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Introductory biophysicsA. Y. 2017-18
4. Basics of statistical mechanics and chemical kinetics
in biophysical processes
Edoardo MilottiDipartimento di Fisica, Università di Trieste
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Extremely short review of statistical mechanics1. Boltzmann factor
Heat exchange
Thermal reservoir
at temperature TPhysical system,
total energy E
exp − EkBT
⎛⎝⎜
⎞⎠⎟
Probability of finding system with energy E is proportional to
Extremely large number of degrees of freedom
Much smaller number of degrees of freedom
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Multilevel statistical system
Consider a macrostate defined by
N1 particles at energy level E1 with degeneracy g1N2 particles at energy level E2 with degeneracy g2... Ni particles at energy level Ei with degeneracy gi...
the number of ways in which we can arrange the identical particles in the M levels is
and when we also include degeneracy, we find that the number of different ways to obtain the macrostate (thermodynamic probability) is
Ω = N!g1
N1
N1!g2N2
N2 !…giNi
Ni !…
N!N1!…NM !
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
We use Stirling’s approximation
and we find
Now the problem is finding the distribution {Ni} that maximizes the thermodynamic probability (this is the distribution that is observed with the highest probability)
lnΩ = N lnN − N( ) + Ni lngi − Ni lnNi + Ni( )i∑ = N lnN + Ni ln
giNii
∑
= N Ni
Nln giNi Ni
∑ N = Nii∑⎛
⎝⎜⎞⎠⎟
lnn!≈ n lnn − n
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Maximization must be carried out constraining both the number of particles and the total energy
For constrained maximization we use the method of Lagrange multipliers and maximize the auxiliary function
lnΩ = N Ni
Nln giNi Ni
∑
N = Nii∑
U = EiNii∑
expression of thermodynamic probability
total number of particles is fixed
total energy is fixed
lnΩ + λN − βU = Ni lngi
Ni Ni∑ + λ Ni
i∑ −β EiNi
i∑
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
lnΩ + λN − βU = N lnN + Ni lngiNii
∑ + λ Nii∑ −β EiNi
i∑
∂∂Nl
lnΩ + λN − βU( ) = ln glNl
−1+ λ − βEl = 0
Nl = gleλ−1−βEl
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Partition function (Zustandsumme)
Ni = gieλ−1−βEi
the partition function is used to determine many other thermodynamicalfunctions
N =X
i
gie��1��Ei = e��1
X
i
gie��Ei ) e��1 =
NPi gie
��Ei=
N
Z
Ni = gie��1��Ei =
N
Zgie
��Ei
Z =X
i
gie��Ei
U =X
i
EiNi =N
Z
X
i
Eigie��Ei = �N
Z
@
@�
X
i
gie��Ei = �N
Z
@Z
@�= �N
@ lnZ
@�
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
The entropy is a measure of the thermodynamic probability
and when we recall the thermodynamic relation
we find
1T= ∂S∂U
S = kB ln⌦ = kBNX
i
Ni
Nln
giNi/N
= kBNX
i
Ni
N(lnZ + �Ei)
= kBN lnZ + kB�U
T =1
kB�
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
A chemical thermodynamics refresher
1. Enthalpy
Recall that a change in internal energy is the sum of the heat absorbed and of the work done by the system
which is the first principle of thermodynamics, and that work can be further subdivided into work due to volume expansion (useless) and all the other work (non-PV work):
ΔU = ΔQ − ΔW
ΔW = PΔV + Δ ′W
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Then
and we can formally restore the form of the first principle by defining the state function enthalpy
so that
ΔU = ΔQ − ΔW = ΔQ − PΔV − Δ ′W
H =U + PV
ΔH = ΔU + Δ PV( ) = ΔQ − Δ ′W
at constant pressure Δ(PV) = PΔV, as in most chemical reactions in the laboratory
if no non-PV work is done on the system, then the enthalpy change corresponds to the heat absorbed by the system
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
2. Helmholtz free energy
According to the second principle of thermodynamics
and therefore
Therefore, when we define we find
ΔQ ≤ TΔS
ΔU = ΔQ − ΔW ≤ TΔS − ΔW = Δ TS( )− SΔT − ΔW
F =U −TS
ΔF = Δ U −TS( ) ≤ −ΔW − SΔT
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Therefore, in isothermal processes (where the system exchanges heat with a large heat bath)
and therefore the work done by the system is less or equal than the decrease of free energy
ΔF = Δ U −TS( ) ≤ −ΔW − SΔT = −ΔW
ΔW ≤ −ΔF
isothermal process
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Therefore, in processes where no work is done or absorbed by the system
i.e.
and this is the condition for a spontaneous process with no work involved.
