1
Supplementary data: Formal determination of the time residency of drugs in
2
membrane
3
To formally determine Eq.2, one will consider that (see Fig.1-SM); (i) the external
4
concentration of drugs is much higher than the internal/cytoplasmic concentration of
5
drugs, such that the global flow of drugs through the membrane follows Fick’s law. In
6
other words, out equilibrium conditions with a unidirectional flow within the membrane
7
towards the cytoplasm are considered here; (ii) the maximal value of the membrane
8
barrier potential, U max , is such that U max / k BT  1 , which allows a stationary regime to be
9
considered with regard to the flow of drugs through the membrane; (iii) drugs located
10
before the maximum value of the barrier potential remain in this location long enough to
11
follow a Maxwell-Boltzmann’s distribution; (iv) the cellular membrane thickness, h , and
12
the membrane diffusion coefficient, D are presumed to be constant; (v) the external
13
concentration of the drugs is low enough so that only a small fraction of drugs are
14
incorporated in the membrane, and therefore the characteristics of the membrane barrier
15
energy are constant and independent of the insertion of drugs.
16
Therefore, using (i) and (ii) the constant flow of drugs, J , is equal to the number of drug
17
molecules into the membrane, N , passing through the cellular membrane (of surface area
18
S cell ) per unit of time and expressed as J   ( N / S cell ) / t . In a stationary regime the
19
flow of drugs is a function of the spatial variation of the drugs’ density, w , within the
20
membrane and of the membrane potential, U (x) , along the x-axis normal to the
21
membrane, written under the form:
22
J   D x w  w xU
(Eq.1-SM)
1
23
Where  is the drug mobility within the membrane and   xU the force applied to the
24
drugs associated with the barrier of potential U within the membrane. As the flow of
25
drugs is constant across the membrane, Eq.1-SM can be rewritten as:
26
J  e U D x weU  JeU  D x weU
(Eq.2-SM)
27
Where U  U / D  U / k BT using Einstein’s relation and eU is introduced to collect
28
Eq.1-SM. Integrating Eq.2-SM over the membrane thickness it follows:
29

J  eU dx   D wineU in  wouteU out

(Eq.3-SM)
h
30
Where win , wout , U in and U out represent the probability density of drugs and the drugs
31
energy when located at the edges of the membrane, in the inner and outer leaflets
32
respectively. Consequently, the “out”-state corresponds to drugs being inserted in the
33
outer leaflet in order to optimize their dehydration and hydration energies, i.e. the non
34
polar part inserted in the membrane and the polar part external to the membrane, and the
35
“in”-state corresponds to the same optimization but in the inner leaflet. By considering
36
the system composed by the inner leaflet and the internal/cytoplasmic compartment, win
37
is similar and proportional to the probability of the drugs, once inserted, to remain in the
38
inner leaflet, which ultimately depends on the difference in their chemical potential,
39
in , between the adsorbed and desorbed, i.e. cytoplasmic, states:
40
win  e   in / k BT / h
41
As one considers out equilibrium conditions, that is that the thermal equilibrium has not
42
been reached, the concentration of drugs in the internal/cytoplasmic compartment is very
43
low, verifying 1  C in . Assuming an ideal gas-like chemical potential for cytoplasmic
(Eq.4-SS)
2
44
drugs, it follows that the major contribution in  in / k BT is  ln Cin  since 1  C in .
45
Thus, in / kBT ~  ln( Cin )   and accordingly,  in / k B T  1 . The latter relation
46
implies that, as the concentration of drugs in the internal/cytoplasmic compartment is
47
very low, the drugs will necessarily desorb the inner leaflet (leaves the “in”-state) to
48
undergo solvation. Furthermore, using both Eq.4-SM and in / kBT  1 , it follows that
49
the density probability of finding drugs in the “in”-state is negligible compared to the
50
density probability of finding drugs in the outer leaflet (which is filled with drugs), i.e.
51
win  e  in / k BT / h  wout . As a result, the first order approximation of Eq.3-SM can be
52
derived as follow:
53
J  eU dx  Dwout eU out 1  o( win / wout )  Dwout eU out
(Eq.5-SM)
h
54
The flow of drugs through the membrane is driven by the difference in the number of
55
drugs between the “out” and “in” states: J   ( N / S cell ) / t  [( N out  N in ) / S cell ] / t 0 ,
56
where t 0 represents the characteristic time for a drug to pass through the membrane. As
57
the number of drugs is proportional to the probability density and the out equilibrium
58
condition implies win  wout , the flow of drugs through the membrane can be reduced to
59
J  ( N out / S cell ) / t 0 . Finally, as the density of drugs, wout , present at the edge of the outer
60
leaflet of thickness h / 2 can be approached by wout  (2 / h)( N out / S cell ) , inserting these
61
expressions into Eq.5-SM leads to Eq.2 in the text, i.e.:
62
t0 
h
h 2  1 U ( x ) U o u t 
dx 
 e
2 D  h 0

(Eq.6-SM)
63
64
3
65
Legend supplementary data
66
Figure-SM: Representation of the drug path through the membrane as a function of
67
the energies involved. “Out”, “In” and “h” are the outer leaflet, the inner leaflet and the
68
membrane thickness respectively. The global flow of drugs, J , between the outer
69
medium and the cytoplasm is driven by the difference in the concentration of drugs, or
70
equivalently, the difference in the chemical potential ( ~ ) of drugs between these two
71
compartments. To effectively traverse the membrane, the drug must incorporate the outer
72
leaflet of the membrane (transition states 12) with a decrease in its energy
73
corresponding to  out , representing the insertion of the drug non polar part. Once
74
incorporated, the drug must bypass the membrane barrier potential to reach the cytoplasm
75
(transition states 23). The dashed area represents the probability density of the drug in
76
the membrane. In effect, because of the membrane barrier energy the probability density
77
is higher in the outer leaflet than in the inner leaflet. As drugs replenish the outer leaflet
78
due to the global flow of drugs between the outer and cytoplasmic compartments, once at
79
the top of the membrane barrier energy (unstable state 3), the drug cannot come back into
80
the outer or inner leaflets (out equilibrium conditions). As a consequence, the drug
81
desorbs the inner leaflet of the cellular membrane to go into the cytoplasm (transition
82
states 34). Note that the desorption of drugs is energetically favoured as: (i) the
83
difference in the drug energy between states 3 and 4 is driven by the difference of the
84
drug chemical potential  in , which is much higher than ~ , which drives initially the
85
global drug flow; (ii) drugs have an affinity toward their cytoplasmic targets.
86
87
4
88
Figure-SM
89
90
91
92
h
Out
In
3
1
out
U max
2
~
4
in
Drug/target
interaction
Global drugs flow
~
J  
5