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Original article / Оригинальная статья УДК 524.8
DOI: https://doi.org/10.21285/2227-2925-2020-10-2-223-231
On aggregation in binary biopolymer systems
© Yuri I. Matveev*, Elena V. Averyanova**
*Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, Moscow, Russian Federation
**Biysk Technological Institute (branch) of the Altay State Technical University, Biysk, Russian Federation
Abstract: The main structural protein of the human eye, which accounts for about 50 % of the mass of all water-soluble proteins comprising the lens, is а-crystallin. Alpha-crystallin functions as a molecular chape-rone, preventing other lens crystallins from interfering in the vital activity. Alpha-crystallins partially or fully stabilise unfolded proteins, preventing the formation of deposits, helping to preserve the lens transparency and reducing the risk of a number of diseases, including cataracts. This biological phenomenon can be considered in the framework of materials science when considering the problem of slowing down the aging processes of polymers. In the present study, methods for slowing down the process of aggregation of а-lactalbumin in solution are considered, using the binary system a-lactalbumin-aA-crystallin as an example. To this end, experimental data on the rate of change of the aggregation process were formalised, i.e. expressed in terms of transition temperatures and plasticisation functions of the components. The proposed expressions make it possible to clarify the concentration dependence of the initial aggregation rate, its order, and also to quantify the effect of the dose of UV irradiation on the aging process of the system. The experimentally obtained result means that an increase in the content of a-crystallin leads to an additional blocking of hydrogen bonds in the surface layers of a-lactalbumin and, accordingly, to an increase in the plasticising effect. In addition, the obtained expression of the activation energy of polymer chain rearrangement helps to account for the influence of infrared radiation on the development of so-called thermal cataracts (usually occurring in glassblowers, steelmakers, blacksmiths, welders, etc.), when the etiological factor consist in infrared rays having wavelengths from 0.74 to 2.50 microns, which freely pass through the cornea and iris without damaging them, and are largely adsorbed by the lens, causing its overheating.
Keywords: a-lactalbumin, crystallin, binary system, aggregation order, plasticisation functions, transition temperatures
Information about the article: Received August 16, 2019; accepted for publication May 29, 2020; available online June 30, 2020.
For citation: Matveev YuI, Averyanova EV. On aggregation in binary biopolymer systems. Izvestiya Vuzov. Prikladnaya Khimiya i Biotekhnologiya = Proceedings of Universities. Applied Chemistry and Biotechnology. 2020;10(2):223-231. (In English) https://doi.org/10.21285/2227-2925-2020-10-2-223-231
Об агрегации в бинарных биополимерных системах
Ю.И. Матвеев*, Е.В. Аверьянова**
*Институт биохимической физики им. Н.М. Эммануэля РАН, г. Москва, Российская Федерация
**Бийский технологический институт (филиал) ФГБОУ ВО «Алтайский государственный технический университет им. И.И. Ползунова», г. Бийск, Российская Федерация
Резюме: Основным структурным белком, на долю которого приходится около 50 % массы всех водорастворимых белков хрусталика человеческого глаза, является a-кристаллин. Одна из основных функций a-кристаллина - шаперонная, обуславливающая необновляемость белков в процессе жизнедеятельности. a-Кристаллины частично или полностью стабилизируют развернутые белки, препятствуя образования осадка, что способствует сохранению прозрачности хрусталика и снижению риска возникновения ряда заболеваний, в том числе катаракты. С другой стороны, это явление можно рассматривать в рамках материаловедения при решении вопроса замедления процессов старения полимеров. В данном исследовании на примере бинарной системы а-лактальбумин-аА-кристаллин рассмотрены способы замедления процесса агрегации a-лактальбумина в растворе. С этой целью экспе-
риментальные данные по скорости изменения процесса агрегации были формализованы, то есть выражены через температуры перехода и функции пластификации компонентов. Предложенные выражения позволяют уточнить концентрационную зависимость начальной скорости агрегации, ее порядок, а также количественно оценить влияние дозы УФ-облучения на процесс старения системы. Физически полученный результат означает, что повышение содержания а-кристаллина приводит к дополнительному блокированию водородных связей в поверхностных слоях а-лактальбумина и, соответственно, к увеличению пластифицирующего эффекта. Кроме того, полученное выражение энергии активации перестроения цепи полимера позволяет учесть влияние ИК-излучения на развитие так называемой тепловой катаракты (обычно имеет место у стеклодувов, сталеваров, кузнецов, сварщиков и людей других профессий), когда этиологическим фактором являются ИК-лучи с длиной волны от 0,74 до 2,50 мкм, которые свободно проходят через роговую и радужную оболочки, не повреждая их, и в значительной степени адсорбируются хрусталиком, что приводит к его перегреву.
