Key words: biophysical modeling, transport phenomena, diffusion. Fick laws. osmosis, thermo-diffusion, diffusion potential

The Biophysical Modeling of the Diffusion in

the Living Systems

Authors: Janos Vincze, Gabriella Vincze-Tiszay

Health Human International Environment Foundation, Budapest, Hungary

Email: [email protected]

Abstract

generate flows for which conservation laws apply. According to the general definition, the diffusion is a mass transportation under the action of a generalising force, which can be the concentration gradient, the pressure gradient, etc. It is about a mass flow, so that the parameter transported complies with the mass preservation law. The diffusion is explained on the basis of the spontaneous movement of the molecules which have the tendency to spread in equal numbers in each subspace tendency which arises from the concentration gradient. The Fick’s first law indicates the sense of development of the diffusion from the higher concentration towards the lower concentration. Fick’s second law establishes that the variation in time of the concentration in a space is proportional with the variation in space of the concentration gradient in that particular timeframe. The diffusion potential exists only transiently, because in the system described above, concentration differences become balanced sooner or later. Diffusion potential can be stable in systems where compartments having different concentration of permeable ions are separated by selectively permeable membranes. It is known that the permeability of biological membranes is ion-specific.

Key words: biophysical modeling, transport phenomena, diffusion. Fick’ laws. osmosis, thermo-diffusion,

diffusion potential

The Biophysics Science

knowledge in the human future. As a borderline science, medical biophysics remains the reference term of any medical act, whether it is preventive, exploratory, curative, recovery or predictive.

Modeling

prediction of normal answers of environment factors and in biophysical reactions and to detect the atypical abnormal responses.

Transport Phenomena

The transport phenomena mean the variation in time and space of generalized forces when they generate flows for which conservation laws apply. [2]

This general and strongly scientific definition of the transportation phenomena has two major merits: 1) particular forms of transportation can be deducted from it (mass transport – diffusion; energy transport – thermal conductibility; impulse transport – viscosity; electric charge transport – electric conductibility, crossed effects and other); 2.) it allows a quantitative characterization of the product exchange, which was impossible based on the previous definitions.

If W – the amount of the transported parameter, for which the conservation law is valid; K – a constant dependent on the type of transportation and the nature of the transported parameter; grad a – the generalized force, then the amount of the parameter (flow) transported through the surface dS in the dt time frame will be given by the relation:

W = K

grad

a

dS

dt





If the transportation takes place only after a direction x, then we obtain the formula:

W = K

 

The differential form is the following:

t

S

x

a

K

W



















Making the proper replacements in the relation above, we obtain the classical laws which describe particular, simple transportation phenomena. [3]

With non stationary transportation we understand those transportations where the value of the flow is modified in time from one point to the other. Making the right replacements in the relationship above we obtain the classical laws which describe the simple non stationary transportation phenomena.

Diffusion

parameter transported complies with the mass preservation law. By replacing in the general relation the generalized force with the dc/dx concentration gradient the K constant with –D, where D is the diffusion coefficient through differentiation, we obtain Fick’s first law:

m = – D

dx

dc

S t

The diffusion is explained on the basis of the spontaneous movement of the molecules which have the tendency to spread in equal numbers in each subspace tendency which arises from the concentration gradient. The minus sign in Fick’s first law indicates the sense of development of the diffusion from the higher concentration towards the lower concentration. [4] The value of the diffusion concentration can also be calculated based on the gas kinetic theory:

D = 1,12





2

m

m

m

m

d

N

N

kT









where: N1 şi N2 – number of molecules of each type in the volume unit; m1 and m2 the mass of the two

molecular species; d – average diameter of the molecules; k – Boltzmann’s constant; T – absolute temperature. From this relation it results that the diffusion coefficient D depends on the nature of gas and it is directly proportional with the temperature and when the temperature is constant, it is reversely proportional with pressure.

If the diffusion must produce through a small orifice, then the diffusion speeds v1 and v2 of the two gases

will be reversely proportional with the radical of the molecular masses m1 and m2:

2 1 1 2

m

m

v

v



which is the mathematical expression of the Graham law.

