2 ]
mitochondrial oxygen concentration
v Pa
first-order filtered Pa
v O2
first-order filtered [O2 ]
vPaCO2
first-order filtered PaCO2
units
initialisation
mM
mM
mM
mV
CuAo,n
a3r,n
1000(10
mM
mmHg
mM
mmHg
[O2,n ]
Pa,n
[O2,n ]
PaCO2n
pHm,n )
n
ctd...
2
variable
vu
General
CMRO2
brief description
first order filtered u
units
none
cerebral metabolic rate of oxygen consumption
JO2
rate of oxygen flux
TOS
absolute tissue oxygen saturation
oxCCO changes in tissue concentration of oxidised CuA
HHb
changes in tissue concentration of deoxyhaemoglobin
HbO2
changes in tissue concentration of oxyhaemoglobin
Hbt
changes in tissue concentration of total
haemoglobin
cm
–
Mitochondria
CuAr
reduced CuA concentration
a3o
oxidised cyt a3 concentration
p
proton motive force
pHm
mitochondrial pH
Volart
arterial volume as a fraction of normal
total blood volume
VolHi
e↵ective fractional mitochondrial volume for protons
f1
rate of reaction 1
f2
rate of reaction 2
f3
rate of reaction 3
G1
free energy associated with reaction 1
G2
free energy associated with reaction 2
k1
forward rate constant for reaction 1
k 1
backward rate constant for reaction 1
Keq1
equilibrium constant for reaction 1
k2
forward rate constant for reaction 2
k 2
backward rate constant for reaction 2
Keq2
equilibrium constant for reaction 2
L
rate of proton re-entry into mitochondrial matrix
LCV
rate of proton re-entry via ATP synthase and associated processes
Llk
rate of proton re-entry via leak channels
initialisation
1
mmol (l tissue)
1
s
1
–
mmol (l tissue)
none
µmol (l tissue)
1
s
1
–
–
–
1
µmol (l tissue)
1
–
µmol (l tissue)
1
–
µmol (l tissue)
1
–
mM
mM
mV
pH units
none
–
–
–
–
–
none
–
mM s 1
mM s 1
mM s 1
mV
mV
s 1
s 1
none
mM 1 s
mM 1 s
none
mM s 1
–
–
–
–
–
–
–
–
–
–
–
–
1
1
mM s
1
–
mM s
1
–
variable table ends.
3
3
Parameter setting
Many parameters in the model are set with reference to other parameters, or in order to ensure correct
“normal” behaviour. Others are directly given values. Throughout, the subscript n indicates a normal
value of some variable or control parameter.
3.1
A table of model parameters with numerical values
Where no units are given, this is because the parameter in question is a dimensionless quantity.
parameter brief description
value and units
Blood chemistry, blood flow regulation and volume
PaCO2n
normal arterial partial pressure of CO2 40 mmHg
PaCO2
arterial partial pressure of CO2
40 mmHg
SaO2n
normal saturation of the arterial 0.96
haemoglobin
SaO2
saturation of arterial haemoglobin
0.96
[Hbtot]
Total haemoglobin concentration in the 9.1 mM
arteries and veins.
[Hbtotn ]
Normal total haemoglobin concentra- 9.1 mM
tion in the arteries and veins.
Pa
arterial blood pressure
100 mmHg
Pa,n
Normal value of ABP
100 mmHg
Pv
venous blood pressure
4 mmHg
Pv,n
normal venous blood pressure
4 mmHg
Pic
intracranial blood pressure
9.5 mmHg
RC
sensitivity of ⌘ to PaCO2
2.2
RO
sensitivity of ⌘ to [O2 ]
1.5
RP
sensitivity of ⌘ to arterial pressure
4.0
Ru
parameter controlling sensitivity of ⌘ to 0.5
u
⌧P a
the time constant associated with vp
5s
⌧CO2
the time constant associated with vc
5s
⌧O 2
the time constant associated with vo
20 s
⌧u
the time constant associated with vu
0.5 s
kaut
control parameter allowing destruction 1
of autoregulation
parameter in the pressure-elastic ten- 62.79 mmHg
coll
sion relationship
parameter in relationship determining 0.1425 mmHg
e0
references
[51]
[51]a
b
a,b
[2]a,b
[2]b
[51]a
[51]b
[51]a
[51]b
[51]b
c
d,f
e
d,f
j
j
j
j
a
[51]
[51]
e
K
nm
rm
parameter controlling sensitivity of e
to radius
exponent in the muscular tension relationship
value of vessel radius giving maximum
muscular tension
10 mmHg
[51]
1.83
[51]
0.027 cm
[51]
ctd...