Ключевые слова: а-лактальбумин, кристаллин, бинарная система, порядок агрегации, функции пластификации, температуры перехода
Информация о статье: Дата поступления 16 августа 2019 г.; дата принятия к печати 29 мая 2020 г.; дата онлайн-размещения 30 июня 2020 г.
Для цитирования: Матвеев Ю.И., Аверьянова Е.В. Об агрегации в бинарных биополимерных системах. Известия вузов. Прикладная химия и биотехнология. 2020. Т. 10. N 2. С. 223-231. https://doi.org/ 10.21285/2227-2925-2020-10-2-223-231
INTRODUCTION
Binary biopolymer systems, such as proteinprotein and protein-polysaccharide formations, are widespread in nature. Moreover, processes occurring in such systems under the influence of various external factors are of practical interest not only for the biological systems themselves as an example of self-regulation, but also for binary systems based on synthetic polymers.
An analytical description of such processes is of significant interest both in the case of medicine (for example, in the development of drugs to slow the development of cataracts), and in the case of polymer materials science when attempting to slow down processes leading to degradation of materials.
Typically, the development of aging processes is influenced by many factors (heat, light, penetrating radiation, oxygen, moisture, aggressive chemicals, mechanical stresses) that cause two types of irreversible chemical reactions in polymers: destruction, when bonds in the main chain of macromolecules break, and structuring when chain stitching occurs. A change in the molecular structure leads to changes in the properties of the polymeric material: elasticity is lost, rigidity increases, dielectric parameters deteriorate, etc. [1].
Here we will focus on the action of only one factor, namely light (photochemical destruction), when the destruction of macromolecules occurs under the influence of ultraviolet (UV) rays having a wavelength A < 400 nm. Polymers containing atomic groups or individual polar groups capable of absorbing light are particularly intensively degrade. In the case of proteins, these are amino acids tryptophan (W) and tyrosine (Y) [2]. The choice of the protein-protein system of the eye lens as an example was motivated by the significant accumulation of experimental material ob-
tained during studies of the aging of this system (development of cataracts) [3, 4], which can be formalised in the framework of polymer physics.
One of the natural mechanisms of inhibition of cataract development in the human eye is associated with the presence of the a-crystallin protein in the lens, which performs a molecular cha-perone role - in this case, preventing the aggregation of ^-crystallin and thereby preventing the development of cataracts [4]. In this case, the a-crystallin molecules permeate into a ^-crystallin structure that has denatured under the influence of UV radiation, permeating it and reducing the ^-crystallin thermal aggregation rate.
The influence of infrared radiation can be seen in the development of so-called thermal cataracts typically occurring in glassblowers, steelmakers, blacksmiths, welders, etc. Here, the etiological factor consists in infrared rays having wavelengths from 0.74 to 2.50 microns, which freely pass through the cornea and iris without causing damage to them, but are then largely adsorbed by the lens, leading to overheating. An increase in the temperature of the lens in turn leads to an increase in the rate of thermal aggregation of ^-crystallin.