0 = ΔW ≤ −ΔF
ΔF ≤ 0
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
3. Gibbs free energy
The Gibbs free energy is like the Helmholtz free energy, for processes where the pressure is held constant:
and we find again that the condition for a spontaneous process, with no work involved, is
G = H −TS =U + PV −TS = F + PV
ΔG ≤ 0
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Let’s summarize it again, just for clarity ...
then
and therefore for transformation at constant temperature and pressure and no non-PV work
ΔU = ΔQ − ΔW
= ΔQ − PΔV + Δ ′W[ ] = ΔQ − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦
≤ TΔS − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦
= Δ TS( )− SΔT⎡⎣ ⎤⎦ − Δ PV( )−VΔP + Δ ′W⎡⎣ ⎤⎦
Δ U + PV −TS( ) ≤ −SΔT +VΔP − Δ ′W
ΔG ≤ 0H =U + PVF =U −TSG = F + PV = H −TS
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
ΔG = ΔH −TΔS ≤ 0 ΔH ≤ TΔS
Edoardo Milotti - Introductory biophysics - A.Y. 2016-17
Concentrations and Gibbs free energy
Entropy of mixing in binary solutions
n1 molecules of solventn2 molecules of solute
N = n1 + n2
Then the number of configurations is
Ω = N!n1!n2 !
Edoardo Milotti - Introductory biophysics - A.Y. 2016-17
Ω = N!n1!n2 !
lnΩ ≈ N lnN − N( )− n1 lnn1 − n1 + n2 lnn2 − n2( )= N lnN − n1 lnn1 − n2 lnn2
= n1 + n2( )ln n1 + n2( )− n1 lnn1 − n2 lnn2
= −n1 lnn1
n1 + n2− n2 ln
n2n1 + n2
= −N X1 lnX1 + X2 lnX2( )X1,2 are the volume fractions
Edoardo Milotti - Introductory biophysics - A.Y. 2016-17
Therefore the entropy change due to mixing is
and, assuming that there is no change in contact energy when the molecules of solvent and solute mix, the corresponding Gibbs free energy change is
ΔSm = kB lnΩ− ln1( ) = −kBN X1 lnX1 + X2 lnX2( )
ΔG = −TΔSm = kBNT X1 lnX1 + X2 lnX2( )= nRT X1 lnX1 + X2 lnX2( )= X1ΔG1 + X2ΔG2
Edoardo Milotti - Introductory biophysics - A.Y. 2016-17
We see that we can associate a free energy to each substance A in solution
and in particular, if we consider the free energy change with respect to standard conditions
and if we let [A]0=1M
volume fractionsconcentrations (mole/l)
1 mole/l
�GA = nART lnXA (nA = nXA)
�GA ��GA0 = nART lnXA
XA0
= nART ln[A]
[A]0
�GA ��GA0 = nART ln[A]
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Chemical kinetics
The elementary reaction
can occur via a sequence of elementary reactions, with intermediates, e.g.,
A→ P
A→ I1→ I2 → P
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Rate equations
The rate at which a reaction proceeds is proportional to the probability of bringing all the reactants in the same place at the same time, i.e., it is proportional to their concentrations, therefore the rate of the general elementary reaction
is aA +bB +…+ zZ→ P
rate = k A[ ]a B[ ]b… Z[ ]z n = a + b +…+ z
order of the reactionrate constant (notice that the rate constant has units adapted to the order of the reaction)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Example: first order reaction
A→ P ⇒
d A[ ]dt
= −k A[ ]A[ ]+ P[ ] = A[ ]0
A[ ]t=0 = A[ ]0P[ ]t=0 = 0
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⇒A[ ] = A[ ]0 exp −kt( )P[ ] = A[ ]0 1− exp −kt( )⎡⎣ ⎤⎦
The concentration of A decreases and it is exactly half the initial concentration when
A[ ] = A[ ]0 exp −kt1/2( ) = A[ ]0 2 ⇒ t1/2 =ln2k
