Download 1 APPENDIX S1 Model of Cellular MKT-077 Uptake When MKT

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APPENDIX S1
Model of Cellular MKT-077 Uptake
When MKT-077 is added to cells suspended in medium at time t=0, the drug enters the
cells and is either free in the cytoplasm or “bound” to cellular components. This “bound” drug
could include drug linked to receptors or drug concentrated within organelles, such as
mitochondria. Intracellular drug concentration is described at any time t by:
CC  Ci  Cb
(Eq A1)
where CC = total intracellular MKT-077 concentration, Ci = concentration of free unbound
MKT-077 in the cytoplasm, and Cb = concentration of MKT-077 sequestered within the
mitochondria or other organelles or bound to other receptors. All concentrations are expressed in
[ng MKT-077/(ml cells)].
Based on previous studies showing large mitochondrial uptake of MKT-077 [15,16], Cb
is assumed to primarily represent drug concentrated in the mitochondria. Assuming that Cb is in
thermodynamic equilibrium with Ci [29], the binding of MKT-077 is described by:
Cb   Ci
(Eq A2)
where β = equilibrium constant (dimensionless). Therefore:
CC  (1   ) Ci
(Eq A3)
Since MKT-077 is most likely not a substrate for drug efflux pumps [14], CC can only
change when MKT-077 enters the cell by diffusion across the membrane. The change in CC is
driven by a combination of the concentration gradient and electrical potential gradient across the
plasma membrane. Using the constant field assumption defined by Goldman [25], the
transmembrane drug flux can be described by the Nernst-Planck equation:
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 C  CM e 

J    i


 1 e

where J = net flux of MKT-077 into the cells per unit surface area of cell membrane [ng/(sec
cm2)], η = membrane permeability (cm/sec), and CM = medium MKT-077 concentration [ng
MKT-077/(ml medium)]. The parameter γ is given by:
 
zF
pmem
RT
(Eq A4)
where z = molecular charge number (z = +1 for MKT-077), F = Faraday constant [96.5 J/(mol
mV)], R = gas constant [8.314 J/(mol K)], T = absolute temperature (K), and ΔΨpmem = potential
difference between the inside of the cell and the extracellular medium.
The rate of change in CC is given by:
 NA 
 C  C M e 
dCC  NAc 
 J   c     i

 

dt
NV
NV
1

e


 c
 c
where N = cell concentration [cells/(ml suspension)], AC = surface area of cell (cm2/cell), and VC
= cellular volume (ml/cell). Assuming that cells are spherical, AC/VC = 6/d, where d = cell
diameter (cm). Therefore:
 C  C M e 
dCC  6 
  k
      i

dt
d 
 1 e

 Ci  CM e 



 1 e

where k = mass transfer coefficient across cell membrane = 6η/d (sec-1).
Since relatively little drug is taken up by the cells compared to the amount in the
medium, CM ≈ CM0, where CM0 = initial MKT-077 concentration in the medium. Thus:
 Ci  C M 0 e
dCC
 k 

dt
 1 e
Substituting Equation A3 into Equation A5 gives:



(Eq A5)
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(1   )
 C  C M 0 e
dCi
 k  i

dt
 1 e
  k
  

  1 e

Ci  C M 0 e 

Integration yields:
k t


(1   ) 1  e  

Ci  CM 0 e 1  e





(Eq A6)
Converting to ng MKT-077/(100,000 cells), Equation A6 becomes:

Ci  100,000 VC C M 0 e  1  e t

(Eq A7)
where

k
(1   ) 1  e 
(Eq A8)
Substituting Equation A7 into Equation A3 yields:
CC  100,000 (1   ) VC e C M 0 1  et 
(Eq A9)
As t approaches infinity, CC reaches a final steady-state value, CC,∞ (ng MKT-077/105 cells), and
Equation A9 reduces to:
CC ,  100,000 (1   )VC e CM 0   CM 0
(Eq A10)
where
  100,000 (1   ) VC e
Substituting Equation A10 into Equation A9 yields:
CC  CC , 1  et 
(Eq A11)
Differentiating Equation A11 with respect to time, t, yields the uptake rate at any time:
dCC
   C C , e  t
dt
From Equation A11:
(Eq A12)
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e t 
C C ,  C C
C C ,
(Eq A13)
Substituting Equation A13 into Equation A12 yields:
 C C ,  C C
dC C
   C C , 
 C
dt
C ,


    C C ,  C C 


(Eq A14)
The initial uptake rate [ng MKT-077/(105 cells min)] is given by:
dCC
dt
   CC , et
t 0
t 0
 k
   CC ,   100,000 