Einstein established that for the dispersed systems in which the molecules of the dissolved substance are higher in comparison with the molecules of the solvent, then between the ray r of the molecules of the diffusible substances (considered to be spherical), viscosity of the solution (η) and the diffusion constant (D) there is the relation:

where: N – Avogadro’s number; R – universal gas constant; T – absolute temperature. Knowing the diffusion coefficient D we can reach the often used formula for the calculation of the molecular weight (M) of the colloidal substances: M = _{2 2} 2

162

N

R









_













D

T

The diffusion is a frequently found phenomenon: with diffusion, the respiratory gas exchange takes place between the air in the alveoli, blood and cellular cytoplasm, the nutritive substances diffuse from the thin intestine into blood and further away in the tissues, the diffusion lays at the basis of the white cells’ chemotactism, the olfaction happens when the molecules of the smelling substances spread in the air reach the cells of the olfactory mucosa. [5]

Starting from Fick’s first law, we can study the time variation of the concentration at the level of a given section. If the gradient is constant, then through the layer of dx thickness of the section considered each number of molecules equal to the number of molecules which leave the layer towards the less concentrated area enters each second from the more concentrated area, hence the concentration in that layer remains constant. But if the concentration varies closer and closer on the direction x, then in time the concentration in the various layers will also change. For the calculation of this variation in time, we must take into account the fact that the entry speed

dn/dt of the substance in the analysed layer in the t moment will be equal to:

dt

dn

x

c

















dx

And the exit speed through the opposite side will differ from the first one, due to the variation of the concentration gradient: c – dc = c – t

x

c

















dx

Hence the exit speed will be equal to:

It means that the variation per second of the amount of substance from the layer of dx thickness is (dn – dn’)/dt, and by dividing these values at the S.dx volume of the layer gives us the sought concentration variation:

dt

dx

S

dn

dn







t

c

















By replacing the values of dn/dt, respectively dn’/dt we arrived to Fick’s second law:

x

t

c

















x

c

















Which establishes that the variation in time of the concentration in a space is proportional with the variation in space of the concentration gradient in that particular timeframe. This law lays at the basis of determination of the diffusion by measuring the distribution of the concentration in the system after a determined period of time.

A particular case of diffusion is dialysis, which consists in the diffusion thought semipermeable membranes of the substances with a molecule smaller than the membranes pores. [6] The dialysis takes place up to the equalising of the concentration in the two parts of the membrane. It is also used in the modern therapy. An example is the artificial kidney whose functioning is based on achieving an extrarenal and extracorporal dialysis which allows the purification of the organism from catabolytes and exo- and endogenous toxins because the depuration could not be performed by the non functional kidney.

Osmosis

The various compartments of the living organisms are separated from one another through membranes with selective permeability. Between two solutions of various concentrations of the same solvent, separated through a membrane impermeable for the solvent, the migration of the solvent takes place from the diluted solution in the concentrated one. This phenomenon is called osmosis. [7] We observe that osmosis is a particular case of diffusion.

If on a part of the semipermeable membrane there is pure solvent and on the other side a solution of this solvent, then the molar concentration of the solvent in the first compartment will be c1 = 1, and in the second

one it will be:

m

m

m

c





where: m0 – number of solvent molls in the volume unit; m – number of the molls of the substance dissolved

the point of view of the solvate, this being the parameter we are measuring, we can state that the osmotic flow always passes from the more diluted solution towards the most concentrated. The solvent diffuses until the equalising of the existing hydrostatic pressure.

The necessary pressure for preventing osmosis is called osmotic pressure, whose value is calculated on the van't Hoff’s law:

π = R . T .

M

c

where: c/M – number of solvate molls; R – universal gas constant, T – absolute temperature. Osmotic pressure can also be determined through cryoscopic determinations:

π = R . T .

c

K

T



where ΔT – the difference between the cryoscopic points of the solvate and solvent; Kc – cryoscopic constant.

Knowing this, we can review the osmosis laws:

– all the solutions of the same concentration, at equal temperatures, have the same osmotic pressure;

– at constant temperature, the osmotic pressure is directly proportional with the value of the solution’s concentration;

– when the concentration is constant, the osmotic pressure varies directly proportional with temperature. The osmosis is frequently found in the living organisms, due to the numerous semipermeable membranes between the various compartments.

The walls of the capillaries, for example, act like such a membrane placed between blood and the interstitial liquid. If the osmotic pressure of a solution coincides with the one of the blood plasma, we talk about an isotonic solution.