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Example: second order reaction
2A→ P ⇒
d A[ ]dt
= −k A[ ]2
A[ ]+ 12P[ ] = A[ ]0
A[ ]t=0 = A[ ]0P[ ]t=0 = 0
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⇒
1A[ ] =
1A[ ]0
+ kt
P[ ] = 2 A[ ]0 − A[ ]( )
The concentration of A decreases and it is exactly half the initial concentration when
2A[ ]0
= 1A[ ]0
+ kt1/2 ⇒ t1/2 =1
k A[ ]0
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Equilibrium constants
We apply these concepts to the reversible chemical reaction
and we note that at equilibrium
i.e. the forward rate is equal to the backward rate, or also
aA + bB cC + dD
k f A[ ]a B[ ]b = kb C[ ]c D[ ]d
C[ ]c D[ ]dA[ ]a B[ ]b
=k fkb
= Keq
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Then, the free energy change for the i-th species with respect to the standard state, per mole, is
and therefore, in a reaction, the Gibbs free energy change splits into parts that take into account the chemical bonds and the concentration changes
ΔG = ΔG0 + cΔGC + dΔGD − aΔGA − bΔGB
= ΔG0 + cRT ln C[ ]+ dRT ln D[ ]− aRT ln A[ ]− bRT ln B[ ]
= ΔG0 + RT lnC[ ]c D[ ]dA[ ]a B[ ]b
= ΔG0 + RT lnKeq
�Gi ��Gi0 = RT ln[i]
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
At equilibrium the free energy change vanishes
ΔG = ΔG0 + RT lnC[ ]c D[ ]dA[ ]a B[ ]b
= ΔG0 + RT lnKeq = 0
Keq = exp − ΔG0
RT⎛⎝⎜
⎞⎠⎟
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Keq = exp − ΔG0
RT⎛⎝⎜
⎞⎠⎟
R ≈ 8.314 J K−1mol−1, RT ≈ 2.5 kJ mol−1@ 300 K( )
this is close to the binding energy of hydrogen bonds in water ≈ 5 kcal/mole ≈ 21 kJ/mole
Exponential dependence on DG0
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Dissociation reactions
AB ! A+B ) [A][B]
[AB]= K
K is called the dissociation constant. K is large when the denominator is small with respect to the numerator (the substance is mostly dissociated).
K is measured in units of concentration. Notice also that when [B]=[AB] (half of B is bound and half is dissociated), then [A] = K.
Finally, it is common to define
pK = � log10 K
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
The dissociation constant of water is important
The concentration of water is omitted by convention.
KW = [H+][OH�]
log 1
0K
W
0.0028 0.0030 0.0032 0.0034 0.0036-15.0
-14.5
-14.0
-13.5
-13.0
-12.5
1/T (K)Plots like this (where the indipendent variable is the inverse temperature) are called "Arrenius plots"
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Redox reactions and the Nernst equation
Example of a Redox reaction (Oxydation – reduction: reduction = acceptance of electrons, oxydation = loss of electrons)
This can be divided in half-reactions (redox couples)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
from
Voe
t & V
oet -
Bioc
hem
istry
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Now consider the generic redox reaction
Just as in the case of the binary reaction
we find
aA + bB cC + dD �G = �G� +RT ln[C]c[D]d
[A]a[B]b
�G = �G� +RT ln[Ared][Bn+
ox ]
[An+ox ][Bred]
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
This is a non-equilibrium situation, where there is transfer of electrons (as in a standard electric battery), and we must use the equation for non-PV work
= Faraday constant
= E.M.F.