EXPERIMENTAL PART
An analytical description of the effect of a^-crystallin on aggregation of £L-crystallin and the effect of UV irradiation on initial adsorption ability of a-crystallin were examined in detail on the example of the a-lactalbumin-a^-crystallin model system. The results of experimental studies given in [5, 6] were used as the initial data together with information on the amino acid composition of the proteins of the system under consideration. a-Lactalbumin has a molecular weight of about 14 000 g/mol; its molecule is comprised of a sin-
gle polypeptide chain consisting of 123 amino acid residues and containing four disulphide bonds. The predominant amino acids of a-lactalbumin are, mg / g: aspartic acid - 18.7; glutamic acid -12.9; leucine -11.5; lysine - 11.5; tryptophan -7.0; isoleucine - 6.8; cystine - 6.4 etc. The eye lens protein a^-crystallin, consisting of 173 amino acid residues (AAR), belongs to the family of small heat shock proteins (sHSPs), having a molecular weight of about 20 000 g/mol.
RESULTS AND DISCUSSION
Refinement of the concentration dependence of the initial aggregation rate and the physical meaning of the parameters included therein
An empirical expression for determining the initial aggregation rate of v as a function of x (phase concentration ratios of a-crystallin and a-lactalbumin) was proposed in [5]:
( v Y/n
= 1 - AC0x,,
(1)
where vo - is the initial aggregation rate when x = 0; n - the aggregation order, which according to [7] is equal to 5.3±0,3; AC0 - initial chaperone adsorption capacity; x = [a-crystallin]/[a-lactal-bumin], [a-crystallin] and [a-lactalbumin] - molar concentrations of a^-crystallin and a-lactalbumin in a solution. In the study of anti-aggregation activity of a^-crystallin the authors of [6] obtained the value ACo = 1.18 with [a-lactalbumin] = 0.5 mg/ml.
In order to establish the dependence of the coefficient AC0 on the characteristic temperatures of the polypeptides of the system a^-crystallin and a-lactalbumin, we use the theory of absolute reaction rates [8] in calculating v:
v = -
kT
iTtfl
exp
AS *
T
exp
AH *
~~k.T
(2)
The values of AS and AH represent changes in the system's entropy and enthalpy during the transition of the rotational polymer isomers (a-lactalbumin and a^-crystallin) into an activated state.
Let us determine the relative concentration of a-crystallin Ca-cryst in the original system as
с
[a — cryst]
[a — cryst] + [a — lact ] 1 + x
(3)
and Ca-lact - like
с
1 + x
(4)
Then the activation enthalpy of the complex a-crystallin and a-lactalbumin AH *cp/ can be represented as
1
* -I *
AH cpi = 1 + x ДНП_,Я„+ 1 + x AH
a-lact
a-cryst.
(5)
Assuming that the changes in the activation entropy of a-crystallin and a-lactalbumin are the same and using expression (2), equation (1) in general form can be written as follows:
exp
(vK)1/n =
or
( AH*,-AH* , Л
cpl a—lact
nkT
(1—y)
exp I у (v/vo)1/n = ^ 1 +x
AH *
(6)
(7)
where y = nkT y = AH aWAH a-
-cryst.
Given that 1 - y << 1, the dependence of
1/n
(v/vo) from x will be non-linear:
= 1 — AC„
where
' 1 + x
ACo = Y (1 - y).
(8) (9)
From expression (9) it follows that for xmax ~5.5 ACo = 1.18, (v/vo)1 /n ~0, i.e. the aggregation process is completely stopped. Accordingly, the dependence (v/vo)1/n = f(x) will differ from the linear (1) already for x = 0.35 and 0.5 (see. Fig. 6 in [5]). Dependence (1) can be obtained from expression (9) when x << 1.
Unfortunately, in [5], measurements for x = 0.5 and x = 0.6 were not performed; however, measurements for x = 0.7 show that in this case
1/n
(v/vo) = 0, i.e. the aggregation process ends earlier Approximating the function (v/vo)1n = f(x) at three points (for x = 0.14, x = 0.35 and x = 0.7), estimate its values at the points x = 0.5 and x = 0.6. The obtained values are given in Table 1. The experimental values are shown in bold type and the corrected corresponding values are indicated in brackets (for x = 0.6) when approximating the Pl(W, x) quadratic parabola.