The osmotic pressure of the isotonic solutions is equal to the one of the blood plasma, the one of the hypotonic solution is lower and the osmotic pressure of the hypertonic solutions is higher than the plasma’s. If the plasma is diluted with a hypotonic solution, the red cells swell osmotically with water, until (in the virtue of the osmosis laws) they break (they lyse). In hypertonic solutions, the red cells loose water and they shrivel. In both cases the red cell’s functionality is compromised.

If a semipermeable membrane interposes between two solutions with different molarity (n1 and n2) the

osmotic pressure will be:

π = (n1 – n2) RT.

When an external pressure (P) is exerted on the solution different from the osmotic pressure (π), then for P

> π the solvent flow goes from the solution towards the solvent and for P<π the solvent flow will have a

different direction.

This phenomenon lies at the basis of the capillary exchanges: at the arterial end of the capillaries the blood's hydrostatic pressure is higher then to the osmotic one and the substances pass in the interstitia; at the venous end the osmotic pressure is lower then to the osmotic one and the substances pass in the blood.

In the case of osmosis, the generalized force is the pressure gradient, which generates a volume flow. The characteristic parameter of the process is the osmotic pressure (π). By making the necessary replacements in the general equation of the transport processes, we obtain the relation:

π =

























.

5

,

0

.

.

c

V

T

R

c

V

T

R

c

M

T

R







Sometimes, we find the notion of osmotic mechanical work. (L). It shows that in the case of existence of an osmotic pressure π, due to which a substance is transported through a membrane which separates the two compartments, an osmotic mechanical work is performed.

When the two solutions of various concentrations (c1 < c2) are separated through a membrane permeable

only for the solvent, this will cross the membrane towards the most concentrated solution, until a difference of hydrostatic pressure equal to osmotic pressure π = p appears. In this case the system itself performed an osmotic mechanical work (L), whose expression can be deducted in the following mode:





dV

p

L

Using the osmosis law, we obtain for osmotic mechanical work:















_

ln

p

p

T

R

n

dV

p

L

The osmosis laws apply only in certain dilutions: in the case of very high or very low concentrations of dissolved substance or at temperatures over 400C. For the electrolytes (which dissociates) it does not apply anymore.

The filtration is the separation of the components of a solution through a porous environment. Particles with a diameter higher than the diameter of the pores of that environment are retained. [8]

Thermo-diffusion

It is one of the most typical non-equilibrium thermodynamic cross-effects. Constant temperature difference results in separation or concentration difference in gases or solutions. Due to the temperature gradient, mass transport can be produced, which is the transfer of thermal energy by flow (convection). [9] This can be described by the following equation:



T

T



dS

dt

dQ











where dQ – the quantity of transferred heat; dt – time interval,  – heat transfer coefficient (J / s • grad • m2); T2

- higher temperature; T1 - lower temperature; dS - common surface of the two parts of different temperatures.

Thermodiffusion is obtained by making a solution in which the dissolved molecules are warm compared to the solvent molecules, and which is poured between the two walls of a long double-walled glass tube. One such experimental solution is to heat the inner wall with flowing steam and keep the outer wall at a lower temperature by cooling it. In this case, it can be observed that the dissolved material is located at a higher concentration along the colder wall, and flows from the warmer spot to the colder one. When the tube is set vertically, the cooler and more concentrated solution flows downward, while the warmer and more dilute solution flows upward. This cross-effect became of practical importance when György Hevesy used it in the 1920s to separate isotopes.

Its biological significance is due to the fact that it allows to interpret the use of specific isotopes of certain cell types. Vincze attributes importance to the capillary thermodiffusion in the concentration of primary urine during urine excretion. The cells that form the tubular system of the loop of Henle produce heat, and therefore the presence such a mechanism may be assumed in the kidneys. The efficiency of thermodiffusion (1–3%) is similar to that of renal function.