(Nernst equation)
�G = �W = �nF�E
FE
�E = �E� � RT
nF ln[Ared][Bn+
ox ]
[An+ox ][Bred]
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Nernst equation is important in calculations of the membrane potentials.
For instance in the case of potassium ions, the following (equilibrium) version of Nernst equation holds
(o = outside the cell, i = inside the cell)
�E� =RT
nF ln[K+]o[K+]i
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
ATP (Adenosine Triphosphate) is a basic element in the energy budget, it is a temporary energy store, and a sort of molecular energy currency
3 phosphate groups
ribose
adenine ring
ATP
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Typically, ATP is unstable, and its reduction to ADP or AMP produces heat.
In the cell environment, the energy released by the removal of one or two phosphate groups, is used to power other reactions (like protein synthesis) that are endothermal (and could not proceed without a source of energy)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
ATP
ADP
AMP+H2O
+Pi
+PPi
ΔG = −30.5 kJ/mol(−7.3 kcal/mol)
ΔG = −45.6 kJ/mol(−10.9 kcal/mol)
Under typical cellular conditions, ΔG is larger, and is approximately −57 kJ/mol (−14 kcal/mol).
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
N.B. we shall meet AMP again, as a building block of nucleic acids ...
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Halobacterium salinarum
Halobacteria are a class of the Euryarchaeota (Archaea) found in water saturated or nearly saturated with salt
Energy production in halobacteria
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
aerial view of the Great Prismatic Spring - Yellowstone
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Charge differences can be generated by charge transport across membranes
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Bacteriorhodopsinis a ~26 kDa transmembrane protein that acts as a light-driven proton pump in Halobacterium salinarum, converting light energy into a proton gradient.
bR is the only protein constituent of the purple membrane (PM), a two-dimensional crystal lattice naturally present as part of the membrane of the bacterium.
view from above
sideview(green lines define the cell membrane)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Top view of the purple membrane patch. The hexagonal unit cell is displayed in the middle of the patch, surrounded by white line defining the unit-cell dimensions.
(from http://www.ks.uiuc.edu/Research/newbr/)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
transmembrane view
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Bacteriorhodopsinhas different conformational states that are spectrally distinguishable
absorption of light quantum
proton pumped into the outer environment
proton from cytoplasm
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
retinal (retinaldehyde): is one of the many forms of vitamin A (the number of which varies from species to species). Retinal is the chemical basis of animal vision.
It is the core functional element of bacteriorhodopsine (and many other light-sensitive molecules).
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
It has been speculated that charge, i.e., proton, transport is mediated by the formation of Grotthuss water wires inside the bRmolecule in the intermediate states.
Recently this has been experimentally confirmed.
E. F
reie
r, S.
Wol
f, an
d K.
Ger
wer
t, “P
roto
n tr
ansf
er v
ia a
tran
sient
lin
ear w
ater
-mol
ecul
e ch
ain
in a
mem
bran
e pr
otei
n”, P
NAS
108
(201
1) 1
1345
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
ATP Synthase, the engine of ATP production
“ATP synthase is one of the wonders of the molecular world. ATP synthase is an enzyme, a molecular motor, an ion pump, and another molecular motor all wrapped together in one amazing nanoscale machine. It plays an indispensable role in our cells, building most of the ATP that powers our cellular processes. The mechanism by which it performs this task is a real surprise.”
(RCSB – Molecule of the month, Dec. 2005)
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
• Two motors, F0 and F1
• F0, powered by flow of protons
• F1, powered by ATP
• Motors are connected, and one can force the other into a generator.
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Materials 2013, 6 5833
thickness and protein-protein/protein-lipid ratio, this finding can be further extended to the engineering multiple-polymersome level life process with a higher architectural complexity.
Figure 6. (a) Schematic representation of ATP-producing polymersome (BR-ATP synthase-polymersome) (adapted with permission from [43]. Copyright 2007 IEEE.). (b) (i) Intravesicular pH change as a measure of proton pumping activity of BR-polymersomes and (ii) photosynthetic ATP production in the BR-ATP synthase-polymersomes. (adapted with permission from [85]. Copyright 2005 American Chemical Society.)