4
parameter
rt
h0
r0
rn
CBFn
brief description
parameter in the muscular tension relationship
vascular wall thickness when radius is
r0
radius in the elastic tension relationship
normal radius of blood vessels
normal cerebral blood flow
AVRn
The normal arterio-venous volume ratio
Volblood,n
normal blood volume as a fraction of
brain volume
Oxygen transport and consumption
CMRO2,n
normal CMRO2
[O2,n ]
normal oxygen concentration in mitochondria
value of [O2 ] at half maximal saturation
nh
hill coefficient for haemoglobin saturation
Mitochondria
Cim
Capacitance of mitochondrial inner
membrane
DNADH
a parameter controlling the e↵ect of activation on glycolytic/TCA cycle flux
Llk,f rac
normal fraction of proton entry into mitochondria which is via leak channels
Volmit
mitochondrial volume fraction
Z
standard physico-chemical constant
(2.303RT /F )
normal mitochondrial inner membrane
n
potential
E0 (NADH) NADH standard redox potential
E0 (UQ)
ubiquinone standard redox potential
E0 (CuA )
CuA standard redox potential
E0 (cyt a3 )
cyt a3 standard redox potential
CuAfrac,n
normal oxidised fraction of CuA
Nt
total mitochondrial NAD pool
Nrat,n
normal mitochondrial NAD/NADH ratio
Ut
total mitochondrial ubiquinone pool
Urat
UQ/UQH2 ratio
Urat,n
normal UQ/UQH2 ratio
value and units
0.018 cm
references
[51]
0.003 cm
[51]
0.0126 cm
h
0.0187 cm
0.01075 ml blood
(ml tissue) 1 s 1
0.333
h
0.04
[6]f
0.034 mmol
(l tissue) 1 s
0.024 mM
[43]f
b,f
a,f
1
[8, 1]b
[28]f
[28]
0.036 mM
2.5
0.00675 mmol
(l mito) 1 mV
0.01
[56]
1
d,f
0.25
[12]f
0.067
59.028 mV
[31]
[32]
145 mV
[38]
-320 mV
60 mV
247 mV
350 mV
0.8
3 mM
9
[38]
[38]
[37]f
[22, 39] f
[17]f
[31, 56]
[29, 57, 44]f
1.35 mM
1
1
[31, 56]
a,f
f
ctd...
5
parameter
cytoxtis
ptot
brief description
concentration of cytochrome c oxidase
in tissue
extra-mitochondrial pH
normal extra-mitochondrial pH
normal mitochondrial pH
constant in the pH bu↵ering relationship
bu↵ering capacity for protons in mitochondria
A constant in the rate of complex V
parameter controlling the ratio of maximal to minimal rates of oxidative phosphorylation
normal complex V flux as a fraction of
maximum possible flux
value of p at which reaction 3 is maximally sensitive to p
parameter controlling sensitivity of k1
to PMF
parameter controlling sensitivity of k2
to PMF
parameter controlling the sensitivity of
reaction 3 to PMF
apparent second-order rate constant for
reaction 3 at p = 0
constant controlling rate Llk
parameter representing the action of
uncouplers
protons pumped by chain
p1
protons pumped by complexes I-III
p2
protons pumped by complex IV during
oxidative phase
protons pumped by complex IV during
reductive phase
pHo
pHo,n
pHm,n
dpH
Cbuffi
pCV 0
rCV
LCV,0
p30
ck1
ck2
c3
k3,0
klk2
kunc
p3
value and units
0.0055 mmol
(l tissue) 1
7.0
7.0
7.4
0.001
references
[15]
0.022 M H+ /pH
[32]
90 mV
5
f,g
0.4
f,g
143.61 mV
f,g
0.01
g
0.02
[42]f,g
0.11
g
f,g
2.5 ⇥ 105 mM
0.038 mV
1
1s 1
1
20 (12 in
model)
12 (4 in
model)
4
4
[31]a
[31]
[32]f
[31]
[13]
[31]
a
simplified
[12, 9]f
simplified
[12, 9]f
[12, 9]f
[12, 9]f
parameter table ends.
