Chirality, phase transitions and their induction in amino acids

Physics Letters B 288 (1992) 153-160 North-Holland

PHYSICS LETTERS B

Chirality, phase transitions and their induction in amino acids Abdus Salam

Department of TheoreticalPhysics, Imperial College,London SW7 2BZ, UK and International Centrefor TheoreticalPhysics, 1-34014 Trieste, Italy Received 11 May

1992

"Atoms such as carbon, oxygen, nitrogen and hydrogen, the major constituents of biological molecules, are less than 0.4 nm in diameter.... The behaviour of small molecules is a reflection of the intrinsic properties of the constituent atoms. Hence it might be expected that the behaviour of large macromolecules can be explained by a knowledge of atomic properties. Since organelles, whole cells and organisms are essentially macromolecular assemblies, it may be possible in time to derive an atomic theory of life" [A.R. Rees and M.J.E. Sternberg, From ceils to atoms - An illustrated introduction to molecular biology (Blackwell, Oxford, 1984 ) p. 3 ]. It has been suggested that chirality among the twenty amino acids which make up the proteins may be a consequence of a phase transition which is analogous to that due to BCS superconductivity [A. Salam, J. Mol. Evol. 33 (1991) 105 ]. We explore these ideas in this paper and show, following Lee and Drell [I.H. Lee and S.D. Drell, in: Fermion masses in the standard model, M.A.B. B6g Memorial Volume, eds. A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991 ) p. 13], that a crucial form for the transition temperature Tc involves dynamical symmetry breaking. The t-quarks or supersymmetry (or something similar which ensures a heavy mass ) appear to be essential if such mechanisms are to hold.

1. T h e Te for BCS s u p e r c o n d u c t i v i t y for m e t a l s is o f the f o r m co e x p [ - 2 / g ~ r f a ( O ) ]. We c o n j e c t u r e d that a s i m i l a r f o r m u l a m a y h o l d for the case o f a m i n o acids chains. In the p r e s e n t p a p e r we shall e x p l o r e this further. First, let us r e v i e w the subject o f BCS s u p e r c o n d u c t i v i t y . T h e best t r e a t m e n t , w h i c h I k n o w o f has b e e n g i v e n by Sakita [ 1 ], f o l l o w i n g on the w o r k o f G i n s b e r g a n d L a n d a u [ 2 ] a n d o f G o r k o v [ 3 ]. T h e i d e a is to start f r o m the F e y n m a n l a g r a n g i a n m e t h o d o l o g y o f writing d o w n the BCS t h e o r y for the superc o n d u c t i n g e l e c t r o n i c system. O n e tries to write d o w n the e q u i v a l e n t G i n z b u r g - L a n d a u e q u a t i o n . F r o m this e q u a t i o n is d e d u c e d the v a l u e o f t e m p e r a t u r e Tc. 2. T h e BCS h a m i l t o n i a n is g i v e n by

H=Ho +HI, 2 (-V) Ho= s~=,~ f dx~u~+(x) ( ~m

-/t)

~Us( x ) ,

H~ = - g e f f j" d x ~'~- ( x ) q/~- (x)~'+ (x)~'~ ( x ) ,

(2.1)

w h e r e gefr is an a t t r a c t i v e c o u p l i n g c o n s t a n t b e t w e e n spin up (~) a n d spin d o w n ($) electrons a n d a n t i e l e c t r o n s is the c h e m i c a l potential. The sign ofge~-is part of the assumption o f the h a m i l t o n i a n w h i c h signifies an a t t r a c t i v e force b e t w e e n C o o p e r - p a i r e d systems o f electrons consist o f one o f the particles being r e p l a c e d by its a n t i p a r t i c l e w i t h a factor o f two w h i c h appears in the m a s s t e r m . We can i n t r o d u c e e l e c t r o - m a g n e t i c i n t e r a c t i o n in a g a u g e - i n v a r i a n t way by the m i n i m a l s u b s t i t u t i o n V--, V - i e A , a n d treat the v e c t o r p o t e n t i a l A ( x ) as an e x t e r n a l source. T h e n , in general, the h a m i l t o n i a n equals 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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PHYSICS LETTERSB

H= ,~,~ dx~u[(x) ( -

~m 1 (V-ieA)2-/~ )

20 August 1992

~s(x)-gefr f dxgt~-(x)~[(x)~,(x)~u,(x),

(2.2)

which is invariant under gauge transformations:

q4(x)-,exp[ieA(x) ]V~(x), d(x)--,A(x) + VA(x).

