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Derivation of equation (9) from main paper
APPENDIX
127
1
REFERENCES Debey, P., Hui Bon Hoa, G., Douzou, P., Godefroy-Colburn, T., Graffc, M. S: GrunbcrgManago, M. (197.5). &ochemietry, 14, 1553-1559. Hapke, B. & Noll, H. (1976). J. Mol. Biol. 105, 97-109. Maaloe, 0. & Kjeldgaard, N. 0. (1966). In Control oj’ illacro?noEecct~r Synthesis, 1’. 70, W. A. Benjamin, Inc., New York. Noll, H. (1969). In Techniques in Protein BiO8ynthe8i8 (Sargent., J. & Campbell, P. N., eds), vol. 2, p. 101, Academic Press, London. Noll, H., Noll, M., Hapke, B. & van Dieijen, G. (1973). In Regulation of Transcription and Tranakxtion, MoabachColloquium No. 24 (Bautz, E.. ed.), pp. 257-311, Springer Verlag, Heidelberg. Noll, M. (1972). Thesis, Northwestern University. Noll, M. & Noll, H. (1972). Nature New BioZ. 238, 225-228. Noll, M. & Noll, H. (1974a). J. Mol. BioZ. 89, 477-494. Noll, M. & Noll, H. (19746). J. Mol. BioZ. 90, 237-251. Noll, M., Hapke, B., Schreier, M. H. & Noll, H. (197%). J. Mol. B&Z. 75, 281-294. Noll, M., Hapke, B. & Noll, H. (1973b). J. MOE. BioZ. 80, 519-529. Schreier, M. H. & Noll, H. (1971). Proc. Nat. Acad. Sci., IT.S.A. 68, 805-809. Spirin, A. S., Sabo, B. & Kovalenko, V. A. (1971). FEBS Letters, 15, 197-200. Sitaehelin, T. & Maglott, D. R. (1971). In Methods in Enzymology, Nucleic Acids and Protein Synthesis (Moldave, K. & Grossman, L., eds), vol. 20, part C, pp. 449-456, Academic Press, New York. Stahli, C. (1975). Thesis, Northwestern University. Walters, J. A. L. I. & Van OS, G. A. J. (1970). Biochim. Biophys. Acta, 199, 453-463. Wishnia, A., Boussert, A., Graffe, M., Dessen, P. & Grunberg-Manago, M., (1975). J. Afol. Biol., 93, 499-515. Zitomrr, R. S. & Flaks, J. 0. (1972). J. Mol. Biol., 71, 263-279.
APPENDIX
Derivation
1
of equation (9) from main paper I.M. KLOTZ
Since both subunits and the 70 S association product are binding Mg2 + reversibly, the following equilibria and association constants apply:
50s + iMg2+ + 5OSMgi KsOSM~ 30 S +jMg2+ = 3OS%, K,,,,, 70s +ZMg2+ = 7OSMg, Kmm .
(Al)
The multiple equilibria between ribosomal species can be represented concisely by the equation “70 %a % LX BOSMg, + YZ SOSMg, + Z 70SMg, K, , W4 0
0
0
where ‘n = total sites available on each respective species of ribosome, and K: is the
128
I.
expcrimrnt’al
equilibrium
co&ant
KIO’I’7 1
Al.
A
tletinrd as l~Z70SMg,J [:EOSMgi] [Z’SOSMg, 1
K& z:
The magnitude of Ki will vary with the concentration of Mg2 + According to Wyman (1964) t’he following general relationship equation (A2) :
can be derived for
(A3) where Ar = r ,,, - rso - raO, and T = average number of Mg2+ ions bound by t’he respective species. If the binding sites for Mg2+ on each ribosomal species are assumed identical and non-interacting, one can show (Klot,z, 1946) that r
r
-
~~,oK,ol?‘W +1
7o - I:+
K,,[Mg2 + 1’
n30K,oCMg2+ 1 _ 3o - i-+ K30[Mg2+]’
r 50 --
(84)
n,oK,o[Mg2 +1 1 -t K,o[Mg2+j’
If we assume furthermore that sites on different ribosomal species are identical, so that, K,, = K,, =
K,,,
= K,,,
(A@
then Ar = q.
