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The basic idea of Peffer Cell Osmotic Pressure Measurement is straight forward^[1]:

1. Get a tank of pure water and set the height of water surface ${\displaystyle Z_{0}}$.

2. Get a tube with a semipermeable membrane at the bottom and opened top.

3. Put the tube into the water tank, and fill it with solution of density ${\displaystyle \rho _{i}}$ up to any height ${\displaystyle Z_{i}}$.

4. Solution has higher osmotic pressure than pure water. Water molecules will go through the semipermeable membrane from water tank into the tube, until the pressure on two sides of the membrane get equivalent. ie. ${\displaystyle \Pi =ck_{B}T}$ is balanced by the extra pressure inside the tube.

5. The increase of pressure inside the tube, which finally balances osmotic pressure, comes from the extra water pressure as the result of rising solution surface inside the tube.

6. Be careful that as water molecule move inside to the tube, the solution will be diluted, which means ${\displaystyle \rho _{initial}\neq \rho _{final}}$; ${\displaystyle \Pi _{initial}\neq \Pi _{final}}$. So the osmotic pressure found, is the osmotic pressure of the solution at equilibrium, but not the solution added to the tube at the beginning.

7. The extra pressure due to raised solution level after equilibrium is reached is given by the equation:

${\displaystyle \Delta P=P_{up}-P_{down}=[P_{atm}+\rho _{f}g(Z_{f}+h_{water})]-[P_{atm}+\rho _{purewater}gh_{water}]=\Pi _{f}=ic_{f}RT}$, where ${\displaystyle h_{water}}$ is the height of tube under the water surface. ${\displaystyle P_{up}}$ is the pressure on the up surface(tube side) of membrane, P_{down} is the pressure on the down surface(tank side) of membrane.

8. ${\displaystyle \Pi _{equilibrium}=ic_{f}RT=Z_{f}\rho _{f}g}$, where g is the gravitational acceleration.

9. By measuring ${\displaystyle Z_{f}}$, ${\displaystyle \rho _{f}}$, we can calculate and find the osmotic potential ${\displaystyle \Pi _{f}}$.

Plant and bacterial cells have stiff cell walls but animal cells don't. So, animal cells react more intensively to Osmotic Shock, and are more interesting to do study with.

Cells are actually very complicated systems, their volumes are high dependent on the environment through many factors. The paper Cell Cellular Pressure and Volume Regulation and Implications for Cell Mechanics^[3] made a basic mathematical model of cell volume and pressure response, combining the influence of cortical tension, water permeation, and ion dynamics. this model can be used to compute cell-shape changes during osmotic shock and predict the response of the cell to externally applied mechanical forces.

1. Cells are spherical

2. Cellular cytoplasm is enclosed by a cortical layer, and one layer of cell membrane. Membrane superstructures and membrane trafficking are neglected, the cell membrane and cortex can be treated as a single mechanical structure.

3. through the cell membrane there are passive Mechanosensitive(MS) Channels, and active Ion Pumps.

Cell membrane is permeable to a variety of ions, small solutes, and water. And there are three main sources that influence cells' volume:

1. Flow of water molecules due to osmotic pressure through cell membrane. Liquid is almost incompressible, the amount of liquid (cytoplasm) inside cells directly relate to cell volume.

2. Ions and osmolyte flow across membrane through passive Mechanosensitive Channels, and active Ion Pumps. The movement of ions and osmolyte changes the osmotic pressure, thus influence cell volume.

3. The actomyosin cortex underneath the membrane, and the active stress generated by molecular motors. The stress will affect cell membrane tension, thus influence the open rate of passive Mechanosensitive Channels, which influence cell volume.

Let ${\displaystyle P_{in}}$, ${\displaystyle P_{out}}$ be the hydrostatic pressure and ${\displaystyle \Pi _{in}}$, ${\displaystyle \Pi _{out}}$ be the osmotic pressure, inside and outside the spherical cell. For high solute concentrations and crowded cellular environments, unlike ideal solution ${\displaystyle \Pi _{in}V=nRT}$, an activity coefficient is needed to include the influence of water flow:

${\displaystyle J_{water}=-\alpha \Delta \Psi }$ ^[4] ${\displaystyle =-\alpha (\Psi _{in}-\Psi _{out})=-\alpha [(P_{in}-\Pi _{in})-(P_{out}-\Pi _{out})]=-\alpha (\Delta P-\Delta \Pi )}$; J_water > 0, water flow into the cell.

Cells are modeled as spherical balls with radius ${\displaystyle r}$, and are totally symmetry:

${\displaystyle {\frac {dV}{dt}}={\frac {d}{dt}}({\frac {4\pi r^{3}}{3}})}$ = amount of water flow into cell through membrane = ${\displaystyle 4\pi r^{2}J_{water}}$

We can get:

${\displaystyle {\frac {dr}{dt}}=J_{water}=-\alpha (\Delta P-\Delta \Pi )}$

The permeability constant ${\displaystyle \alpha }$ is related to both the basal permeability of the cell membrane and water flux through specialized water channels.