To identify the causes of the discrepancy of function (9) with constant ACo from the experimental dependence, we will dwell in more detail on expression (8). It was shown in the work [9] that the activation energy of the rearrangement of the polymer chain can be expressed in terms of its glass transition temperature. In this case,
x
1
Values of the plasticisation function Pl(W) for 0.35 <x < 0.7 Значения функции пластификации Pl(W) при 0,35 < x < 0,7
Table 1 Таблица 1
X 0.140 0.349 0.500 0.600 0.701
(vN0)yn 0.840 0.661 0.520 0.400 (0.467) 0.002
PI(W, x) 1.000 1.000 0.989 0.982 (0.971) 0.947
const • (T )
Y = -i-— , and y = (Tg )a-lact /(Tg )a-crist
nT
and the expression (8) takes the following form:
const • (T )
ACo = -(1 - (Tg)a,act /(Tg)a.c„st) (10)
nT
It follows from expression (10) that, in the case of the model system under consideration, y - const. However, y may change when switching from a model system to a real one. If in the model system a decrease (Tdn)a_iact is achieved due to the plasticising effect of dithiothreitol (DTT) [6], then with UV irradiation of pL-crystallin, plasti-cising effect is driven by the destruction of tyro-sine and tryptophan due to photoionisation and subsequent formation of radicals and solvated electron. Therefore, in the case of a model system
y = (Tgo)a-aact Pl(W /(Tg) a-crist ,
where Pl(W) - is the plasticisation function, which depends on the content of DTT in the solution -W; Pl(W) = (Tg) a-lаct/(Tgo)a-lаct, (Tgo)a- act - glass transition temperature of a-lactalbumin when DTT is absent; (Tg)a-lact - glass transition temperature of a-lactalbumin at a concentration of DTT - W. In the case of a real system, the plasticisation function will depend on W and radiation doses D, i.e. Pl(W, D).
From the work [10] it follows that in the case of egg and milk proteins, which include the protein system studied in [5], the ratio Tg/ Tdn can be considered as a constant value, slightly dependent on protein type. Therefore, in expression (10), the ratio (Tg)a-lact / (Tg)a.crist can be written
as (Tg)a-lact / (Tg) a-crist = = (Tdn)a-lact / (Tdn) a-crist.
Below, we perform a series of numerical estimates for the model system a-lactalbumin-aA crystallin in order to determine constants in expression (10). In Table 2 and 3 amino acid compositions of a^-crystallin and a-lactalbumin and the expressions used to evaluate their denatura-tion temperatures are presented. Since the considered proteins have a below-critical degree of polymerisation, this fact was considered using the expressions proposed in [11]. The performed estimates give (Tdn)exp = 63 °C, Tg = 46 °C for a^-crystallin, (Tdn)ca/ = 47 °C - for a-lactalbumin,
and const = 121.5 for n = ~5 (it will be shown below that for the systems under consideration
1 / n
n = 5.5). However, for calculating (v/v o) the main thing to know is the ratio const / n in expression (10).
Thus, in the case of the considered model system, expression (9) will take the following form:
I —I = 1 - 24.85(1 - 0.9525Pl(W, x))—.
Using expression (11) and the values (v/vo)1 / n and x given in the table 1, we obtain the values of the plasticisation function Pl (W) with the corresponding values of x (see Table 1). The function Pl(W, x) will be approximated by a quadratic parabola: Pl(W, x) = 1 - ax - bx2. The coefficients a and b can be found from the values of Pl (W, x) for x= 0.5 and x= 0.7. As a result, the dependence Pl (W, x) will take the following form:
Pl(W,x) = 1 + 0.113x - 0.27x2.
