Biochimica et Biophysica A cta, 1162 (1993) 209-216 © 1993 Elsevier Science Publishers B.V. All rights reserved 016%4838/93/$06.00
209
BBAPRO 34430
Structure and function of protamines: an 1H nuclear magnetic resonance investigation of the interaction of clupeines with mononucleotides Gabriella D'Auria, Livio Paolillo, Roberto Sartorio and Silvana Wurzburger Department of Chemistry, University 'FedericoH', Naples (Italy) (Received 4 August 1992)
Key words: Protamine; NMR, 1H-; Chpeine; Clupeine-nucleotide interaction
Protamines form a class of low-molecular-weight proteins that protect the chromosomal DNA in the spermatic cells of eukaryotic organisms. Protamines are located in the small and/or large groove of DNA where they complex the DNA nucleotides. Very little is known up to date on the role and specificity of binding of the various protamine fractions belonging to a single eukaryotic species. In the present paper, a detailed investigation on the complexation properties of the protamine fractions (elupeines) extracted from herrings has been carded out by means of proton nuclear magnetic resonance and ultraviolet absorbtion data. In particular, the binding properties of the clupeine fractions with purinic (5'dAMP) and pyrimidinic (5'dCMP) mononucleotides have been measured and analysed at different clupeine concentrations. The results indicate that, contrary to previous preliminary hypothesis, the three clupeine fractions exhibit quite comparable binding properties toward mononueleotides. In addition it has been found that nucleotides can induce a conformational transition of the disorder-order type in the chpeine molecules and this property is concentration and temperature dependent. It is concluded that, as far as specificity is concerned, the clupeine fractions seem to possess the same behaviour toward mononucleotides.
Introduction
Protamines form a class of low-molecular-weight proteins deeply involved in the protection of chromosomal DNA in the spermatic cells of eukaryotic organisms [1]. Many efforts have been devoted to the elucidation of the role played by protamines in protecting the genetic material from duplication or generic modification during the meiotic process. Unfortunately, besides a general picture of the protamine-DNA complex that locates protamine molecules in the small [2,3] or large groove of DNA [4], no detailed information is available on the specificity of the interaction with DNA. Some papers have appeared on the subject mainly using the protamine fractions named clupeines, extracted from herrings and easily separated and purified by wellestablished procedures [5]. Experiments have been performed using optical methods [6,7], fluorescence [8] and nuclear magnetic resonance [9,10]. Attempts to correlate experimental data to structural properties have not convincingly demonstrated that the hetero-
Correspondence to: R. Sartorio, Department of Chemistry, University 'Federieo Ir, via Mezzoeannone 4, 80134 Naples, Italy.
geneity of chemical composition is linked to different binding properties of the clupeine fractions. Published nuclear magnetic resonance data [9,10] seem to suggest a different behaviour of the clupeine fractions in their interaction with mononucleotides. These data appear, however, rather preliminary. Further experimental work, before drawing any conclusive statements on the clupeine binding properties, seems appropriate. For this reason, we have undertaken a detailed investigation of clupeine-nucleotide interactions using not only NMR techniques but also optical spectroscopy. In particular, we have carefully examined the behaviour of the three main clupeine fractions at different molar concentrations in the presence of purinic (2'-deoxyadenosine 5'-monophosphate) and pyrimidinic (2'-deoxycytidine 5'-monophosphate) deoxynucleotides. UV and NMR data will be discussed in terms of multiple equilibria in solution in which self-association and binding phenomena play a predominant role. Materials and Methods
Clupeine fractions YI, YII, and Z, whose amino-acid sequences are reported in Table I, were chromato-
210 TABLE I
Amino-acid sequences of the three clupeine fractions YI, }111and Z
YI
YII
Z
1 5 Ala Arg Arg Arg Arg Ser Ser Ser Arg 20 Pro Arg Arg Arg Thr Thr Arg Arg Arg
10 Pro Ile Arg Arg Arg 25 Arg Ala Gly Arg Arg
15 Arg 30 Arg Arg
I 5 Pro Arg Arg Arg Thr Arg Arg Ala Ser 20 Arg Pro Arg Arg Val Ser Arg Arg Arg
10 Arg Pro Val Arg Arg 25 Arg Ala Arg Arg Arg
15 Arg 30 Arg
1 5 Ala Arg Arg Arg Arg Ser Arg Arg Ala 20 Arg Arg Pro Arg Arg Val Ser Arg Arg
10 Ser Arg Pro Val Arg 25 Arg Arg Ala Arg Arg
15 Arg 30 Arg Arg