1  e


 VC e CM 0

(Eq A15)
Modeling the Relationship between MKT-077 Uptake and Oxygen Consumption Change
To quantify the effect of MKT-077 on oxygen consumption, we tested several models
relating MKT-077 uptake to the change in oxygen consumption. Since it is reasonable to assume
that the change in oxygen consumption rate, q [ml O2/(105 cells min)], would be related to the
amount of MKT-077 sequestered by the mitochondria or the rate at which MKT-077 was taken
up by the mitochondria, we tested the following relationships:
dq
   b0
dCb
dq
   b1 Cb
dCb
dq
 dC 
   b2  b 
dCb
 dt 
(Eq A16)
(Eq A17)
(Eq A18)
where dq/dCb = the change in cellular oxygen consumption for a given change in mitochondrial
MKT-077 concentration {ml O2/[(ng MKT-077) min]}, αb0 = proportionality constant {ml
O2/[(ng MKT-077) min]}, αb1 = proportionality constant {[ml O2 (105 cells)]/[(ng MKT-077)2
min]}, and αb2 = proportionality constant {[ml O2 (105 cells)]/[(ng MKT-077)2]}. The models
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represented by Equations A16, A17, and A18 are designated the “constant”, “uptake”, and “rate”
models, respectively.
Since total cellular MKT-077 concentration, CC, is measured experimentally, these
equations can be expressed as a function of CC. From equations A2 and A3:
  
 CC
Cb  
 1  
  
 dCC
dCb  
1




Substituting these expressions into Equations A16-A18 yields:
dq
  0
dCC
(“constant” model; Eq A19)
dq
   1 CC
dCC
(“uptake” model; Eq A20)
dq
 dC 
  2  C 
dCC
 dt 
(“rate” model; Eq A21)
where α0 = proportionality constant {ml O2/[(ng MKT-077) min]}, α1 = proportionality constant
{[ml O2 (105 cells)]/[(ng MKT-077)2 min]}, and α2 = proportionality constant {[ml O2 (105
cells)]/[(ng MKT-077)2]}, and are defined as:
  

 i  
1  
n 1
 bi , i  0, 1, or 2
From Equation A12:
1
1
1
 
t
dCC
 CC , e dt
(Eq A22)
Substituting Equations A11, A12, and A22 into Equations A19-A21 and simplifying yields:
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dq
  0 CC , e t
dt
(“constant” model; Eq A23)
dq
 1  C C2, e t  1  et 
dt
(“uptake” model; Eq A24)
dq
   2  2 C C2, e 2 t
dt
(“rate” model; Eq A25)
If we assume the recording starts at time t* and MKT-077 is added to the system at time
t=0, the oxygen consumption rate can be expressed as a function of time. Before addition of
MKT-077, the consumption rate is constant:
q(t )  q1 , t *  t  0
(Eq A26)
where q1 = basal O2 consumption rate [ml O2/(105 cells min)]. After addition of MKT-077, the
three models given in Equations A23-A25 describe the consumption change. Integration of
Equations A23-A25 yields:
q(t )  q1  0 CC , 1  et , t  0
 1 CC2 ,
q(t )  q1  
 2

2
 1  e t  ,


   CC2 , 
 1  e 2 t ,
q(t )  q1   2

2


(“constant” model; Eq A27)
t0
t0
(“uptake” model; Eq A28)
(“rate” model; Eq A29)
where q(t) = O2 consumption rate as a function of time after addition of MKT-077 [ml O2/(105
cells min)].
Equations A26-A29 can be substituted into the oxygen mass balance equation for this
closed system:
 dP 
k O 2     N q( t )
 dt 
(Eq A30)
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where kO2 = oxygen solubility (Henry’s Law constant) of the suspension [ml O2/(ml suspension
mm Hg)], P = pO2 in the media (mm Hg), t = time (min), N = cellular concentration (1x106
cells/ml suspension), and q(t) = oxygen consumption rate defined by Equations A26-A29. By
integrating this function, pO2 can be expressed as a function of time and three unknown
parameters (P*, q1, and α0, α1, or α2):
 N 
 q1  t  t *  , t *  t  0
P(t )  P *  
 kO 2 
 N 
 N 0 CC , 
 q1  t  t *   

P(t )  P*  
k
k
O2
 O2 


(Eq A31)
 1
t 
t   1  e  , t  0
(“constant”; Eq A32)
 N 
 N   1 CC2 ,   1  e t 

*
 t 
 q1  t  t   
 

P(t )  P  
3  e t  , t  0 (“uptake”; Eq A33)


2

 kO 2 
 kO 2   2  
*
 N 
 N     2 CC2 ,   1  e 2 t
 q1  t  t *   
 
P(t )  P *  
  t 
2
2
 kO 2 
 kO 2  


 , t  0 (“rate”; Eq A34)

where P* = initial pO2 at time t* (mm Hg). The equations for t > 0 are only valid for pO2 ≥ 20
mm Hg (see text).
The oxygen consumption can also be expressed as a function of intracellular MKT-077
concentration, CC, rather than time. Substituting Equation A13 into Equations A26-A28 and
simplifying yields:
q(CC )  q1   0 CC , 0  CC  CC ,
 
q(CC )  q1   1  CC2 , 0  CC  CC ,
 2 
(“constant” model; Eq A35)
(“uptake” model; Eq A36)
  2 
2
2
q(CC )  q1  
 CC ,  CC   CC , , 0  CC  CC ,
2




(“rate” model; Eq A37)
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where q(CC) = O2 consumption rate as a function of total intracellular MKT-077 [ml O2/(105
cells min)].