Thermo-osmosis

When a temperature gradient through the membrane creates a material flow, it is referred to as thermo-osmosis. Raising the temperature of a more concentrated solution compared to more dilute solution in an osmometer, the number of water molecules returning to the colder and more dilute solution is increased by the greater velocity resulting from this rise in temperature, thus achieving a dynamic equilibrium of the correspondent temperature gradient even though the warmer solution is more concentrated, and the colder one is more dilute. [10] The reduction in vapor tension in solution due to dissolution can be compensated by increasing the temperature, and it is even possible to migrate water from the concentrated and warmer solution to the colder and more dilute solution, thus diluting the more dilute solution at the expense of thermal energy.

ln ₂ 1 0

T

T

R

p

p

_



_



where p0 – the higher temperature; p1 – the vapor pressure corresponding to lower temperature,  – heat of

evaporation of water; R – the universal gas constant; T – the temperature. Taking into account that T is generally low, we find that in the osmometer (ideally in the case of a semipermeable membrane) the hydrostatic pressure difference (p) is in balance with the difference between the osmotic pressure and the thermo-osmotic pressure: p =

v

T

R 

T

T

v

N

N









n

n

n

N





n

n

n

N





where N1 – the molar fraction of water in the more dilute solution; N2 – the same in the more concentrated

solution:

Thus, in such an ideal case, a temperature difference of l C0 can maintain equilibrium with a maximum difference of 3.1 osmoles. The actually measurable values are smaller. Vető achieved significant experimental results in connection with thermo-osmosis. Thermo-osmosis is a cross-effect that allows the interpretation of certain phenomena of biological water transport. Vincze explained the mechanism of urine and saliva excretion this way.

Diffusion potential

When electrolyte solutions of different concentrations came into contact, the ion diffusion may lead to an electrical potential gradient, a so-called diffusion potential. The process can be imagined as follows. Diffusion always moves from the higher concentration spot to the lower concentration spot. [11] If the mobility of the anion and the cation is the same, they will diffuse together and no tension will develop. However, if their mobility is different, the more mobile ion will advance in diffusion, dragging the other, less mobile ion. [12] This creates a certain order in the distribution of the heterogeneous charge carriers, and that is why there is a potential difference between the different concentration ranges. An electric double layer is formed at the border of the two solutions.

The diffusion potential exists until the concentration difference is leveled out. Assuming a concentration gradient of dc/dx, the potential gradient (dE/dx) is given by:

where uk and ua – the mobility of cations and anions, while the other notations are the same as those used in the

Nernst equation. Integration gives the following equation:





















u

u

u

u

F

z

T

R

E

E

E

c

c

dE provides the potential difference that occurs in the contact layer of c1 and c2 solutions.

Thus, it can be stated that the diffusion potential is also a cross-effect, in which case the driving force of the flux is not only the chemical potential gradient, but the force resulted from the electric potential gradient should also be considered. [13] Of course, the effects of different gradients are difficult to distinguish, and these phenomena cannot be clearly detected in living organisms. We must accept simple transport processes as well as cross-effects (thermodiffusion, thermo-osmosis, diffusion potential, Donnan equilibrion) as idealized model cases. [14]

References

[1] Hoppe, W., Lohmann, W., Markl, H., Ziegler H.: Biophysics. Springer-Verlag, Berlin, 1983.

[2] Vincze, J.: The Transport Phenomena and Cross-effects in the Living Systems. NDP P., Budapest, 2001. [3] Vincze, J.: Biophysics of the Apparatuses. NDP P. Budapest, 2020.

[4] Vincze, J.. Medical Biophysics. NDP P., Budapest, 2018.

[5] Kedem, O., Katchalsky, A. Mechanism of Active Transport. Trans. Faraday Soc. 1963, 59:1918–1930. [6] Vincze, J., Vincze-Tiszay G.: Biophysical model of active transport through vesicles. Medical Research

Archives, 2020, 8., (4):1–9.

[7] Sarson, E., Cobelli, C.: Modelling Methodology for Physiology and Medicine. Elsevier P., 2014.

[8] Vincze, J.: The Biophysical Modeling of the Apparatuses in the Human Organism. Lambert Academic Publishing, Berlin, 2020.

[9] Renote, I., Malhotra, V.: Protein transport by vesicles and tunnels. J. Cell Biology, 2019, 218(3): 737–740. [10] Vincze, J., Vincze-Tiszay G.: The Biophysical Modeling of the Transport Phenomena in the Living

Systems. Technium, 2020, 2(4): 134–140.

[11] Vincze, J.: The Cross-effect in the Living Systems. Med. P. Budapest, 1984.

[12] Vincze, J., Vincze-Tiszay G.: The Biophysical Modeling of Crossing-Effects in Living Organisms. J. Med. Healtcare, 2020, 2(2):1-3.

[13] Vincze, J. Biophysical vademecum. NDP P., Budapest, 2021.