NANO EXPRESS Open Access

A Method for Decomposition of the BasicReaction of Biological Macromolecules intoExponential ComponentsYu. M. Barabash* and A. K. Lyamets

Abstract

The structural and dynamical properties of biological macromolecules under non-equilibrium conditionsdetermine the kinetics of their basic reaction to external stimuli. This kinetics is multiexponential in nature.This is due to the operation of various subsystems in the structure of macromolecules, as well as the effectof the basic reaction on the structure of macromolecules. The situation can be interpreted as a manifestationof the stationary states of macromolecules, which are represented by monoexponential components of the basicreaction (Monod-Wyman-Changeux model) Monod et al. (J Mol Cell Biol 12:88–118, 1965). The representation of

multiexponential kinetics of the basic reaction in the form of a sum of exponential functions A tð Þ ¼X

i¼1

naie

−ki tÞ�

is a multidimensional optimization problem. To solve this problem, a gradient method of optimization with softwaredetermination of the amount of exponents and reasonable calculation time is developed. This method is used to analyzethe kinetics of photoinduced electron transport in the reaction centers (RC) of purple bacteria and the fluorescenceinduction in the granum thylakoid membranes which share a common function of converting light energy.

Keywords: Multidimensional optimization, Decomposition of the curve, Exponential function, Electron-conformationalstates of biological macromolecules, Multi-component structure, Reaction center, Purple bacteria, Electron transport,Fluorescence of chlorophyll, Configuration variability

BackgroundIt is known that the kinetics of electron transport in theRCs of purple bacteria [1–3] and the shape of the curve ofchlorophyll fluorescence in the complex of photosystem IIin thylakoid grana are multiexponential in nature [4]. As arule, this is explained by a multilevel organizationalstructure of macromolecules and a manifestation of self-regulation effect on the structural and dynamic propertiesof biological macromolecules in non-equilibrium operat-ing conditions. This effect is determined by certain hiddenparameters among which structural variables thatcharacterize the state of macromolecules can act. Whilethese parameters are difficult to measure directly inexperiment, they impact the kinetics of basic reaction inmacromolecules along with control parameters (substrateconcentration, light intensity, etc.). Due to changes in the

structure of macromolecules, a feedback in the basic reac-tion occurs. This feedback links the recurring elementaryacts of the basic reaction. The elementary acts lead to sig-nificant changes in the structure of macromolecules. Theymodify the very elementary reaction and provide adaptivefunctioning of the macromolecules. The effect of self-regulation can occur with different amounts of structuralvariables. These factors make it difficult to analyze thebasic reaction, and the kinetics of this reaction assumes amultiexponential character. Therefore, there is a problem,using the experimentally measured kinetics of basic reac-tion, to identify the stationary states and self-consistent re-organizations of macromolecules, which are reflected bymonoexponential components of the basic reaction.

MethodsMultidimensional Optimization MethodThe representation of multiexponential kinetics curve forthe basic reaction in the form of a sum of exponential

* Correspondence: ybar@ukr.netInstitute of Physics, National Academy of Science of Ukraine, 46, Nauky Ave,Kyiv 03039, Ukraine

© The Author(s). 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Barabash and Lyamets Nanoscale Research Letters (2016) 11:544 DOI 10.1186/s11671-016-1758-1

terms A tð Þ ¼X

i¼1

naie

−kitÞ�

with restriction (ki > 0,

ai > 0), is a multidimensional optimization problem withidentification difficulties. There are various optimizationalgorithms (gradient, evolutionary, stochastic methods,etc.) which have different sorting strategies. The choice ofalgorithm is related to the ambiguity of the behavior ofobjective function, the existence of local extrema of ob-jective function, and the curse of dimensionality (obtainingquality results within a reasonable time) [5, 6].To analyze the kinetics of photoinduced charge trans-

port in the RCs of purple bacteria and induce the fluor-escence in the granum thylakoid membranes, a gradientoptimization algorithm using the methods of linear alge-bra was developed. This algorithm allows to determinethe number of exponents and their parameters whichcorrespond to the global minimum of the objective func-tion expansion (root-mean-square error). It also allowsexcluding the impact of initial values of parameters andstep size of their changes on the results of calculation.The algorithm does not require searching the values ofexponential weights (ai) during optimization that pro-vides a reasonable time of calculations.The algorithm includes two reciprocal steps. At the first

step, the problem of determining the numerical represen-tation (ai) of a given decomposition basis with the best ap-proximation to the analyzed curve is solved. At thesecond step, an optimization of the parameters of decom-position basis (n, ki, (ai)) is carried out. Let us briefly con-sider the first step. One can assume that the analyzedsignal x(t) with a finite energy belongs to a metric spacewith the scalar product (x, y) = ∫x(t)y(t)dt. The signal x(t)is uniquely (best) represented as a linear combination [7]

x tð Þ ¼X

i¼1

nαiφi tð Þ; ð1Þ

where φi(t) is a system of n linearly independent functions,and αi form the desired numerical representation x(t).This basis does not need to be orthogonal. Let us multiplyx(t) by φj(t). Then the relation between the signal x and itsnumerical representation a in a matrix form is