a Control
parameter. See discussion below.
value with wide range in normal variation.
c Fit to data from [45].
d Set heuristically: The parameter is needed in the model, but we have not yet gathered and analysed
sufficient data to set it accurately.
e Set from data in [24, 25] as described in the text.
f Further discussion in the text.
g The parameters
p30 , c3, ck1 , ck2 , LCV,0 , rCV and pCV 0 have collectively been set to give the
b Typical
6
behaviour shown in Figures 9 and 10 in the main text, and to give the qualitative behaviour during
functional activation in [53] and [33]. They have not been optimised to fit any particular data-set.
h The parameters r , r and R are set following the methodology described in Section E below.
n
0
P
j Accurate setting of the time constants requires data with good time resolution gathered from studies in
the contexts of hypoxia, hypercapnia, changes in mean arterial blood pressure and functional activation.
Currently these are set heuristically.
3.2
Oxygen transport and consumption
Directly set parameters
CMRO2,n = 0.034 mM s 1 . This value can be calculated from the value of 3.4 mL/100 g brain/min
quoted by a number of authors, e.g. [43].
[O2,n ] = 0.024 mM. Physiological mitochondrial O2 concentrations of 20 30 µM are quoted in [8].
[1] uses a value of 0.0262 mM for “intracellular oxygen”, but the context is mitochondrial.
Hbtot, Hbtotn = 9.1 mM. The value of 2.27 mM Hb used in the BRAINCIRC model equates to
a blood haemoglobin concentration of about 154 g/L, consistent with the normal physiological range.
Multiplying by 4 gives the number of O2 binding sites.
Calculated parameters
The quantities
and nh in the relationship between dissolved oxygen and haemoglobin bound
oxygen are calculated as follows: Traditionally the shape of saturation curves is represented by Hill
equations of the form
[O2 ]nh
SO2 = n
h + [O ]nh
2
where SO2 is oxygen saturation. This solves to give
[O2 ] =
✓
SO2
1 SO2
◆
1
nh
.
A value of nh = 2.5 and half maximal oxygen partial pressure of 26 mmHg is found to fit the data well
in [28]. Assuming the solubility of O2 in blood is 0.0014 mM mmHg 1 gives = 0.036 mM.
The quantity DO2 (in s 1 ) is chosen to given normal oxygen supply at normal blood and mitochondrial oxygen concentrations, i.e.
JO2,n
DO2 =
.
(25)
([O2,c,n ] [O2,n ])
JO2,n is set to CMRO2,n , the normal value of CMRO2 . In order to calculate [O2,c,n ] we first need to
calculate normal venous oxyhaemoglobin from normal arterial oxyhaemoglobin and delivery from the
conservation equation
JO2,n
SvO2n = SaO2n
.
CBFn [Hbtotn ]
Taking “typical” capillary haemoglobin to be
ScO2n = (SaO2n + SvO2n )/2,
we can calculate typical capillary dissolved oxygen concentration from the dissociation curve
[O2,c,n ] =
✓
ScO2n
1 ScO2n
7
◆
1
nh
.
3.3
Mitochondria
Set parameters: general
Volmit = 0.067, and Z = 59.028 mV are taken from [31]. It should be noted that the volume fraction
of mitochondria is likely to show variation between tissues and individuals.
ptot = 20 (or 12). This is the total number of protons pumped during the passage of four electrons
from an initial reducing substrate to oxygen. The value of ptot depends on which reducing substrate is
used. Where the reducing substrate is NADH, ptot is taken as 20 (i.e. 10 protons per NADH molecule).