graphically purified from a commercial mixture of clupeine sulphate (Sigma, grade III) following the procedure suggested by Ando [5]. 2'-deoxyadenosine 5'monophosphate (5'-dAMP), 2'-deoxycytidine 5'-monophosphate (5'-dCMP) were analytical grade products and used without further purification. N M R solutions were prepared at three different protamine concentrations (Cp nearly 1, 2 and 3- 10 -3 M). To these solutions increasing amounts of nucleotides were added up to 0.2 M concentration (CN). N M R measurements were carried out on a Bruker AM-400 (400 MHz) located at the 'Centro Interdipartimentale di Metodologie Chimico-Fisiche' of the University 'Federico II' of Naples. The probe temperature was 298 K. All chemical shifts are referred to TSP (3-(trimethylsilyl)propionic-2,2,3,3-d 4 acid, sodium salt, Aldrich). N M R samples were always pH-controlled (pH 6), although no deuterium isotope correction was made for the experimental readings. UV measurements were carried out at 298 K with a Perkin-Elmer spectrophotometer model 320. Absorbance was determined, in the range 250-320 nm, for the system 5'-dAMP-phosphate buffer (0.05 M (pH 6)) as a function of 5'-dAMP concentration ( 2 . 1 0 - 5 - 2 • 10-1 M) and for the system clupeine-5'-dAMP-phosphate buffer (0.05 M (pH 6)) at constant protamine concentration (1 • 10 -3 M) and at increasing amounts of nucleotide (1" 1 0 - 2 - 3 • 10 -1 M). In the concentration range used, the protamine solution is completely transparent. Results and Discussion
UI/ analysis In order to define the effect that clupeines have on the auto-association equilibrium of 5'-dAMP in aqueous solution, a preliminary study on the 'binary' system (nucleotide-phosphate buffer) was carried out. In Fig. 1 (top), the apparent molar extinction coefficient e at 260 nm (hm~x) is reported vs. the logarithm of the
nucleotide concentration• For a concentration larger than 1 . 1 0 -2 M, a large hypochromic effect, indicative of the onset of self aggregation, due to the stacking interaction of the nucleotide bases, is observed. The association constant can be calculated from the experimental data by assuming a single-step association equilibrium: n N ¢~ N .
Ks=[Nn]/[N] n
(1)
where N stands for nucleotide. The algebra of the model, the numerical treatment, as well as the K s obtained for different n values, are reported in the Appendix. (The calculated value of K s for n = 2 is 2.4 m o l - 1 dm3). When clupeine at 0.8.10 -3 M concentration is added, the hypochromic effect rises at a larger nucleotide concentration (C N > 1 • 10-1 M). This is shown in Fig. 1 (bottom, curve b) where e26o is drawn vs. the 5'-dAMP concentration for the 'ternary' system that includes clupeine fraction YI. Similar curves are obtained for clupeine fractions YII and Z. The constant value of the extinction coefficient (14.4 • 10 3 cm -1 M -1) exhibited in a wide range of C N / C e ratios is indicative that no nucleotide self-association (stacking) appears to occur in the presence of protamines up to a 1 • 10 -3 M concentration. The observed e260 trend is consistent with the hypothesis of an electrostatic interaction between protamine molecules and nucleotides in the monomeric form. This process would in fact reduce the number of free monomeric nucleotide molecules in solution shifting the onset of autoassociation to a larger nucleotide concentration. N M R analysis While the UV data yield direct information only on the nucleotide interactions, N M R parameters are informative on the nucleotide and protamine components at the same time. The N M R analysis consists in
211 ZOO00
amino acids. The chemical shift changes of the arginine ~CH 2 protons have, then, been used to gain information on the interaction occurring between the protamine molecules and the negatively-charged nucleotides (as model compounds for the vastly more complex DNA system). For our purposes, clupeine concentrations of nearly 1, 2 and 3" 10-3 M were chosen for the NMR experiments. One of these (the less concentrated solution) has been used for the UV measurements and, thus, the spectrophotometric results are directly comparable to the NMR data. Fig. 2 shows the behaviour of the chemical shifts of the Arg ~ C H 2 protons of fraction YI about 2 . 1 0 -3 M when increasing amounts of 5'-dAMP and 5'-dCMP are added to the clupeine solutions. (Similar trends are obtained with the two other clupeine fractions). Before discussing the chemical shift trends of the A r g ~ C H 2 protons of the three clupeine fractions with nucleotide concentration, it seems important to consider carefully the behaviour of the Arg proton resonance intensities.