φ1;φ1ð Þ φ2; φ1ð Þ… φn; φ1ð Þφ1;φ2ð Þ φ2; φ2ð Þ… φn; φ2ð Þ

:::

φ1; φnð Þ φ2; φnð Þ… φn; φnð Þ

26666664

37777775

a1a2:::an

26666664

37777775¼

x; φ1ð Þx; φ2ð Þ:::

x; nð Þ

; Ga ¼ β;; ð2Þ

Let us introduce reciprocal basis functions θi(i = 1, 2,3,…, n) which are represented by linear combina-

tions φi θj� � ¼

Xk¼1

nΓ jkφk tð Þ; . The scalar products of the

bases equal φi; θj� � ¼ δij; i:e

Xk¼1

nΓjk φi;φkð Þ ¼ δij;

Γ � G ¼ I→Γ ¼ G−1� ��

in matrix form. Using a commonbasis, it is possible to rewrite (2) in the following form:

αi ¼ x; θið Þ; i ¼ 1;2;…;n ð3Þ

where αi are the required values of exponential weights.Therefore, at the first step the number (n) and discrete

values of decrements (ki) of the basic exponential func-tions φk are specified, and a matrix of their scalar prod-ucts G is formed. Then we calculate the inverse to Gmatrix Г and determine their reciprocal basis θj. Usingthe expression (3), we find the values of the weights ofexponential functions αi, which best fit the analyzed sig-nal. According to the array of possible decrements, thedecrements of exponential functions (ki) are discrete innature. This leads to errors in the optimization, whichcorrespond to the difference between the left and rightside of the expression (1), i.e. between the analyzed sig-nal and its approximation.Now the first step is finished. At the second step we

find the optimal parameters of the decomposition basis.We select one exponent with an initial value of decre-ment and, using the operations of the first step, go overthe decrement values k1 and determine the optimalvalues of the parameters (α1opt, k1ont, a1 > 0) of the firstexponent, which provides the minimum error (disper-sion) of approximation. Then we add the second expo-nent and use the first step of optimization. At fixedvalue of the first exponent, k1opt, we examine k2 anddetermine the optimal values for two exponents(a1opt, k1ont, a2opt, k2ont). After that we go back to thefirst exponent and at fixed value of the second exponent,k2opt go over k1 and determine new optimal values of bothexponents. Then we return to the second exponent and atnew value of k1opt examine k2 and determine new valuesof both exponents. The procedure is repeated until thedispersion of approximation stops to change. Then weadd the third exponent and determine k3opt for it withfixed decrements of previous two exponents, k1opt andk2opt. After that we repeat the above process ofoptimization for two exponents with fixed value of thethird exponent k3opt. Then we adjust k3opt for new valuesof k1opt and k2opt. Next, we adjust k1opt and k2opt at fixedvalue of k3opt. The procedure is repeated until the disper-sion of approximation stops to change. Then we add thefourth exponent, and so on. The optimization processends when the next set of the components of decompos-ition at the next step will have greater dispersion than theprevious set, and the latter will be used.Computer software for the above algorithm has been

developed. The program menu is shown in Fig. 1a. Theanalyzed signal should be in text format in two columns,with the point as decimal mark. The first column in-cludes time (X axis) in seconds, and the second columncontains signal amplitude in arbitrary units. The signalamplitude should continuously reduce with time. To runthe program, one has to enter in the menu the maximal

Barabash and Lyamets Nanoscale Research Letters (2016) 11:544 Page 2 of 5

number (<8) of decomposition exponents and clickStart/continue. As a result, the directory with filesappears.If the original file has correct format, the decomposition

of the initial signal into exponential components starts.After that the parameters of exponents will appear in atable. The decomposition results can be saved in two textfiles: (i) as an approximation signal for visual control, bymeans of Save Ai*Exp(-Ki) button, or (ii) as a table of ex-ponential parameters, by clicking Save Ai = f(Ki). Figure 1bshows the normalized plot of the kinetics of absorption re-covery of RC solution after switching off the excitationlight (reference signal) and the approximation curve forthe experimental signal. A good correlation between theexperimental signal and its decomposition into three com-ponents (dispersion = 0.00015) is observed. The accuracyof decomposition is ±2.5%.Figure 2 shows a characteristic behavior of the object-

ive function of approximation of electron transportkinetics in RC for different modes of photoexcitation.This character of the objective function allows theoptimization without the risk of falling into a local mini-mum, and significantly reduces the computation time(to 5 min).