When the reducing substrate is succinate and the simplified model is being run, this number is decreased
to 12 [12, 31].
p1 = 12 (or 4), p2 + p3 = 8, following [12], and close to the values in [31]. [9] suggests that equal
numbers of charges are transferred during the oxidative and reductive phases of the cytochrome-coxidase reaction, giving p2 = 4 and p3 = 4. When the reducing substrate is succinate p1 becomes
4.
cytoxtis = 0.0055 mmol (l tissue) 1 [15]. Given the large changes in tissue cytochrome-c-oxidase
content during development, it is possible that there is some physiological variation in this quantity,
giving rise to quantitatively di↵erent oxCCO signals in di↵erent individuals.
CuAfrac,n = 0.8. The value of 0.82 is given for adult rat brain in [17]. The lower value of 0.673 for
piglet is also quoted in [46]. In both cases there is a large standard deviation. The value of 0.8 is also
consistent with the in vitro data for cytochrome c in [54], on the assumption that the redox states of
cytochrome c and CuA are close. In any case, we expect the CuA centre not to be fully oxidised in
normal circumstances.
n = 145 mV consistent with values in [38].
pHn = pHm,n pHo = 0.4 [35]. pHo = 7. This value of normal pH outside the mitochondria
is used in [31] and the BRAINCIRC model. In the reduced model, pHo can be seen as a control
parameter. In the full model, various stimuli such as changes in PaCO2, or hypoxia, should be able to
influence pHo . Currently these pathways are omitted.
dpH = 0.001 and Cbuffi = 0.022 M H+ /pH unit, are taken from [31].
Cim = 6.7500 ⇥ 10 3 mmol (l mito) 1 mV 1 [56]. This parameter has an important influence on
how a stimulus-induced change in p (e.g. via activation) translates into changes in pH and
.
Calculated parameters: general
pHm,n = pHo + pHn = 7.4. Normal intramitochondrial pH is chosen to be 0.4 pH units greater
than extramitochondrial pH. This is close to values obtained in simulations of [32] at normal parameter
values and consistent with the pH value in [35].
Normal PMF: pn =
n + Z pHn .
The total concentration of cytochrome-c-oxidase and hence CuA in mitochondria, cytoxtot is set
from the total concentration in tissue cytoxtis :
cytoxtot = cytoxtis /Volmit .
The normal oxidised fraction of CuA , i.e. CuAfrac,n is used to calculate CuAo,n and CuAr,n : via
CuAo,n = CuAfrac,n cytoxtot ,
CuAr,n = cytoxtot
CuAo,n .
Set parameters: reactions 1, 2 and 3
In this section E and E0 refer to reduction potentials and standard reduction potentials of half
reactions respectively.
E0 (NADH) = 320 mV from [38].
Nt = 3 mM. This value for the total mitochondrial NAD pool is taken from [31].
8
Nrat,n = 9. The normal NAD/NADH ratio is chosen to 9:1, i.e. mitochondrial NAD is assumed to
be 10 percent reduced in normal circumstances. A wide range of values from about 5:1 to about 20:1
can be found in the literature, e.g. [29, 57, 44].Our value is within this range.
DNADH = 0.01 mV. The extent to which demand activates glycolysis and the TCA cycle, and thus
a↵ects the NAD/NADH ratio, is obviously important. For example [36] suggests that an important
e↵ect of activation is glycolytic. The current value of DNADH is chosen heuristically so that changes in
demand have a small e↵ect on the redox state of NADH. Increase in this parameter from 0.01 to 0.1
causes reduction of all modelled elements of the chain during functional activation.
E0 (UQ) = 60 mV is the value for ubiquinone in [38].
Ut = 1.35 mM. This value for the total mitochondrial ubiquinone pool is taken from [31].
Urat = 1. This parameter is only needed for the simplified model and can be defined as the
UQ/UQH2 , assumed to be determined by the experimental context.