E (280 rim)
111000'
16000 ¸
o o
14000
12000
10000
8000
IoeC:N 6000
-'s
20000
:4
-'3
-'z
°=1
E (Z60 nm)
O
18000
a
Ob
16000
14000
12000
Proton resonance intensities •
Proton resonance intensities are very important in discussing the interactions between clupeine fractions and mononucleotides. In fact, a specificity in the interaction of clupeine fractions YI and Z with purinic nucleotides has been postulated [9,10] as a consequence of the severe intensity loss suffered by the clupeine resonances after addition of 5'-dAMP and 5'-dGMP. This hypothesis has never been completely tested at different protamine concentrations and at different temperatures. Clupeine fractions a r e in fact
I
0 0
1oo0o
8O00 cN (moi4) o,0
o11
o:2
03
o;,
o.s
Fig. 1. Top: concentration-dependence of molar extinction e26o of 5'-dAMP in phosphate buffer 0.05.10 -3 M ((pH 6.0) T = 298 K). Bottom: concentration-dependence of molar extinction e~0 of 5'dAMP (a), in phosphate buffer 0.05.10 -3 M ((pH 6.0) T = 298 K) and (b), in phosphate buffer 0.05" 10 -3 M and clupeine YI 0.8" 10 -3. M. e~0 = dm3 mol- 1cm -z.
o,os 4~u~. Am,X:HZ pm
•
YI-5'dCMP
0,00"
O0 o O -0,05'
the acquisition of clupeine spectra with and without nucleotides, studying the modifications of parameters, such as chemical shifts and line shapes. The proton NMR spectrum of clupeines in D 2 0 shows resonances only in the aliphatic region. Therefore, only backbone a C H and side-chain protons can be observed. The important features of the aliphatic proton region of clupeine fractions have been discussed elsewhere [10]. Besides the obvious a C H backbone resonances whose chemical shift changes reflect structural modification, side-chain protons may b e relevant to the interpretation of the interaction processes. Clupeine fractions bear about 20 positively-charged Arg residues out of 30
O YI-S'dAMP
•
O
Ooo 0
-0,10'
0 0
-0,1 5-
-0,20 0,0
011
01Z
0,3
Fig. 2. Relative chemical-shift changes (ASot,s) of the Arg 8CH 2 resonances of clupeine YI 1.71.10 -3 M at increasing amounts of: (o), 5'-dAMP and (o), 5'dCMP.
212
III"
0
Z=0.71 10^-3 mol/I
•
Z=1.71 10,'-3 mol/I
['1 Z=2.70 10^-3 mol/I
1'01"~0
0 0
0,5.
ii0
[:]
II
E]
I-I
•
I-I
[] []
CN (mol/I) 0,0
0,00
0,;s
030
035
0,20
Fig. 3. Relative intensity changes of the Arg 8CH 2 resonances, for the system clupeine Z-5'-dAMP, at different clupeine concentrations. (0), 0.71'10 -3 M; (e), 1.71-10 -3 M and ([:]), 2.70.10 -3 M. I ° = Arg 8CH 2 resonance intensity at CN = 0.