Correction Pl(W, x) for x = 0.6 gives Pl(W, x) = 0.971.
The empirically-obtained result means that an increase in the content of a-crystallin leads to an additional blocking of hydrogen bonds in the surface layers of a-lactalbumin and, accordingly, to an increase in the plasticising effect. In this case, plasticisation occurs in the area of action of the Zhurkov mechanism [12], i.e. due to blocking of hydrogen bonds.
Similarly, if the dependence (v/vo)1/n= f (x) it is possible to perform estimates for the system a- and pL -crystallins irradiated with UV. It should be noted that x = 0.7 is the limit value xiim, which does not depend on the model substance, since all measurements are carried out at physiological temperature, the molecular chaperone properties of a-crystallin are studied.
On the reason of the temperature decrease of p-crystallin denaturation and the initial adsorption ability of the chaperone AC° under UV irradiation Under UV irradiation of p-crystallin, two amino acids (tryptophan and tyrosine) are exposed, which p-crystallin contains significantly more than a-crystallin. The resulting amino acid radical interacts with neighbouring peptide chains of the
Parameters of aA-crystallin Параметры aA-кристаллина
Table 2 Таблица 2
AAR ni AV,, AJ n,AV, A3 Ф, Tdni, K Ф/ / Tdni
M 2 117.1 234.2 0.0132 527 2.500-10"5
S 19 72.1 1369.9 0.0772 667 11.570• 10"5
Q 5 111.9 559.5 0.0315 408 7.720-10"5
A 6 64.7 388.2 0.0219 1141 1.920-10"5
K 8 127.4 1019.2 0.0574 282 20.350- 10-5
T 9 89.1 801.9 0.0452 447 10.110 • 10-5
V 10 98.8 988.0 0.0557 410 13.584- 10-5
P 10 87.2 872.0 0.0492 452 10.880- 10"5
G 9 47.6 428.4 0.0242 898 2.690-10"5
D 15 90.7 1360.5 0.0767 814 9.420-10"5
L 14 115.9 1622.6 0.0915 347 26.370- 10-5
I 9 115.9 1043.1 0.0588 347 16.940- 10-5
E 10 107.8 1078.0 0.0608 529 11.490- 10-5
F 14 140.4 1965.6 0.1108 519 21.350- 10"5
W 1 170.1 170.1 0.0096 432 2.220-10"5
N 2 94.8 189.6 0.0107 521 2.050-10"5
H 7 119.5 836.5 0.0472 670 7.045-10"5
R 12 146.6 1759.2 0.0992 277 35.810- 10-5
Y 6 147.0 882.0 0.0497 457 10.870- 10-5
C 2 82.2 164.4 0.0093 1147 0.811-10-5
Note. I nA Vi = 17532,9A ; A V, - van der Waals volume of the i-th amino acid residue; n, - the number of amino acid residues of the i-th type; ф = nA V/I nA Vi; Tdn-1 = I tyiTdni1; Tdn = Tm« = 443 K = 170 оС; N = I ni = 174 - degree of polymerisation; critical degree of polymerisation Nc = 346. (Tdn)exp = 63 оС. If using the expression Tdn= Tm, „ - k • Nc/ Nnuc, where k = 20.8, we define Nnuc globules aA- crystallin, Nnuc = 67, and accordingly the degree of polymerisation of the edge Ncon = 107.
Примечание. InAV, = 17532,9A3; AV,- ван-дер-ваальсовый объем i-го аминокислотного остатка; n, - число аминокислотных остатков i-го типа; ф, = nAV/I nAVi; Tdn-1 = I tyTdni1; Tdn = Tm« = 443 K = 170 оС; N = I n, = 174 - степень полимеризации; критическая степень полимеризации Nc = 346. (Tdn)exp = 63 оС. Если с помощью выражения Tdn = Tmr - k Nc/Nnuc, где k = 20,8, определим Nnuc глобулы aA-crystallin, Nnuc = 67, и соответственно степень полимеризации опушки Ncon = 107.