Urat,n = 1. This parameter is only needed for the simplified model to calculate a normal equilibrium
constant for reaction 1 and can be defined as the UQ/UQH2 ratio such that at normal values of
membrane potential and oxygen, we get flux fn through the chain (in contexts where mitochondria are
fed on succinate).
Simulations of [32] suggest that in normal circumstances the UQ/UQH2 ratio is considerably less
than the NAD/NADH ratio. When the value of supply (via parameter kDH in that model) is adjusted so
that NAD/NADH ratio becomes 9, the UQ/UQH2 ratio is approximately 0.7 (at this value cytochrome
c is about 85 percent oxidised, also consistent with our assumptions). It is not easy to derive from
experimental work with succinate-fed mitochondria such as [54] accurate values of the UQ/UQH2 ratio:
However given that cytochrome c redox states in [54] are close to the CuA states observed in vivo, and
hence presumably cytochrome c states in vivo, it is reasonable to assume that the UQ/UQH2 ratio
might also be similar. Further, simulations suggest that the model behaviour is insensitive to the
precise value of Urat and Urat,n – a 10 percent change in either causes no more than a 1 percent change
in baseline flux.
E0 (CuA ) = 247 mV. This value is chosen to be somewhat less positive than the cyt c potential [37].
E0 (cyt a3 ) = 350 mV, similar to the values in [22, 39]. The higher value in [31] is needed because
in this model all proton pumping occurs prior to reduction of cyt a3 . The cyt a3 redox potential is
assumed to be independent of the reduction state of CuA, and vice versa, which is an oversimplification
given the variety of redox interactions in the enzyme [39, 10].
k3,0 = 2.5 ⇥ 105 mM 1 s 1 is taken from [13]. As shown below this can be used to calculate a value
of a3frac,n ' 0.99.
The seven parameters ck1 , ck2 , c3, p30 , LCV,0 , rCV and pCV 0 between them control how the
rates of reactions 1,2, and 3, and ATP synthase respond to changes in p . As they are specific to
the structure of the model, they are not easily derivable from the literature. However they influence
model behaviour during any process which a↵ects p. The values chosen combine to give the behaviour
described in the sections on functional activation and hypoxia.
ck1 = 0.01 is chosen heuristically. By setting this to be low, the e↵ect of uncoupling is primarily
on the reverse rate of electron transfer in reaction 1. It is possible that this parameter should change
value depending on whether the full model or the simplified model is simulated.
ck2 = 0.02 is chosen heuristically, but again in a range where the e↵ect of uncoupling is stronger on
the reverse rate of electron transfer in reaction 2, following the simple model presented in [42].
c3 = 0.11. This parameter controls the maximum sensitivity of f3 to changes in p and its value
is set heuristically.
p30 = pn 25 mV. Simulations suggest that in order to get observed behaviour the rate f3 should
be sensitive to changes in p at normal membrane potentials. The point of maximum sensitivity is
chosen to be somewhat lower than normal PMF.
9
Calculated parameters: reactions 1, 2 and 3
Normal reaction rates are set from CMRO2,n by defining
f1,n = f2,n = f3,n = fn ⌘ CMRO2,n /Volmit .
The quantity E1 is set in di↵erent ways depending on the situation we are trying to model. In
general
E1 = E0 (CuA ) E(R)
where R is some reducing substrate. For the in vivo situation where the reducing substrate is primarily
NADH, this becomes
E1 = E0 (CuA ) E0 (NADH) + CNADH
where CNADH = Z/2 log10 (NADH/NAD). Defining Nrat = NAD/NADH with normal value Nrat,n , we
allow demand to influence the NAD redox state by setting
Nrat =
Nrat,n
2D
u NADH
.
The parameter DNADH controls the sensitivity of the NAD redox state to changes in demand. From
Nt , Nrat,n and Nrat we get the NADH concentrations NADHn = Nt /(1 + Nrat,n ), and NADH = Nt /(1 +
Nrat ).
For readability, the full forms of E1 for reducing substrates NADH and UQH2 are:
E1 (NADH) = E0 (CuA )
E0 (NADH) +
E1 (UQH2 ) = E0 (CuA )
✓
Z
Nrat,n
log10
2D
2
u NADH
E0 (UQH2 ) +
◆
Z
log10 (Urat ) .