known to be scarcely soluble in water and it is important to establish clearly the effect of solubility a n d / o r aggregation on the reduction of the clupeine peak intensities. In Fig. 3, the intensities of the Arg ~iCH2 proton resonances are plotted varying C N values for the 5'dAMP-clupeine Z system. Similar plots have been obtained for the other two clupeine fractions at the three different clupeine concentrations mentioned above. In all cases, new interesting experimental observations can be made: (i), the intensities of Arg 8CH 2 resonances decrease with increasing nucleotide concentration. This phenomenon becomes much more evident going to the more concentrated protamine solutions (Fig. 3). (ii), A second upfield-shifted Arg 8CH z peak is observed at 1.7 and 2.7.10 -3 M clupeine concentrations (Fig. 4, main peak I, second peak II). This second peak is slightly broader and is reminiscent of phenomena occurring in order-disorder transitions in polypeptides [11]. The intensity trends of the main Arg 8CH 2 peaks for the three clupeine fractions are qualitatively similar although somewhat reduced in the case of clupeine YII. The area loss at low C N is proportional to the clupeine concentration; lower, then, at the clupeine concentration about 10 -3 M. In all cases when CN increases above the saturation value (R > 1), a recovery (Fig. 4, curve d) of the main Arg peak intensity and a simultaneous decrease of the second upfield shifted Arg peak are observed. In addition, it seems important to point out that, upon nucleotide addition, solutions begin to loose their transparency and this phenomenon becomes much
more evident at higher clupeine concentrations. After a few hours, the more concentrated samples deposit an oily precipitate (hardly detectable at low clupeine concentrations). The experimental observations described above can be interpreted as follows: (i), when nucleotide is added to the clupeine solutions, the occurring interactions favour a conformational transition of the protamine molecules to a more ordered structure that is less soluble in solution. A similar phenomenon has been described [12] for the coil-to-helix transition promoted by nucleotides in poly-L-arginine. The transition depends on the charge density on the clupeine molecules and, therefore, it is nucleotide-concentration-dependent. (ii), When C N increases beyond a certain value, the conformational equilibrium is shifted again to the more favoured coil form. The critical CN/C,~rg value for the transition (Carg- total molar concentration of Arg residues), which is always higher than 1, demonstrates that the bound nucleotide molecules play an important role in the conformational equilibrium.
/d
C
Ar8 (}CH2
~:4
3:3
3:2
3:,
2'H
~'o
2:9
£8
2"H
2:7
ppm
~:6
Fig. 4. Superimposed spectra (400 MHz) of the clupeine Z in D 2 0 (2.7.10-3 M, 298 K) in the presence of different amounts of 5'dAMP. R = C N / C A , s . (a), R = 0.6; (b), R = 1.20; (c), R = 1.80; (d), R = 6.70. Arg 8CH 2 I main peak; Arg 8CH 2 II second peak; 2 ' H and 2"H nucleotide resonances.
213
Chemical shift analysis
TABLE II
As it was pointed out earlier [9,10], all clupeine fractions exhibit a shift of their Arg 8CH 2 resonances with both nucleotides. The magnitude of the shift is always larger for purinic nucleotides as it clearly results from Fig. 2. The higher chemical shift change in the addition of purinic vs. pyrimidinic nucleotides has been considered indicative of a stronger interaction of the formers with the clupeines. In order to assess the validity of this hypothesis, it seems appropriate to prove that binding parameters extracted from the experimental saturation curves are markedly different. Assuming a 1:1 interaction between mononucleotide molecules and polar arginine side-chains (considered as independent binding sites), a single microscopic binding constant can be considered for all arginine residues. Neglecting the stacking equilibrium 2N ** N2, the binding process can be defined as:
Binding constants K b and their S.D. of 5'-dAMP and of 5'-dCMP on clupeines at different concentrations C e of the three fractions YI, YI1 and Z (T = 298 K (pH 6))
Arg + N ~, Arg-N
Kb = [Arg-N]/[Arg].[N ]
The (AS)max obtained through Eqns. 6 and 7 are -0.120+0.010 ppm for 5'-dAMP and -0.060+0.010 ppm for 5'-dCMP, respectively. System
Cp (M)
Kb (mol- 1dm 3)
5'dAMP-YI