Results and DiscussionProblem Solving and DiscussionThe developed software has been tested for an analysisof the kinetics of electron transport in various modes ofRC photoexcitation and induction of chlorophyll fluores-cence in four leaves of winter wheat.The RCs were studied on wild-type isolated bacteria

Rhodobacter sphaeroides. The chromatophores ofbacteria were solubilized by detergent LDAO (lauryldi-methylamine N-oxide), and RCs were separated fromother membrane components by means of hydroxyapa-tite column chromatography. The isolated RCs wereslurry suspended in a 0.01-M Na-P buffer, pH 7.2, with0.05% LDAO [8]. The RC absorption kinetics wasstudied by means of exposure of RCs to light pulses ofdifferent durations and intensities. The results were sat-isfactorily approximated by 2 ÷ 4 exponential functions.Although the absorption kinetics corresponded to the

kinetics of electron transport, the achievement of dark ab-sorption values during RC relaxation after turning off thelight did not guarantee the end of the structural changes inRCs. In this regard, the RC absorption kinetics was studiedunder photoexcitation by two successive light pulses withduration of 100 s and intensity of 7 mW/cm2; the intervalbetween the pulses was varied from 0 to 2500 s (Fig. 3).The present studies showed that at intervals between RC

photoexcitation pulses smaller than 40 s the weight a1 of fastcomponent of RC recovery increased, the weight a2 of inter-mediate component decreased, and the weight a3 of slowcomponent slightly increased; the decrements k1,>k2,>k3 rap-idly dropped, that indicated on the impact of previous RChistory. At intervals timp = 40 ÷ 100 s a slow change of pa-rameters was observed. At timp > 100 s the parameters ofexponents reached their initial values. At intervals greaterthan 700 s, the parameters (ai, ki ) of RC recovery afterturning off the second pulse did not change, and thepulses of photoexcitation were independent of each other.The efficiency of the developed algorithm was also

tested in the study of fluorescence induction in the PSII

Fig. 2 Dependence of objective function (dispersion) of decompositionon the decrement k3 at the final stage of optimization (after k1 and k2were decrement)

Fig. 1 a Menu of the program for signal decomposition into sum ofexponential functions. b Plots of experimental and approximationsignals (both normalized)

Barabash and Lyamets Nanoscale Research Letters (2016) 11:544 Page 3 of 5

chlorophyll complexes of four types of winter wheat of dif-ferent ages. The shape of chlorophyll fluorescence induc-tion curve in the leaves of plants depends on severalprocesses that determine the energy distribution. A signifi-cant role is played by the energy exchange processes inthe light-harvesting antenna complex, photochemicaltransformation, and electron outflow from the reactioncenters. Blocking the electron transport from the reactioncenters leads to a simplification of the shape of fluores-cence curve. In this case the shape depends mainly on theenergy transfer in antenna. Several subsystems (units) canbe distinguished in the PSII structural organization [9, 10]:a unit of photoelectric converters that perform the photo-chemical reaction, an antenna unit, and block of smallprotein subunits. The antenna unit includes pigment-proteins of internal antennas, external minor antennas,and external LHCII antennas with trimers (S, M, L). ThePSII complex has three dimers (C2–dimers of core-

complexes) which include the reaction centers with in-ternal antennas, as well as trimers (S, M) of the extenalLHCII antenna. The induction of fluorescence wasmeasured in a classical mode in cut leaves adapted todarkness. The electron transport was blocked by DCMU(3-(3,4-dichlorophenyl)-1,1-dimethylurea) which was fedinto the tissues together with the buffer mixture throughthe section in the bottom of the leaves.It was shown that there are only three components in

the exponential decomposition of the kinetic curve offluorescence induction for the leaves of plants of differ-ent genotypes grown in normal conditions, and the sumof these components well correlates with the experimen-tal curve (Fig. 4).Table 1 shows the results of decomposition of induc-

tion curves of chlorophyll fluorescence for four types ofwinter wheat.The values of exponential components for four leaf

genotypes are close to each other. The ratio of the weightsof exponential components with decrements (k1 > k2 > k3)is approximately a1:a2:a3 = 10/5/1. Electron microscopicstudies of thale cress (Arabidópsis thaliána) and spinach

Fig. 4 Experimental curves of fluorescence induction for the leavesof four types of winter wheat (Pdl, Per, Ods, Dos) (red); the results ofapproximation by three exponential components (black)

Table 1 The results of decomposition of induction curves ofchlorophyll fluorescence for four types of winter wheat