Parameters of a-lactalbumin Параметры a-лактальбумина
Table 3 Таблица 3
AAR n, AV, A3 nAV, A3 Ф, Tdni, K Ф/ / Tdni
M 3 117.1 234.2 0.0132 527 2.500-10"5
S 9 72.1 1369.9 0.0772 667 11.570- 10"5
Q 7 111.9 559.5 0.0315 408 7.720-10"5
A 5 64.7 388.2 0.0219 1141 1.920-10"5
K 12 127.4 1019.2 0.0574 282 20.350- 10-5
T 8 89.1 801.9 0.0452 447 10.110- 10-5
V 8 98.8 988.0 0.0557 410 13.584- 10"5
P 2 87.2 872.0 0.0492 452 10.880- 10"5
G 7 47.6 428.4 0.0242 898 2.690-10"5
D 13 90.7 1360.5 0.0767 814 9.420-10"5
L 17 115.9 1622.6 0.0915 347 26.370- 10-5
I 9 115.9 1043.1 0.0588 347 16.940- 10-5
E 7 107.8 1078.0 0.0608 529 11.490- 10"5
F 6 140.4 1965.6 0.1108 519 21.350- 10"5
W 4 170.1 170.1 0.0096 432 2.220-10"5
N 8 94.8 189.6 0.0107 521 2.050-10"5
H 4 119.5 836.5 0.0472 670 7.045-10"5
R 1 146,6 1759.2 0.0992 277 35.810- 10-5
Y 4 147.0 882.0 0.0497 457 10.870- 10"5
C 8 82.2 164.4 0.0093 1147 0.811-10"5
Note. I nAV, = 17732,7A3; Tdn = Tm,«. = 457 K = 184 "С; N = I n: = 142 - degree of polymerisation; Nc = 360; NnUc = (Nnuc)aA-crys ■ (N) a-iact/(N)aA-crys = 54; Ncon = 88. If using the expression Tdn= Tm « - k ■ Nc/ Nnuc, where k= 20.8, we define (Tdn)cai globules a-lactalbumin, (Tdn)cai = 47°С.
Примечание. I nAV, = 17732,7A3; Tdn = Tm,« = 457K = 184 оС; N = I n, = 142 - степень полимеризации; Nc = 360;
Nnuc = (Nnuc)aA-crys . (N) a-lact/(N)aA-crys = 54; Ncon = 88. Если С ПОМОЩЬЮ выражения Tdn = делим (Tdn)cal глобулы a-лактальбумина, (Tdn)cai = 47 ОС.
Tmr - k Nc/Nnuc, где k = 20,8, опре-
protein molecule . As a result, it is possible to form ring structures along the macromolecule chain, which will be a mixture of ring and linear sections with a significantly lower degree of polymerisation (the number of amino acid residues) than the initial chain. The latter will lead to the temperature of denaturation of such a chain decreasing. Therefore, by selecting a dose of radiation, it is possible to achieve the transition of ^-crystallin to a disordered state at physiological temperature. This is the difference between the conversion of a-lactalbumin to a disordered state (this occurs due to plasticisation of DTT) from ^-crystallin.
Moreover, since a-crystallin also contains tryptophan and tyrosine, although in smaller quantities than ^-crystallin, UV radiation can affect (Tg )a-crist through its decline on the basis of the mechanism mentioned above. The latter is confirmed by the data given in [5]. The dependence found in [5] is well-described using equation (10) if instead of (Tg )a-cnst write (Tg )a-crist Pl (D) (hereinafter the meaning of Pl (D) will be clarified).
The value of AC0/AC0inact = 0 corresponds to the condition (Tg)a-iact/(Tg)a-crist = 1. From this the glass transition temperature of irradiated a-crystallin can be determined, which helps to calculate the change in its effective degree of polymerisation upon the formation of ring structures [11].