KdWz2 +1 1 + K,,JMg’+]
K,,[Mg2 +1 - ‘jL5’ 1 + KM,[Mg=j-
h’,,Wg2 +1 jL3’ 1 + K,,Wg2 +1 (A6)
Thus, from equations (A3) and (A6) one obtains
din Ki d In [Mg2 +]
_^ (AWdMg2 +:I 1 + KmWg2’l
b47)
In a range of [Mg2 ‘1 where KiMg[Mg2 +] B 1
648) This result is superficially analogous to that from an analysis which represents the equilibria by 50 S + 30 S + nMg $70 S.Mg,
and obtains
the following
relationship:
K .I ~- ~[7081--~ a
[50 S] 130 Sl [Mg2 + 1”’
log Kk = log K,
+ )t log [Mg” +],
dlnK, d In [Mg2+] (Zitomer
& Flaks,
= ”
1972). REFERENCES
Klotz, 1. M. (1946). Arch. Biochen~., 9, 109-117. Wyman, J. (1964). Advan. Protein Chem., 19, 223. Zitomctr, R. 8. & Flaks, J. G. (1972). J. Mol. Viol.,
BPPENDl
71, 263-279.
X 2
Solution of the differential equation describing the exchange process M. NOLL In t’he following I derive the differential equation describing the spontaneous exchange of [l*C]subunits with vacant couples. x,, and that of Let the input of 70 S ribosomes correspond to a concentration [‘*Cl50 S subunits active in couple formation to a concentration y,,. Describing the equilibrium of vacant couples with subunits by 70 s
+
50 s + 30 s>
VW
and assuming that after adding bhe radioactive 50 S particles the new equilibrium is reached within a short time compared to the time needed for exchange to equilibrium with t,he labeled subunits, the change of t’he concentration x of labeled 70 S ribosomes at a time t is dx
dt
:
- k,x + k,(y, - s)(zo
ace);
(AlO)
x, denotes the total concentration of labeled and unlabeled 70 S ribosomes and may be obtained from the dissociation equilibrium according tjo equation (A9) : (All)
127
1
REFERENCES Debey, P., Hui Bon Hoa, G., Douzou, P., Godefroy-Colburn, T., Graffc, M. S: GrunbcrgManago, M. (197.5). &ochemietry, 14, 1553-1559. Hapke, B. & Noll, H. (1976). J. Mol. Biol. 105, 97-109. Maaloe, 0. & Kjeldgaard, N. 0. (1966). In Control oj’ illacro?noEecct~r Synthesis, 1’. 70, W. A. Benjamin, Inc., New York. Noll, H. (1969). In Techniques in Protein BiO8ynthe8i8 (Sargent., J. & Campbell, P. N., eds), vol. 2, p. 101, Academic Press, London. Noll, H., Noll, M., Hapke, B. & van Dieijen, G. (1973). In Regulation of Transcription and Tranakxtion, MoabachColloquium No. 24 (Bautz, E.. ed.), pp. 257-311, Springer Verlag, Heidelberg. Noll, M. (1972). Thesis, Northwestern University. Noll, M. & Noll, H. (1972). Nature New BioZ. 238, 225-228. Noll, M. & Noll, H. (1974a). J. Mol. BioZ. 89, 477-494. Noll, M. & Noll, H. (19746). J. Mol. BioZ. 90, 237-251. Noll, M., Hapke, B., Schreier, M. H. & Noll, H. (197%). J. Mol. B&Z. 75, 281-294. Noll, M., Hapke, B. & Noll, H. (1973b). J. MOE. BioZ. 80, 519-529. Schreier, M. H. & Noll, H. (1971). Proc. Nat. Acad. Sci., IT.S.A. 68, 805-809. Spirin, A. S., Sabo, B. & Kovalenko, V. A. (1971). FEBS Letters, 15, 197-200. Sitaehelin, T. & Maglott, D. R. (1971). In Methods in Enzymology, Nucleic Acids and Protein Synthesis (Moldave, K. & Grossman, L., eds), vol. 20, part C, pp. 449-456, Academic Press, New York. Stahli, C. (1975). Thesis, Northwestern University. Walters, J. A. L. I. & Van OS, G. A. J. (1970). Biochim. Biophys. Acta, 199, 453-463. Wishnia, A., Boussert, A., Graffe, M., Dessen, P. & Grunberg-Manago, M., (1975). J. Afol. Biol., 93, 499-515. Zitomrr, R. S. & Flaks, J. 0. (1972). J. Mol. Biol., 71, 263-279.