0.92.10 -3 1.90"10 -3 3.20.10 -3 1.10.10 -3 1.80.10 -3 2.73"10 -3 0.71.10- 3 1.70-10 -3 2.70"10 -3 1.90.10-3 1.56.10 -3 1.90.10 -3
(9 +4).102 (8 -I-1).101 (4 +2).101 (6 +2).102 (6 -t-1).101 (3 -I-2)'101 (1.4-t- 0.6)" 103 (6 + 1).10 t (4 +1).101 (1.5 + 0.5)" 10 t (8 + 1)'101 (3 +0.5).101
5'dAMP-YII
5'dAMP-Z
5'dCMP-YI 5'dCMP-YII 5'dCMP-Z
(2)
Then the 8ob ~ of Arg 8 C H 2 c a n be expressed at a constant CArg and at varying C N as: 8obs(Cs) = 8(Cn = ~)" [Arg-N]/CAre + 8 (Cn ----"0)" [Axg]/CAre (3)
where 8(C N = oo) and 8(CN = 0) are respectively the extrapolated values of 8ohs of Arg 8CH 2 with nucleotide going from extremely concentrated solutions to zero; CArg = [Arg-N] + [Arg]. Rearranging Eqn. 3:
The effect of clupeine concentration on the 5'-dAMP binding process has been further examined considering that meaningful information cannot be extracted from a single concentration. The K b values are plotted against Cp in Fig. 5. The following considerations can be made: (i), a single curve describes the K b dependence on Cp, indicating a similar behaviour of the protamine fractions with respect to the binding process. (ii), A de-
[Arg-N]/CArg = (8obs(C N) -- 8 ( C N = 0))/(8(C N =~)- a ( C s = 0)) =
Aaobs/(Aa)max
log Kb
(4)
O YI-S'dAMP
Solving for [Arg-N] in Eqns. 2 and 3: AS,y,~ = (K b" [ N ] / 1 + K b" [N])" (AS)n~
(5)
that can be rearranged in the linear form as: 1/(ASobs) = 1/(AS)max + 1/(K b"[N]" (AS)max)
(6)
where [ N ] : C N -- (At$obs//(A~)max). CArg
•
YI-S'dCMP
[]
YII-S'dAMP
•
YII-S'dCMP
a
Z-S'dAMP
•
Z-S'dCMP
't0 (7)
Eqns. 6 and 7 can be solved by an iterative leastsquare method yielding the binding constant values reported in Table II where the three clupeine fractions are shown to exhibit rather similar binding constants with both purinic and pyrimidinic 5'-dAMP and 5'dCMP mononucleotides. The different NMR chemical shift changes upon nucleotide addition, i.e., higher chemical shift for purinic nucleotides do not appear, then, correlated to a different binding affinity.
CN (mot/I) 0
o.ooo
o. o2
o. o3
o. o,
0.o0s
Fig. 5. Dependence of the binding constants on nuc|eotide concentration for the three clupeine fractions and both 5'-dAMP and
5'-dCMP. (o), YI-5'dAMP; (o), YI-5'dCMP; (n), YII-S'dAMP; (II), YII-5'dCMP; (/,), Z-5'dAMP; (A), Z-5'dCMP. Kb= tool- tdm 3. Bar length represents the error.
214 tectable change of K b upon clupeine concentration is found. (This could be interpreted in terms of a reduced average number of Arg residues available to the interaction with nucleotides as it is pointed out by the area reduction of the Arg resonances). Complementary information is gained from the 5'dAMP proton resonances. In the absence of protamine, (R = 0), the chemical shift changes with concentration exhibited by 5'-dAMP HI', H2 and H8 protons are known to be indicative of the self-association stacking process [13]. In the presence of protamine at constant concentration, chemical shift changes, as reported in Fig. 6 in the case of H8 in the presence of clupeine Z, are observed. This trend is different at the three protamine concentrations. The experimental trends could only be interpreted by knowing the effects of the possible chemical processes occurring in solution. Nucleotide molecules, in the presence of protamines, can, in fact, participate to three different processes: (i), self-association outside the polycation domain (stacking); (ii), binding to protamines and (iii), self-interaction in the polycation domain. All these processes are both nucleotide and protamine concentration dependent and are supposed to be fast on the NMR time-scale. Therefore, the experimental 6oh~ value for each proton is: ~obs= X x i ' ~ i where x i represents the molar fraction of the ith species and 6i its chemical shift. Limiting chemical shift values for free nucleotide molecules in monomeric and stacked form, ~f and 5fs, respectively, could be obtained from binary nucleotide solutions in the two extreme limits of 1 / C N ---, 0 and