2
The normal value of E1 is
E1,n = E0 (CuA )
E(Rn ),
where E(Rn ) is the reduction potential at normal concentration of R. So when the reducing substrate
is NADH,
E1,n = E0 (CuA )
E0 (NADH) +
Z
log10 (1/Nrat,n ),
2
and similarly for succinate as the reducing substrate
E1,n = E0 (CuA )
E0 (UQH2 ) +
Z
log10 (1/Urat,n ) .
2
From the normal values of E1 and p, it is possible to calculate a normal equilibrium constant for
reaction 1, and hence normal value of k1 :
Keq1,n = 10
1/Z(p1 pn /4 E1,n )
,
k1,n = fn /(CuAo,n
CuAr,n /Keq1,n ) .
When an in vivo situation is modelled with NADH as the main reducing substrate, we choose
k1,0 = k1,n NADH/NADHn where k1,n is the value of k1 at normal p and NADH levels. A similar
methodology is applied when the reducing substrate is succinate.
E2 is set as:
E2 = E0 (cyt a3 ) E0 (CuA ) .
10
From the values of E2 and normal value of p, it is possible to calculate a normal equilibrium constant
for reaction 2, and hence normal value of k2 :
Keq2,n = 10
1/Z(p2 pn /4 E2 )
,
k2,n = fn /(CuAr,n a3o,n
CuAo,n a3r,n /Keq2,n ) .
k3 is calculated from measured values of k3,0 , i.e.
k3 =
k3,0 (1 + exp(c3 p30 ))
.
exp(c3 p30 )
a3r,n is calculated by inverting the definition of f3 at normal values of all quantities:
a3r,n =
fn (1 + exp[ c3( pn
k3 [O2,n ] exp[ c3( pn
p30 )])
.
p30 )]
This gives rise to a normal oxidised fraction of cyt a3 as a3frac,n = (cytoxtot a3r,n )/cytoxtot ' 0.99.
Set parameters: Complex V and proton leak
Three parameters determine the shape of response of the ATPase to changes in PMF: LCV,0 , pCV 0
and rCV . The functional form of LCV is currently chosen to match qualitatively the function in [20].
As the function used in [20] is both complex and an approximation to a previous approximation in [35],
rather than attempting to use it directly we chose instead a simple class of increasing functions which
saturates at both large and small values of p. As mentioned above, the parameter values are chosen
to be consistent with the behaviour described in the sections on functional activation and hypoxia, but
are not necessarily the unique parameter values giving reasonable model behaviour.
LCV,0 = 0.4. This is the normal rate of ADP phosphorylation as a proportion of maximal phosphorylation rate – combined with pCV 0 and rCV it determines the maximum slope of the flux- p
relationship for the ATPase. It is set so that at normal values p exerts some control over the rate of
oxidative phosphorylation.
pCV 0 = 90 mV and rCV = 5. [35] ignores the possibility of reversibility of the enzyme (this
e↵ectively sets rCV ! 1). Here rCV is given a finite value, following [20] (from which a value of
rCV ⇡ 3 can be inferred). The graphs in [20] can also be used to derive a somewhat higher value
of pCV 0 than used here. Investigating the e↵ect of the shape of the flux- p relationship on the
behaviour of the model is an important task for the future.
Llk,f rac = 0.25. [27] has the very high value of 0.85 for synaptosomes. The value chosen here lies
in the range given for most tissues in [12]. However the e↵ect of this parameter on model behaviour is
clearly important for future exploration.
klk2 = 0.038 mV 1 , taken from [31].
Calculated parameters: Complex V and proton leak
The normal value of L, the inward proton current through the membrane, is set to
Ln = ptot CMRO2,n /Volmit ,
since ptot is precisely the number of protons pumped during reduction of one molecule of oxygen. A
quantity Llk,f rac is defined as the normal fraction of L passing through leak channels giving LCV,f rac =
1 Llk,f rac as the normal fraction passing through Complex V. This gives normal value of Llk and
LCV :
Llk,n = Llk,f rac Ln ,
LCV,n = LCV,f rac Ln .