In living cells, cell membrane not only act like a semipermeable membrane, where water molecules can go through. There are also many passive mechanosensitive(MS) channels^[5], and active ion pumps^[6][7], which enable the cell to actively control the influx and efflux of ions and other osmolytes. To reduce complexity, this model only include one species of MS channel and one species of ion pump.

MS channels act as emergency valves to release solutes in response to hypotonic shocks. Their opening probability follows a Boltzmann Function^[8]. Therefore the ion flux across MS channels is proportional to: ${\displaystyle NP_{open}\Delta c/h_{0}}$; N is the total number of MS channels, ${\displaystyle P_{open}}$ is the opening probability, ${\displaystyle \Delta c}$ is the concentration gradient of ions, and ${\displaystyle h_{0}}$ is the membrane thickness. Approximating Boltzmann Function by a piecewise linear function.

The ion flux due to MS channels is modeled as:

${\displaystyle J_{1}=J_{MS}={\begin{cases}0&{\mbox{if }}\sigma \leq \sigma _{c}\\-\beta (\sigma -\sigma _{c})\Delta \Pi &{\mbox{if }}\sigma _{c}<\sigma <\sigma _{s}\\-\beta (\sigma _{s}-\sigma _{c})\Delta \Pi &{\mbox{if }}\sigma \geq \sigma _{s}\end{cases}}}$.

${\displaystyle \beta }$ is a rate constant, which contains ${\displaystyle RT}$ and ${\displaystyle h_{0}}$. ${\displaystyle \sigma _{c}}$ is the threshold stress, blow which ${\displaystyle J_{1}}$ is zero. ${\displaystyle \sigma _{s}}$ is the saturating stress, above which all MS channels are open. ${\displaystyle \Delta \Pi =RT\Delta c}$.

MS channels release ions passively in the direction of the concentration gradient, whereas ion pumps, are used by cells to actively import ions against the concentration or electrochemical gradient^[9][10]. Ion Pumps requires energy from ATP, enzymatic reactions, or sunlight to operate. The free energy change during the pumping action is ${\displaystyle \Delta G=RTlog(c_{in}/c_{out})-\Delta G_{a}}$, where ${\displaystyle c_{in}}$ and ${\displaystyle c_{out}}$ are the ion concentrations inside and outside the cell, ${\displaystyle \Delta G_{a}}$ is the input energy required. The ion flux across transporters can be modeled as ${\displaystyle J_{2}=-\gamma '\Delta G}$, ${\displaystyle \gamma '}$ is a permeation constant. Assuming ${\displaystyle c_{in}-c_{out}\ll c_{in}}$ and do Taylor Expansion.

The ion flux by active ion pump is modeled as :

${\displaystyle J_{2}=J_{pump}=\gamma (\Delta \Pi _{c}-\Delta \Pi )}$.

${\displaystyle \Delta \Pi _{c}=\Pi _{out}\Delta G_{a}/RT}$ is a critical osmotic pressure difference, which is given by ${\displaystyle \Delta G=0}$. It should be noted that the expression and activity of ion transporters are influenced by other regulators and ATP/ADP concentrations ^[11].

The total ion and osmolytes flux into the spherical cell is:

${\displaystyle {\frac {dn}{dt}}=4\pi r^{2}(J_{1}+J_{2})}$

${\displaystyle n}$ is the total number of ions and osmolytes inside the cells. If ${\displaystyle J_{1}+J_{2}>0}$, more ions and osmolytes flow into the cell, than flow out.

This consider the situation where the cell membrane is perfectly attached to the cell cortex^[12], and dynamics of membrane superstructures and membrane trafficking are neglected^[13]. Therefore the cortex and membrane is treated as a single layer. For spherical cells, from mechanical force balance, the overall cortical stress is given by ${\displaystyle \sigma =\Delta Pr/2hg}$, where h is the cortical thickness. Based on elastic cortex models ^{[14][15][16][17]}, the constitutive law of cortex can be written as:

${\displaystyle \sigma ={\frac {K}{2}}({\frac {S}{S_{0}}}-1)+\eta {\frac {1}{S}}{\frac {dS}{dt}}-\sigma _{a}}$

Where ${\displaystyle K}$ is the elastic modulus, ${\displaystyle \eta }$ is the viscosity of the cortex, ${\displaystyle S}$ and ${\displaystyle S_{0}}$ are the deformed and reference surface areas ${\displaystyle S=4\pi r^{2}}$.