C N ~ 0 . On the contrary, the other two 6/ values related to nucleotide molecules simply bound to the protamine a n d / o r self-associated on the protamine matrix, 6b a n d / o r 5bs, respectively, cannot be evaluated. However, some qualitative considerations can be drawn from the trends of the binding phenomenon with nucleotide concentration (Fig. 5). In infinitively-diluted 5'dAMP solutions and in the presence of protamine the binding constant K b affects the distribution of nucleotide molecules between free and bound monomeric states. Since K b is protamine-concentration-dependent, extrapolation of 6ohs for C N ~ 0 yields different ~i values as it appears in the curves reported in Fig. 6. At high nucleotide concentrations (1/CN ~ 0), the molar fraction of the bound nucleotide molecules, because of the saturation to the protamine, decreases and the chemical shift of the nucleotide protons tends to correspond to that of the binary system. This is indeed confirmed in the plots of tSobs vs. C N at several concentrations as it is shown in Fig. 6 for clupeine Z. Conclusions
The hypothesis that the slightly different chemical composition of the three clupeine fractions may play a role in their specificity of binding to DNA, has been the object of some papers in the past few years [9,10]. In particular, the unique Ser-Ser-Ser distribution in clupeine YI has been postulated to be responsible of a higher binding specificity towards purinic mononucleotides. The experimental evidence on which this
8,60 c~H8 ppm
8,55
•
Z 0,0 mol/I
o
Z 0,71 10^-3 mall
•
Z 1,71 10^-3 mol/I
[]
Z 2,70 10^-3 mol/I
mlm
mm
•
•
rm
8,50
O o
•oe~
o
[] 0
m• o 0
[]
8,45
n
CN 8,40
o.oo
030
o, o
o.;o
(111o111)
0.40
Fig. 6. D e p e n d e n c e of chemical shift of the 5'-dAMP H8 proton on mononueleotide concentration in the presence of different amounts of clupeine Z. ( • ) , C z = 0; (©), C z = 0.7 1.10-3 M; (e), C z = 1.71.10-3 M; ([]), C z = 2.70-10 -3 M.
215 hypothesis relies, is based essentially on NMR phenomena; such as different chemical shift trends of arginine side-chain protons upon nucleotide addition and a large area loss of the same proton peaks in the presence of purinic 5'-dAMP and 5'-dGMP [9,10]. In this paper, we show that a careful combined use of NMR and UV spectroscopy can help in elucidating the complex interactions occurring in solution. This satisfactorily explains the role of protamines in terms of binding specificity to mononucleotides The UV analysis at 0.8 M clupeine concentration demonstrates that the nucleotide self-aggregation phenomenon is shifted to higher nucleotide concentration in the presence of clupeine fractions. This is interpreted in terms of monomeric nucleotide molecules binding the charged Arg side chains of the clupeines through electrostatic interactions. NMR data at the same clupeine concentration (less than 1.0.10 -3 M) enable the extraction of binding constants from the NMR curves (Figs. 5, 6 and Table II). It can be observed that both purinic (5'-dAMP) and pyrimidinic (5'-dCMP) nucleotides exhibit essentially similar binding constants in the presence of the three clupeine fractions, the binding being always a little lower for pyrimidinic nucleotides. At higher clupeine concentrations, the binding constants are smaller but they are still quite similar for the three fractions. The area-loss phenomenon of the clupeine Arg side-chain resonances observed at different clupeine concentrations can be interpreted as due to a conformational transition in the presence of nucleotides. On the basis of what is known about the disorder-order transitions in model polypeptides such as poly-L-Arg [12], we suggest that clupeines, too, exhibit this behaviour. The second upfield-shifted Arg resonance (Fig. 4) is a convincing proof of this hypothesis. In addition, not only the turbidity and eventually the observed oily precipitate on standing, but also the concentration effect (higher at higher protamine concentration) confirm the hypothesis on the origin of the second protamine ordered conformation. On the basis of the reported experimental data an understanding of the interactions existing in solution can be proposed. At concentrations lower than 10 -3 M, a predominant electrostatic interaction exists between clupeines and nucleotides. Nucleotides tend to self-associate only after the binding to the clupeines has occurred. At higher clupeine concentrations, nucleotide molecules induce a structural transition in the clupeines with an ensuing partial precipitation. In this case, the auto-aggregation of clupeine molecules, virtually absent at lower concentrations, occurs. In all cases the phenomena here described involve the three clupeine fractions even though to a slightly different extent. We conclude then, that the examined clupeine fractions do not exhibit any net difference of
binding specificity towards purinic and pyrimidinic mononucleotides.