APPENDIX
Derivation
1
of equation (9) from main paper I.M. KLOTZ
Since both subunits and the 70 S association product are binding Mg2 + reversibly, the following equilibria and association constants apply:
50s + iMg2+ + 5OSMgi KsOSM~ 30 S +jMg2+ = 3OS%, K,,,,, 70s +ZMg2+ = 7OSMg, Kmm .
(Al)
The multiple equilibria between ribosomal species can be represented concisely by the equation “70 %a % LX BOSMg, + YZ SOSMg, + Z 70SMg, K, , W4 0
0
0
where ‘n = total sites available on each respective species of ribosome, and K: is the
128
I.
expcrimrnt’al
equilibrium
co&ant
KIO’I’7 1
Al.
A
tletinrd as l~Z70SMg,J [:EOSMgi] [Z’SOSMg, 1
K& z:
The magnitude of Ki will vary with the concentration of Mg2 + According to Wyman (1964) t’he following general relationship equation (A2) :
can be derived for
(A3) where Ar = r ,,, - rso - raO, and T = average number of Mg2+ ions bound by t’he respective species. If the binding sites for Mg2+ on each ribosomal species are assumed identical and non-interacting, one can show (Klot,z, 1946) that r
r
-
~~,oK,ol?‘W +1
7o - I:+
K,,[Mg2 + 1’
n30K,oCMg2+ 1 _ 3o - i-+ K30[Mg2+]’
r 50 --
(84)
n,oK,o[Mg2 +1 1 -t K,o[Mg2+j’
If we assume furthermore that sites on different ribosomal species are identical, so that, K,, = K,, =
K,,,
= K,,,
(A@
then Ar = q.
KdWz2 +1 1 + K,,JMg’+]
K,,[Mg2 +1 - ‘jL5’ 1 + KM,[Mg=j-
h’,,Wg2 +1 jL3’ 1 + K,,Wg2 +1 (A6)
Thus, from equations (A3) and (A6) one obtains
din Ki d In [Mg2 +]
_^ (AWdMg2 +:I 1 + KmWg2’l
b47)
In a range of [Mg2 ‘1 where KiMg[Mg2 +] B 1
648) This result is superficially analogous to that from an analysis which represents the equilibria by 50 S + 30 S + nMg $70 S.Mg,
and obtains
the following
relationship:
K .I ~- ~[7081--~ a
[50 S] 130 Sl [Mg2 + 1”’
log Kk = log K,
+ )t log [Mg” +],
dlnK, d In [Mg2+] (Zitomer
& Flaks,
= ”
1972). REFERENCES
Klotz, 1. M. (1946). Arch. Biochen~., 9, 109-117. Wyman, J. (1964). Advan. Protein Chem., 19, 223. Zitomctr, R. 8. & Flaks, J. G. (1972). J. Mol. Viol.,
BPPENDl
71, 263-279.
X 2
Solution of the differential equation describing the exchange process M. NOLL In t’he following I derive the differential equation describing the spontaneous exchange of [l*C]subunits with vacant couples. x,, and that of Let the input of 70 S ribosomes correspond to a concentration [‘*Cl50 S subunits active in couple formation to a concentration y,,. Describing the equilibrium of vacant couples with subunits by 70 s
+
50 s + 30 s>
VW
and assuming that after adding bhe radioactive 50 S particles the new equilibrium is reached within a short time compared to the time needed for exchange to equilibrium with t,he labeled subunits, the change of t’he concentration x of labeled 70 S ribosomes at a time t is dx
dt
:
- k,x + k,(y, - s)(zo
ace);
(AlO)
x, denotes the total concentration of labeled and unlabeled 70 S ribosomes and may be obtained from the dissociation equilibrium according tjo equation (A9) : (All)