4.2. Reverse Osmosis Water Purification Membrane
4.2.1. Background
Aquaporins (Aqps) are membrane water channels, playing important roles in regulating water transport in cells and thus contributing to the water homeostasis of organisms [95]. From a practical application point of view, E. coli aquaporin-Z (AqpZ) with a histidine-tag has advantages due to large-scale protein production and single-step purification (Ni-NTA or Talon resin) [48]. AqpZ forms a tetramer (70–80 kDa, see Figure 7a) which can transport water across the membrane in the presence of an osmotic gradient. It is noted that osmotic water permeability coefficient (Pf) increases with increasing protein content, and decreases with a protein-to-lipid weight ratio >0.02, possibly through protein-to-protein interaction discussed in Section 3 (Figure 7b). AqpZ has been reported to selectively transport only pure water molecules across cellular membranes with a high water permeability (P ≥ 10−13 cm3·monomer−1·s−1) and a low Arrhenius activation energy (Ea = 3.7 kcal·mol−1) [48]. This corresponds to a water transport rate of about 3 × 109 water molecules/monomer/s. This exceptionally high water transport capability and sharp water selectivity of AqpZ make AqpZ-embedded polymer
from
Cho
i and
Mon
tem
agno
: “Re
cent
Pro
gres
s in
Adva
nced
Nan
obio
logi
cal
Mat
eria
ls fo
r Ene
rgy
and
Envi
ronm
enta
l App
licat
ions
”, M
ater
ials
6 (2
013)
582
1
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
Biosynthesis within a bubble architecture
(a) (b)
(c) (d)
Figure 1. Encapsulation of polymer vesicles in a bubble aqueous channel. (a) Schematic representation of a proteo-polymeric vesiclereconstituted with BR and F0F1-ATP synthase. (b) The procedure for embedding BR–ATP synthase–polymer vesicles in the bubble waterchannel. (c) TEM micrograph of BR–ATP synthase–polymer vesicles. (d) Schematic diagram of a dry foam structure of bubbles withBR–ATP synthase–polymer vesicle incorporation. When spherical bubbles come together, dry aqueous foam formation takes place, formingpolyhedra. Plateau borders contain most of the aqueous solution.
solution. Bubbles were blown using a pipette with a smallaperture by expelling air. During bubble formation, detergentmolecules self-assemble to form monolayers on the innerand outer surfaces of a water channel containing BR–ATPsynthase–polymer vesicles. To test for the incorporation andfunction of polymer vesicles in the aqueous channels, weprepared foams by filling a UV cuvette with bubbles. Asfreshly formed foam showed free drainage during the initialstages, the cuvette was kept inverted in the dark to remove thepolymer vesicle solution not incorporated within the bubbles.
The self-assembly of detergent molecules in the bubblearchitecture was confirmed by measuring the pH of theaqueous layer before and after blowing bubbles by trappingthe fluorescent pH probe pyranine [15, 20]. Initial pHreadings of 7.1 in the detergent solution increased to about7.4 after forming the bubbles (data not shown). Since thepH of the water channel is related to the concentration ofdetergent molecules, we attribute this pH increase to detergentmolecules’ self-assembly into monolayers of the bubbles.Before taking any measurements, we confirmed the formationof dry foam, where bubbles take the form of polyhedra withnanoscale liquid films and Plateau borders (figure 1(d)) [21].To minimize gravity-induced vertical drainage from the waterchannels, foam samples were sealed and continually rotated ata speed of 20 rpm.
It is noteworthy that the pH values of the foam’s aqueouschannels decreased gradually (figure 2). Over the courseof 60 min, a pH change of 0.045 units was observed.We attribute this acidification to the detergent moleculesentering the water channel during the coalescence process.When bubbles coalesce, detergent molecules composing thecommonly shared water channel of neighbouring bubblesare supposed to self-assemble into monolayers of the newlymerged larger bubble. However, some detergent may bereleased into the water channel. This suggests that theinstability of the bubbles in this study induced a gradualacidification by increasing the detergent concentration withinthe water channels over time.