Acknowledgement Financial support of the Italian Ministry of University (MURST) is kindly acknowledged.
Appendix According to the single-step association equilibrium n N c* N n
the constant K s expressed as a function of the molar fraction of monomer and n-mer species results: Ksn = [ N n l / [ N ] n = a , , / ( n "
a'~" C n ~ 1 ) = (1 -- Ot I)//(?I • 0/~" C ~ - 1)
(A-D where a 1= [N]/CN;
(A-2)
ot n = n " [ N n ] / C N
(A-3)
C N = I N ] + n[Nn]
The absorbance A, using the Lambert-Beer law, is A = E 1" [N] + n .E n- [Nn]
(A-4)
where el is the molar extinction coefficient of monomeric species and ~n is the molar extinction coefficient of a single nucleotide in n-meric species. Then: E = A/CN
= el" a t + En" otn = el" ol 1 q- e n" (1 - a l )
(A-5)
Let us denote each experimental result with a pair C N i ) , i = 1, 2 . . . . m . For the i th experiment, we have an unknown value of the molar fraction a u = [N]JCNi and the corresponding theoretical value E i = E 1 " Otli + E n " ( 1 -- O t l i ) provided by the model. The unknowns of the problem are n, E, and a u. Notice that, instead of estimating a l i for all i, the actual aim of our experiment is to calculate the association constant Ks:
(El,
Ksn = (1 - O t l i ) / n " ( f f l i ) n" ( C N i ) n - 1
(A-6)
Once that e~, (E i, CNi), i = 1, 2,..m, are given, then e n, n, Ksn can be calculated as the solution of the following constrained least-square problem m i n i m i z e : II E i - (~1" °/li -[- En" (1 -- a l i ) )
II2
n,~n,gsn
subject to n > 0; 1000 < ~n < e l ; 0 < Ksn < 2000
(A-7)
216 where au, i = 1, 2,..m is the real solution in (0,1) of the polynomial equation: n "gsn'(CNi)
n - 1 .ot~ i + °tli _ 1 = 0
(A-8)
obtained through Eqns. A-1 and A-5. This problem can be easily solved using a standard numerical minimization routine. References 1 Subirana, J.A. (1983) in The Sperm Cell (Andr6, J., ed.), Proc. Fourth Int. Symp. Spermatol., Martunus Nijhoff, Dordrtecht. 2 Feughelman, M., Langridge, R., Seeds, W.E., Stokes, A.R., Wilson, H.R., Hooper, C.W., Wilkins, M.H.F., Barclay, R.K. and Hamilton, L.D. (1955) Nature 175, 834-838. 3 Suau, P. and Subirana, J.A.J. (1977) Mol. Biol. 117, 909-926.
4 Warrant, R.W. and Kim, S.H. (1978) Nature 271, 130-135. 5 Suzuki, K. and Ando, T. (1968) J. Biochem. 63, 403-405. 6 Nishi, N., Tsunemi, M., Hagiwara, K., Tokura, S. and Tsutsumi, A. (1986) in Peptide Chemistry 1985 (Kiso, Y., ed.), pp. 245-250, Protein Research Foundation, Osaka. 7 Nishi, N. (1988) in Peptide Chemistry 1987 (Shiba, T. and Sakakibara, S., eds.), Protein Research Foundation, Osaka. 8 Arellano, A., Wehling, K. and Wagner, K.G. (1984) Int. J. Biol Macromol. 6, 249-254. 9 Andini, S., Bonora, G.M., Ferrara, L., Paolillo, L., Toniolo, C. and Wurzburger, S. (1986) Biochim. Biophys. Acta 866, 216-221 10 Paolillo, L., Andini, S., Ferrara, L. and Wurzburger, S. (1984) Int J. Quant. Chem. 26, 873-888. 11 Jardeztky, O. and Roberts, G.C.K. (1981) in NMR In Moleculal Biology, Academic Press. 12 Rifkind, J.M. and Eichorn, G.L. (1979) Biochemistry 9, 17531761. 13 Ts'O, P.O.P. (1970) in Fine Structure of Proteins and Nucleic Acids (Fasman, G.D. and Timasheff, S.N., eds.), pp. 50-189, Marcel Dekker, New York.