The constant Llk0 is calculated from normal proton motive force
Llk0 =
Llk,n
exp(klk2 pn )
11
1
.
pn and Llk,n :
The quantity kCV is set by defining the control parameter LCV,0 (see above):
1 exp( kCV ( pn
1 + rCV exp( kCV ( pn
LCV,0 ⌘
pCV 0 ))
,
pCV 0 ))
(normal u is 1 has been assumed) and inverting to give:
1
kCV =
pn
pCV 0
1 LCV,0
ln
1 + rCV LCV,0
!
.
The constant LCV,max is calculated as
LCV,n
.
LCV,0
LCV,max =
4
Responses of TOI, CM RO2 and CBF during functional activation
DNADH !0.00
70.8
DNADH !0.005
ck1 !0.00
ck1 !0.005
ck1 !0.008
ck1 !0.009
ck1 !0.01
ck1 !0.011
ck1 !0.012
ck1 !0.015
ck1 !0.02
ck1 !0.05
ck1 !0.1
70.8
DNADH ! 0.008
DNADH !0.009
70.6
70.6
DNADH !0.01
DNADH !0.011
DNADH !0.012
70.4
70.4
TOI !
TOI !
DNADH !0.015
DNADH !0.02
70.2
DNADH !0.05
70.2
DNADH !0.1
70.0
70.0
69.8
69.8
69.6
69.6
0
5
10
15
20
25
30
0
5
10
Time!s"
15
20
(a)
30
(b)
70.8
ck2 ! 0.02
c3!0.099
c3!0.088
c3! 0.077
c3!0.066
c3!0.055
c3!0.044
c3!0
c3!0.11
c3!0.117
70.8
ck2 ! 0.04
70.6
ck2 ! 0.2
70.6
ck2 ! 2
70.4
ck2 ! 20
TOI !
70.4
TOI !
25
Time!s"
70.2
70.2
70.0
70.0
69.8
69.8
69.6
0
5
10
15
20
25
30
0
Time !s"
5
10
15
20
25
30
Time!s"
(c)
(d)
Figure 1: Showing the e↵ect of changing parameters on TOI signal during functional activation. In
all graphs functional activation was modelled b varying the parameter u from 1 to 1.2 for a 10 second
time period. In each case the black line represent current parameter value. In alphabetical order the
figures show The change in oxCCO with ?? DN ADH, (b), ck1 , (c) ck2 and (d) c3.
12
0.0355
DNADH !0.00
ck1 !0.00
ck1 !0.005
ck1 !0.008
ck1 !0.009
ck1 !0.01
ck1 !0.011
ck1 !0.012
ck1 !0.015
ck1 !0.02
ck1 !0.05
ck1 !0.1
DNADH !0.005
DNADH ! 0.008
DNADH !0.01
CMRO2 !Μmol g-1 min-1"
CMRO2 !Μmol g-1 min-1"
DNADH !0.009
0.0350
DNADH !0.011
DNADH !0.012
DNADH !0.015
DNADH !0.02
DNADH !0.05
0.0345
DNADH !0.1
0.0350
0.0345
0.0340
0.0340
0
5
10
15
20
25
0
30
5
10
15
(a)
25
30
(b)
ck2 "0.01
ck2 "0.016
ck2 " 0.018
ck2 "0.02
ck2 "0.022
ck2 "0.024
ck2 "0.3
ck2 "0.04
ck2 "0.2
0.0350
0.0348
0.0346
c3!0.099
c3!0.088
c3! 0.077
c3!0.066
c3!0.055
c3!0.044
c3!0
c3!0.11
c3!0.117
0.0352
0.0350
CMRO2 !Μmol g-1 min-1"
0.0352
CMRO2 !Μmol g-1 min-1"
20
Time !s"
Time!s"
0.0344
0.0342
0.0348
0.0346
0.0344
0.0342
0.0340
0.0340
0
5
10
15
20
25
30
0
5
10
15
Time!s"
Time!s"
(c)
(d)
20
25
30
Figure 2: Showing the e↵ect of changing parameters on CM RO2 signal during functional activation. In
all graphs functional activation was modelled b varying the parameter u from 1 to 1.2 for a 10 second
time period. In each case the black line represent current parameter value. In alphabetical order the
figures show The change in oxCCO with (a) DN ADH, (b), ck1 , (c) ck2 and (d) c3.