For spherical cells:

${\displaystyle \sigma ={\frac {K}{2}}({\frac {r^{2}}{r_{0}^{2}}}-1)+\eta ({\frac {2}{r}})({\frac {dr}{dt}})-\sigma _{a}}$

In typical osmotic shock experiments, cells' volume change happen in the order of minutes, which means the ${\displaystyle \eta ({\frac {2}{r}})({\frac {dr}{dt}})}$ term is much smaller than the other terms in the constitutive law:

${\displaystyle \sigma \approx {\frac {K}{2}}({\frac {r^{2}}{r_{0}^{2}}}-1)-\sigma _{a}}$; elastic cortex model^[18][19][20].

After applying a hypotonic shock to a cell, which is initially in equilibrium.(${\displaystyle \Delta P=P_{in}-P_{out}}$; ${\displaystyle \Delta \Pi =\Pi _{in}-\Pi _{out}}$)

a) As the media's osmotic pressure ${\displaystyle \Pi _{out}}$ decrease, ${\displaystyle {\frac {dr}{dt}}=J_{water}=-\alpha (\Delta P-\Delta \Pi )}$ changes from zero to positive. Water molecules move into cell through the cell membrane, both the cell radius and cell volume increase.

b) The increase of cell volume(radius) will increase the hydrostatic pressure, the cortical stress, and stretch the cell membrane. The stretching on cell membrane increase the number of open passive MS channels ${\displaystyle NP_{open}}$, then increase the efflux of ions and osmolytes. ${\displaystyle {\frac {dn}{dt}}=4\pi r^{2}(J_{1}+J_{2})}$ changes from zero to negative. Therefore, the total number of ions and osmolytes inside cell ${\displaystyle n}$ decrease.

c) The decrease of ${\displaystyle n}$ decreases the ${\displaystyle \Delta \Pi }$, cortical stress, and efflux rate of ions. ${\displaystyle \sigma \approx {\frac {K}{2}}({\frac {r^{2}}{r_{0}^{2}}}-1)-\sigma _{a}}$ decrease. Until ${\displaystyle \Delta \Pi <\Delta P}$, then water molecules start to flow out of the cell, cell volume(radius) start to decrease, the ${\displaystyle \Delta \Pi }$ start to decrease.

d) Finally, cell will reach equilibrium and return to their original volume. The net change on cells due to hypotonic shock are the decrease of total number of ions and osmolytes inside cell, and the decrease of osmotic pressure ${\displaystyle \Pi _{in}}$ inside cell.

After applying a hypertonic shock to a cell, which is initially in equilibrium.(${\displaystyle \Delta P=P_{in}-P_{out}}$; ${\displaystyle \Delta \Pi =\Pi _{in}-\Pi _{out}}$)

a) As the media's osmotic pressure ${\displaystyle \Pi _{out}}$ increase, ${\displaystyle {\frac {dr}{dt}}=J_{water}=-\alpha (\Delta P-\Delta \Pi )}$ changes from zero to negative. Water molecules move out of cell through the cell membrane, both the cell radius and cell volume decrease.

b) The decrease of cell volume(radius) will decrease the hydrostatic pressure, the cortical stress, and "shrink" the cell membrane. The shrinking on cell membrane decrease the number of open passive MS channels ${\displaystyle NP_{open}}$, then decrease the efflux of ions and osmolytes. If the hypertonic shock is sufficiently larger that ${\displaystyle \sigma <\sigma _{c}}$, then all MS channels will be closed, ${\displaystyle J_{1}=0}$. Then ${\displaystyle {\frac {dn}{dt}}=4\pi r^{2}(J_{1}+J_{2})}$ changes from zero to positive, since ${\displaystyle J_{1}=0}$ and ${\displaystyle J_{2}=J_{pump}>0}$. Therefore, the total number of ions and osmolytes ${\displaystyle n}$ increase inside cell.

c) The increase of ${\displaystyle n}$ increases the ${\displaystyle \Delta \Pi }$, cortical stress, and efflux rate of ions. ${\displaystyle \sigma \approx {\frac {K}{2}}({\frac {r^{2}}{r_{0}^{2}}}-1)-\sigma _{a}}$ increase. Until ${\displaystyle \Delta \Pi >\Delta P}$, then water molecules start to flow into the cell again, cell volume(radius) start to increase, the ${\displaystyle \Delta \Pi }$ start to increase.

d) Finally, cell will reach equilibrium and return to their original volume. The net change on cell due to hypertonic shock are the increase of total number of ions and osmolytes inside cell, and the increase of osmotic pressure ${\displaystyle \Pi _{in}}$ inside cell.

Normally cells adapt to hypotonic shocks faster than hypertonic shocks. One possible reason is that during the hypertonic shock, all of the MS channels might be closed, then the speed of recovery will be limited^[21].