vom Can Ben y te Ke | a4 ‘ UNIVERSITY OF PITTSBURGH ia /} ae PITTSBURGH 13, PENNSYLVANIA DEPARTMENT OF BIOPHYSICS January 20, 1959 Dr. Joshua Lederberg Department of Genetics Stanford University Palo Alto, California Dear Dr. Lederberg: Thank you very much for your prompt reply to my letter of December 15 and for your post-card informing me of the derivation by Armitage of the equations for the distribution of numbers of mutants whose growth rate is different from the parent. A typed copy of the appendices aS, enclosed which re- place$ the handwritten copy I sent you. My equations for the average number of mutants check exactly with those of Armi- tage. There is a mistake in the handwritten copy of Appen- dix I: the mutation rate per unit time should equal X= pt eX instead of toa in order to make the definition of p in Appendix I the same as the definition of c& in Appendix It. This correction has been made in the typewritten copy. This, changes the value of & from 3.0 x 1074 to 4.3 x 10-4 in Fig. 4 (a corrected figure is enclosed) and a corresponding change should be made in the last line on page 5. Since Fig. is a log plot this change makes little aifference. As the equation stands now it is not only identical to that of Armitage but also reduces to the Luria-Delbrtick equa- tion when the growth rates are equal, as it must. A bonus of the Armitage paper is that the variance as well as the mean has been calculated for the case of unequal growth rates. Theory predicts a greater variance for a greater difference in growth rate, and this effect is found experimentally. As far as my coming to your laboratory is concerned, I appreciate your consideration of this possibility. I feel that collaboration between us on the genetics of pili would be very productive and I am hoping that this will eventually be possible. Dr. Lederberg -2- January 20, 1959 You encouraged me to investigate other opportunities at Stanford and elsewhere because of limitations of space and the size of your group. Dr. Hubert Bloch, chairman of the Micro- biology Department of the University of Pittsburgh Medical School has offered me an appointment in his department, which I will probably accept if things at Stanford don't work out for 1959-60. Although this appointment is quite attractive I would prefer to come to Stanford and work in your group. I have told him that I am waiting to hear from you and that it will be at least several weeks before I can expect any answer, As to an opening on a collaborative basis with Biophysics, Biochemistry, or Medical Microbiology, Biophysics is a distinct possibility. My degree is in Biophysics and I have had further experience in Radiobiology, Electron Microscopy, and Electro- phoresis. However, I do not know to whom I should write at Stanford; perhaps you could refer me to the person who is or- ganizing this division or maybe you would rather check on this possibility yourself. I am enclosing the page proof of a chapter on electrophoresis I recently wrote with Dr. Lauffer which might be of interest to the Biophysics group. Let me say again that I appreciate your interest in my work and your prompt response to my application in spite of the many distractions of organizing a new department. Sincergly yours, Youbet Charles C. Brinton, Jr. CCB: 1f Appendix i Derivation of the equations for the average numbers of pt and p” cells in a culture when the growth rates of the two forns are different. Assumptions and Definitions + . . division Pp’ —~» P mutetion rate « «o per bacteriuu per > —, zt mutation rate = 0 tot = generation tine of pv celis Pp” = gencretion time of p* cells N+ = miver of pt celle at tine t 57 = omuuber of p™ celis at time t 4: chronological time. At to 0, Nit 1 and No- = 0. (sey = (=) (= dt “total at 7 mitation at division In the aosence of mutetion: ah ef ty , N.