Type of winterwheat

Weighted contributionsof exponentialcomponentsa1, a2, a3%

Rateconstantsk1, k2, k3c−1

Chla/b

Ratioa1/a2/a3

Pdl 59, 29, 7 400, 87, 11 3.59 8/4/1

Per 67, 23, 3 300, 57, 5 3.52 22/8/1

Ods 67, 22, 4 300, 66, 12 3.91 17/6/1

Dos 77, 22, 4 370, 71, 13 3.60 19/6/1

Fig. 3 a, b The parameters of exponential components of kinetic curve of RC recovery ai and ki after the exposure to second light pulse as afunction of the interval between two successive pulses of RC photoexcitation

Barabash and Lyamets Nanoscale Research Letters (2016) 11:544 Page 4 of 5

(Spinácia olerácea) [11] showed that the ratio of the de-tected three configurations of PSII pigment-protein com-plexes was 8:2:1, that is close to the above ratio a1:a2:a3(see Table 1).

ConclusionsA computer program for the decomposition of the basicreaction kinetics of biological macromolecules into ex-ponential components has been developed. The programwas tested for analysis of the kinetics of photoinducedcharge transfer in the reaction centers (RCs) of purplebacteria and the induction of fluorescence in granumthylakoid membranes. An identification of the kineticsof structural changes in RC in the process of photoexci-tation was carried out with this program. It was foundthat after photoexcitation of RC, its structure had themost non-equilibrium character within the first 40 safter switching off the light. The dark state was achievedby RC in the process of relaxation after turning off thelight, and dark values of absorption were achieved afteradditional exposure of RC without light for 700 s.Examination of fluorescence induction of PSII com-

plexes in four genotypes of winter wheat of different agesshowed that only three exponential components werepresent in the kinetics of induction, with the ratio ofweights a1:a2:a3 = 10/5/1. This value is close to the ratio of8:2:1 for three configurations of PSII pigment-proteincomplexes, obtained by electron microscopic studies ofthale cress (Arabidópsis thaliána) and spinach (Spináciaolerácea). This indicates on possible correlation betweenthe exponential components of fluorescence induction ofPSII complexes and the composition of pigment-proteincomplexes of granum thylakoid membranes. It providesan opportunity to replace expensive electron microscopicinvestigations by much easier studies of the kinetics offluorescence induction in granum thylakoid membranes.

FundingNone.

Authors’ ContributionsYuMB took part in the formulation of problems and ways to solve it. AKLparticipated in the task and designed the manuscript according to therequirements of article publication. Both authors read and approved thefinal manuscript.

Competing InterestsThe authors declare that they have no competing interests.

Received: 24 September 2016 Accepted: 25 November 2016

References1. Monod J, Wyman J, Changeux JP (1965) J Mol Cell Biol 12:88–1182. Kleinfeld D, Okamura MY, Feher G (1984) Electron-transfer kinetics in

photosynthetic reaction centers cooled to cryogenic temperatures in thecharge-separated state: evidence for light-induced structural changes.Biochemistry 23:5780–5786

3. Shaytan KV, Uporov V, Lukashev Ye P, Kononenko AA, Rubin AB (1991) MolBiol 25:695–705, in Russian

4. Kochubey SM, Yu M, Barabash ID, Panas (2015) Multi-component structureof induction curve of chlorophyll fluorescence in wheat leaves//XXII GalynaPuchkovska International School-Seminar “Spectroscopy of Molecules andCrystals” September 27 – October 4, 2015. Chynadiyovo, Zacarpattia,Ukraine, p 34

5. Glushkov VM (1974) Introduction to Automated Control Systems. Tekhnika,Kiev, p 320, in Russian

6. Gen M, Cheng R (1997) Genetic algorithms and engineering design. NewJersey: John Wiley & Sons; 1997. p 352

7. Franks LE (1974) Signal theory-Moscow: Soviet Radio. Russian, p 3448. Zakharova NI, Churbanova IY (2000) Biochemistry 65:1819. Golbeck J, van der Est A, Eds (2014) The biophysics of photosynthesis. New

York: Springer. p46510. Kochubey SM (2001) Organization of photosynthetic apparatus of higher

plants. Kiev, “Alterpress”, Russian, p 20411. Boekema EJ, van Breemen JFL, van Roon H, Dekker JP (2000) Arrangement of

photosystem II supercomplexes in crystalline macrodomains within thethylakoid membrane of green plant chloroplasts. J Mol Biol V 301:1123–1133

Submit your manuscript to a journal and benefi t from:

7 Convenient online submission

7 Rigorous peer review

7 Immediate publication on acceptance

7 Open access: articles freely available online

7 High visibility within the fi eld

7 Retaining the copyright to your article

Submit your next manuscript at 7 springeropen.com

Barabash and Lyamets Nanoscale Research Letters (2016) 11:544 Page 5 of 5