As the first step, we monitored BR’s proton pumpingactivity in the polymer vesicles. The generation of a photo-induced electrochemical proton gradient was measured bytrapping pyranine inside the polymer vesicles. Fluorescencewas first measured after incubation in the dark and any photo-induced intensity change was measured after illumination witha 5.0 W green LED (λ = 570 nm). Intravesicular pHmeasurements were performed in buffer solution using BR–polymer vesicles and BR–ATP synthase–polymer vesicles.Both systems in buffer solution showed an increase in theinternal pH with illumination (figure 3). That is, the generationof a photo-induced proton gradient resulted in alkalinization
2199
Biosynthesis within a bubble architecture
(a) (b)
(c) (d)
Figure 1. Encapsulation of polymer vesicles in a bubble aqueous channel. (a) Schematic representation of a proteo-polymeric vesiclereconstituted with BR and F0F1-ATP synthase. (b) The procedure for embedding BR–ATP synthase–polymer vesicles in the bubble waterchannel. (c) TEM micrograph of BR–ATP synthase–polymer vesicles. (d) Schematic diagram of a dry foam structure of bubbles withBR–ATP synthase–polymer vesicle incorporation. When spherical bubbles come together, dry aqueous foam formation takes place, formingpolyhedra. Plateau borders contain most of the aqueous solution.
solution. Bubbles were blown using a pipette with a smallaperture by expelling air. During bubble formation, detergentmolecules self-assemble to form monolayers on the innerand outer surfaces of a water channel containing BR–ATPsynthase–polymer vesicles. To test for the incorporation andfunction of polymer vesicles in the aqueous channels, weprepared foams by filling a UV cuvette with bubbles. Asfreshly formed foam showed free drainage during the initialstages, the cuvette was kept inverted in the dark to remove thepolymer vesicle solution not incorporated within the bubbles.
The self-assembly of detergent molecules in the bubblearchitecture was confirmed by measuring the pH of theaqueous layer before and after blowing bubbles by trappingthe fluorescent pH probe pyranine [15, 20]. Initial pHreadings of 7.1 in the detergent solution increased to about7.4 after forming the bubbles (data not shown). Since thepH of the water channel is related to the concentration ofdetergent molecules, we attribute this pH increase to detergentmolecules’ self-assembly into monolayers of the bubbles.Before taking any measurements, we confirmed the formationof dry foam, where bubbles take the form of polyhedra withnanoscale liquid films and Plateau borders (figure 1(d)) [21].To minimize gravity-induced vertical drainage from the waterchannels, foam samples were sealed and continually rotated ata speed of 20 rpm.
It is noteworthy that the pH values of the foam’s aqueouschannels decreased gradually (figure 2). Over the courseof 60 min, a pH change of 0.045 units was observed.We attribute this acidification to the detergent moleculesentering the water channel during the coalescence process.When bubbles coalesce, detergent molecules composing thecommonly shared water channel of neighbouring bubblesare supposed to self-assemble into monolayers of the newlymerged larger bubble. However, some detergent may bereleased into the water channel. This suggests that theinstability of the bubbles in this study induced a gradualacidification by increasing the detergent concentration withinthe water channels over time.
As the first step, we monitored BR’s proton pumpingactivity in the polymer vesicles. The generation of a photo-induced electrochemical proton gradient was measured bytrapping pyranine inside the polymer vesicles. Fluorescencewas first measured after incubation in the dark and any photo-induced intensity change was measured after illumination witha 5.0 W green LED (λ = 570 nm). Intravesicular pHmeasurements were performed in buffer solution using BR–polymer vesicles and BR–ATP synthase–polymer vesicles.Both systems in buffer solution showed an increase in theinternal pH with illumination (figure 3). That is, the generationof a photo-induced proton gradient resulted in alkalinization
2199
Scheme of a liposome with BR and F0/F1 “Bubbles” seen with electron microscopy from
Cho
i and
Mon
tem
agno
: “Bi
osyn
thes
is w
ithin
a b
ubbl
e ar
chite
ctur
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Artificial photosynthesis and ATP production in biomimetic materials
Edoardo Milotti - Introductory biophysics - A.Y. 2017-18
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