- Ps Ng po gltftp~) in N,- = In Ny p- + (t/t,-) In2, an ~ in 2 “~ in 2 # Ny ty , dt division ty- in the absence of mutation: B+ = No p+ qlt/tpt) K+ = 2(t/tp+) since No pt = 1 when t= 0, or Rot = e(t/tyt) ane If a is emall enough so that the fraction of pt cells lost by mutation 1s always small compared to the totai number of at cells, this forma may be used to compute the number of pt celle at any time. This assumption is valid at 37° up to several hundred generations since a @’2x 1074 . Since «a is defineé os the number of mutants occurring per bacterium per divisicn, the rate of mutant production per division €quais the nmunmpes cf parent celis timesu. The rate of mutant production per unit time is preportional to the raunber of parent ceils times the station rate per unit time. The mitaution rate per unlit clae equals uw Giviaed uy tik genera-~ tion tlue of the parent cell, times In ev. * aN Nyt a (u/t,+) Ine \Tat” / mutation 7 or _ el &/t,*) in 2 Inq \ at / autation tp Therefore: aX, - a Ine (in 2/t_+) t ln 2 dt tpt to- This is a readlly soluble differentiel equation of the form + f(xy = G(X) where In Nowe Xe t, #(t) = - “tye ; alt) = tee (1n 2/t +) t ¥ Tf time & measured m ynits of ivision eucles of backhia rathey than CWnmoloyicaily, the unit od time becoues tor divided by Ina dud this Cavation ‘reduces 0 Eq.(3) % Lurig and P delbriex (ii). The solution of the general equation is: y eF[ [Fsax + ¢ | (Margenau end Murphy, p. 41) where F = F(x) = f f(t) a. In our ease, F = F(t) = rY ing | ling , p> “to he Oe Noe we gh end INE Po (etytye) an 2 Lalnd,(t/ty+) In 2, P L t+ The integra: inside the braekete may be transformed into tc i po s je dquo» e% . a Ing ett 2 (U1 /tgel + fifty l) t gt In 2 (11 Ata} - Ui stp= 1) Also, since a = QO when t « 0, Integra. t |n re os tot in 2 (Ti7eye} - TH7TS=1) Therefore: dpc of S/tp~) In 2 (in 2 [1/tp+) = Li/tp-) & 1) i> = tot eee (12 /to+) ~ [1/t5-] “This eavation reduces to the Luiy- Delbruck (1) eguatim (6\ when the generation | times O+ the iw forms are eYuu | Inz 4 whe pe Np- m ih t Ny = Cote aka 4, E ddatim (bo) (5° mM. at Ne . ee fuel cavati waa be writtu woe simply as t + nz tpr In2 t,- No i ox (e r ~ C P ) , | — + tpn ov €a, (20) hits eaqvatio 45 exaetly cq va | Tt Eg. (0)A of Armitage (16) where : nd _ dnt _ Ine W=Q X=), Ye =O, Spt Pte 9 tpt » ) » J -~ a Appenain 2 Derivation of the equation for mutation rate in terus ef the fraction of parallel clones naving zero imitants (the “gero-point” wethod). The provauiiity of a asitation per bacteriw: per division =e Lee The provabliity of not naving e autation per bacteriua per divisional «- «a, The nmwiber of divisions that have occurred to produce a Glone = N- i where N is the total number of oelin in the clone, (This nwaber is correct whether or not the clune nas divided synchronously. ) in omder to have no oitanta in &@ clone, there cust not have veen a mutetion during any of the divisions. Since tie probvablilty of an event which depends on the siualtancous Puiflliwent of several separate events ia the product of the probavliities of the separate events, the probevility of navirng no mutante isa pe (l- ays When Nis iarge, p = {1 - aj¥ ing = Nin (i - c) whens ip < 107°, in pe - Me and a ® =(1n 5/N). ‘Wis eQvatim is exacel edual to the combined eeyats ms (4) aud G) oF Leriq aod Delbvkek (u) foy the ane where No tS Small compared to Ne. —2 O-—T— T_T T IT TT LOG & —5 0.1 1 1 1 | py 1 1 | t.1_1 1 3 20 30 40 50 TEMPERATURE (CENTIGRADE) Fig. 4 + The pt —»p~ mutation rate, “x » aS a fimction of incubation temperature. Imtire clones arising from presumptively single Pt cells are plated and the fraction, P, of clones containing zero mutants is determined. O is then estimated from the formula: X = = where N 1s the average total number of cells in the clones (Appendix 2), This method of determining & is entirely independent of any growth rate difference between parent and mutant. The experimental points are the open circles. The solid circle is the value of used for plotting the theoretical curve of Fig. 2.