Since, upon irradiation, there is a decrease in (Tg)a-crist, then introducing the plasticisation function Pl(D) = (Tg )a-crist/Tg )a-crist,inact , where D is the dose of UV irradiation; Pl(D) <1, and using expression (10), AC0 /ACoinact can be written as follows:
ACo/ACo nc = Pl(D) f 1-0-9525P/(W,x)/PKP)>| ^ g) ' ^ 1-0.9525Pl(W, x) J
From expression (11a), the dependence Pl(D) can be determined from the experimental values of ACo/ACoinact given in [4]. In Table 4 the calculated values of Pl (D) are given. For Pl (D*) = 0.9525, where D* = 32.5 J/cm, Pl(W, x)= 1, the value ACo/ACc,Jnact = 0, a change in the effective degree of polymerisation of the chain of a-crystallin AN under UV irradiation will be ~ 4 units (per 4 amino acid residues). When evaluating AN, the Fox - Flory equation was used, which is usually used to calculate Tg polymers with a degree of polymerisation below critical. Most of the proteins are polymers of this kind, including a- and ^-crystallin.
Approximation of the data presented in Table 4 gives the following dependency:
P(D) = 1 - 0.244D + 0.169D2.
It should be noted that while an increase in the content of a-crystallin leads to a decrease in the rate of aggregation, then UV irradiation leads to inhibition of this process (an increase in the aggregation rate). Thus, considering the effect of irradiation, equation (11) can be written as follows:
y/n
V| = 1 -24.85PI(D)(1 -0.9525Pl(W±x) X . (11b) Vo J V A Pl(D) 1 + x
Determination of the aggregation order by calculation
Equation (1) includes the aggregation order parameter n, which, according to [7], is equal to n = 5.3±0.3. Show that the found value n can be estimated using the data of A.N. Kolmogorov cited in [13], where was first proposed the equation that describes the kinetics of crystallisation of metals, which later became widely used in the description
Table 4
The values of the plasticisation function Pl(D ) for а-crystallin depending on the dose of UV-irradiation at 0 <x < 0,15
Таблица 4
Значения функции пластификации Pl(D) для а-кристаллина в зависимости от дозы D УФ-облучения при 0 < x < 0,15
Parameter Value
D 0.038 0.077 0.154 0.231 0.308 0.407
ACo!ACojnact 0.83 0.70 0.35 0.40 0.20 0.06
Pl( D ) 0.987 0.977 0.951 0.955 0.940 0.929
Note In the considered range of x, the function Pl(W, x) = 1, D = D/D*.
Примечание. В рассматриваемом диапазоне изменения x функция Pl(W, x) = 1, D = D/D*.
''Владимиров Ю.А., Потапенко А.Я. Физико-химические основы фото-биологических процессов: учеб. пособие для мед. и биол. спец. вузов. М.: Высш. шк., 1989. 199 с.
of nucleation processes of various structures in food, biological and pharmaceutical materials [14].
Usually in the modern notation [14] the equation of A.N. Kolmogorov, which abroad is called the Avrami equation, has the following form:
1-a = exp(-Kctn ),
(12)
where a is the degree of structural order (crystal-linity) of the solution; Kc is speed constant; n is a parameter characterising nucleation (in our case, the aggregation parameter).
A.N. Kolmogorov wrote the right side of
equation (12) asexp(—— c Q),, while
Q = j a(t')(jk (r)dr)3 dt',
(13)
where k(t) is the rate of increase of the crystallised mass; a(t) is the probability of formation per unit volume unit time interval of one crystallisation centre [13].
A.N. Kolmogorov showed that, given the various functions a (t) and k (t), one can obtain Q = const f . Two cases were considered:
- when a(t) and k(t) do not depend on time, i.e. a(t)=a= const, and k(t)= 1, and then f ~t4;
- when all crystallisation centres are formed at the beginning, and then f ~t3.