1.06
1.06
DNADH !0.00
DNADH !0.005
DNADH ! 0.008
DNADH !0.009
DNADH !0.01
DNADH !0.011
DNADH !0.012
DNADH !0.015
DNADH !0.02
DNADH !0.05
DNADH !0.1
CBF !normalised"
1.04
1.03
ck1 !0.00
ck1 !0.005
ck1 !0.008
ck1 !0.009
ck1 !0.01
ck1 !0.011
ck1 !0.012
ck1 ! 20
ck1 !0.015
ck1 !0.02
ck1 !0.05
1.05
1.04
CBF !normalised"
1.05
1.02
1.03
1.02
1.01
1.01
1.00
1.00
0
5
10
15
20
25
0
30
5
10
15
20
25
30
Time!s"
Time!s"
(a)
(b)
1.06
c3!0.099
c3!0.088
c3! 0.077
c3!0.066
c3!0.055
c3!0.044
c3!0
c3!0.11
c3!0.117
1.06
1.04
1.03
1.04
CBF !normalised"
1.05
CBF !normalised"
1.05
ck2 !0.01
ck2 !0.016
ck2 ! 0.018
ck2 !0.02
ck2 !0.022
ck2 !0.024
ck2 !0.3
ck2 !0.04
ck2 !0.2
1.03
1.02
1.02
1.01
1.01
1.00
1.00
0
5
10
15
20
25
0
30
5
10
15
20
25
30
Time!s"
Time!s"
(c)
(d)
Figure 3: Showing the e↵ect of changing parameters on CBF signal during functional activation. In
all graphs functional activation was modelled b varying the parameter u from 1 to 1.2 for a 10 second
time period. In each case the black line represent current parameter value. In alphabetical order the
figures show The change in oxCCO with (a) DN ADH , (b), ck1 , (c) ck2 and (d) c3.
13
5
CM RO2 responses for coupled and uncoupled mitochondria
0.8
5
4
CMRO2 !a.u."
CMRO2 !a.u."
0.6
DNADH !0.0
0.4
DNADH !0.005
DNADH ! 0.009
3
DNADH !0.0
DNADH !0.005
DNADH ! 0.009
DNADH !0.01
DNADH !0.011
DNADH !0.012
DNADH !0.015
DNADH !0.02
DNADH !0.1
2
DNADH !0.01
DNADH !0.011
DNADH !0.012
0.2
DNADH !0.015
1
DNADH !0.02
DNADH !0.1
0
0.0
0
5
10
0
15
5
10
15
O2 ΜM
O2 ΜM
(a)
(b)
5
0.8
4
CMRO2 !a.u."
CMRO2 !a.u."
0.6
ck2 !0.0
ck2 !0.01
ck2 !0.016
ck2 ! 0.018
ck2 !0.02
ck2 !0.022
ck2 !0.024
ck2 !0.3
ck2 !0.04
0.4
0.2
3
ck2 !0.0
ck2 !0.01
ck2 !0.016
2
ck2 ! 0.018
ck2 !0.02
ck2 !0.022
ck2 !0.024
1
ck2 !0.3
ck2 !0.04
0
0.0
0
5
10
0
15
5
10
15
O2 ΜM
O2 ΜM
(c)
(d)
Figure 4: Showing the a↵ect of parameter changes on the CM RO2 response to [O2 ] conc. in both coupled and uncoupled mitochondria. (a) and (b) show changes in coupled and uncoupled mitochondria
respectively with DN ADH variation and ??and (d) show changes for coupled and uncoupled mitochondria with ck2 respectively. As previous the reducing substrate is set as succinate, and u is set to be 0.4.
For uncoupled the parameter kunc is raised to 1000 from 1 giving a four fold increase in max CM RO2 .
14