In our case, the decrease in the rate of aggregation of a-lactalbumin occurs due to the capture of molecules of a-lactalbumin by molecules of a-crystallin when they converge and mutually penetrate each other. In this case, k(t) will be
proportional to the size of the diffusion zone of their mixing, i.e. t1/2 [15].
For a(t)=a=const, and k(t)~t1/2 from equation (13) we obtain tn ~t5 5 , i.e. in equation (1) n= 5.5, which, within the limits of measurement accuracy, corresponds to the data of [7].
CONCLUSIONS
Thus, as a result of the transformations of the equation for calculating the initial aggregation rate of a-crystallin, the physical meaning of the coefficients included in it was revealed, which allows us to determine ways to slow down the aggregation process in the binary system a-lactalbumin-a^-crystallin.
Due to the fact that a-crystallin has a high degree of polydispersity, both aggregates of various sizes and individual molecules of a-crystallin will be present in its solution. It was shown that the enhancement of the chaperone-like action of a-crystallin is achieved either by increasing the concentration of individual molecules with an increase in its total content in the system, or by external action on aggregates of a-crystallin, causing them to decompose into individual molecules. This property can be used in the development of drugs that slow cataract development, either by selecting substances that contribute to the decomposition of a-crystallin aggregates, or by causing plasticisation of ^-crystallin and increasing initial adsorption capacity of chaperone.
The calculation equation of the initial molecular chaperone adsorption capacity (10) also allows us to consider the influence of infrared radiation on the development of the so-called thermal cataract typically occurring in glassblowers, steelmakers, blacksmiths, welders, etc.
0
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Contribution
Yuri I. Matveev, Elena V. Averyanova carried out the experimental work, analyzed the experimental results and prepared the text of the manuscript. Yuri I. Matveev, Elena V. Averyanova have equal author's rights and bear equal responsibility for plagiarism.
Conflict of interests
The authors declare no conflict of interests regarding the publication of this article.
The final manuscript has been read and approved by all the co-authors.
INFORMATION ABOUT THE AUTHORS
Yuri I. Matveev,
Cand. Sci. (Physics and Mathematics), Senior Scientist,
Emanuel Institute of Biochemical Physics Russian Academy of Sciences, 4, Kosygin St., Moscow, 119334, Russian Federation, e-mail: [email protected]
Elena V. Averyanova,
Cand. Sci. (Chemistry), Associate Professor, Department of Biotechnology, Biysk Technological Institute (branch) of the Altay State Technical University, 27, Geroi Sovetskogo Soyuza Trofimov St., Biysk, 659305, Russian Federation, e-mail: [email protected]
Technical Report) // Pure and Applied Chemistry. 2011. Vol. 83. Issue 8. P. 1567-1617.
15. Чалых А.Е. Диффузия - метод исследования полимерных систем // Высокомолекулярные соединения. Серия С. 2001. Т. 43. N 12. С.2304-2328.
Критерии авторства
Матвеев Ю.И, Аверьянова Е.В. выполнили экспериментальную работу, на основании полученных результатов провели обобщение и написали рукопись. Матвеев Ю.И, Аверьянова Е.В. имеют на статью равные авторские права и несут равную ответственность за плагиат.
Конфликт интересов
Авторы заявляют об отсутствии конфликта интересов.
Авторы прочитали и одобрили окончательный вариант рукописи.
СВЕДЕНИЯ ОБ АВТОРАХ
Матвеев Юрий Игнатьевич,
к.ф.-м.н., старший научный сотрудник, Институт биохимической физики им. Н.М. Эммануэля РАН, 119334, г. Москва, ул. Косыгина, 4, Российская Федерация, e-mail: [email protected]
Аверьянова Елена Витальевна,
к.х.н., доцент кафедры биотехнологии, Бийский технологический институт (филиал) ФГБОУ ВО «Алтайский государственный технический университет им. И.И. Ползунова», 659305, г. Бийск, ул. им. Героя Советского Союза Трофимова, 27, Российская Федерация, e-mail: [email protected]