(2.3)

The partition function based on the appropriate lagrangian corresponding to the above hamiltonian is given by #

Z:fD~uD~exp[-~drfdx~s(~- r 0

v

1 (V-ieA)2-#)~s] 2m

where we have written a non-relativistic equation for the fermion (electron) ~uin the theory: #

exp(geff~d'rfdx~,r(x)ffj+(x)~+(x)~,t(x)). o

(2.4)

v

Note that the sign before gefr has changed to plus g~fr instead of minus g~frin the hamiltonian formulation. This is because AH= - A L for the case of potentials which do not contain time derivatives of the fields themselves. The four-fermion interaction can be expressed in terms of a complex auxiliary Higgs scalar field ~:

c ~ Dq)DO*exp ( +x2 f dax ~*O+g~[-f2tcf d4x((/~O+~,v/~O*)) ,

(2.5)

where we have used the notation # ~ d4x= ~ dT~ d3x.

(2.6)

0

v

The constant C is given by

C= f Df~DO*exp(+x2 f d4xq)*~),

(2.7)

where x is a constant with dimension of mass. We shall introduce a source for the ~ field in the form 1 Z[J'J*]=-cfD~udftD(~DO*exp[-~d4x#s( 0

X exp ( + x 2 f

(V-ieA)2 ] 2 m - / t ) spI'

~*~d4xWgl~r2tcfdax(or~+fb+~NrO*+j*Fp+cP*j)).

(2.8)

The partition function is given by

Z=Z[J,j*] b=j*=o •

(2.9)

The action which has appeared in (2.8) is L dr where

L=Lo +LI

(2.10)

, 2m

154

(2.11)

Volume 288, number 1,2 Lt =

I

PHYSICS LETTERSB

20 August 1992

( ie e2 ) dx - ~m [~Vg/s-(VqTs)~'slA+ ~m ~s~'sA2+g~ZX(~TqT+0+qJ~Tq~*) '

(2.12)

Using the temperature dependent Fourier decomposition ~q(x, z ) = (flV) -~/2 ~ ~u~(k, n) e x p [ i ( k . x - ~ , z ) ] , k

(2.13)

we obtain the free action // f d z L o = Y~ [G(k, n)(--iG+ogk)G(k, n)+x2~*(k, n)q~(k, n ) ] .

(2.14)

k,n

0

Therefore, the propagator of the electron with four momentum (k, n) is given by 1

ogk--i~.'

k2

ogk=2--mm- / t '

~.=~(2n+l),

(2.15)

and the propagator of ~ by 1

x~.

(2.16)

Notice that the propagator of ~ to this order does not depend on four momentum, because the lagrangian (2.11 ) does not contain the kinetic energy term. 3. Calculation of the loop diagram. We shall compute one-loop diagram with two external ~lines. The (temperature dependent) leading terms are p

qo=O

-q

gerrK2

q

#v

~

1 (i~. - o9p) ( - i ~ - o9p,)

=A+Bq2+ ....

(3.1)

q-p=p,

where geffK2 S" 1 A=--~---,~+o9~

_ ~ geffK2 1 1 . -~ (2z03fd3Pc2+og-------~.

(3.2)

Introducing density of states a (09) from the ansatz 1

(270 3 f dBp~ f dogpa(ogp),

(3.3)

we get

A= ~n ge~2 i do9 0"(o9) geff~2 ; do9 -O¢29 ~.[_ ~ ~n 7 0"(0 ) ~2 ={_(.02' --oo --oo

(3.4)

where O)p~-p 2/2m_/~ = 0 is the Fermi surface and we have approximated by using density of states at the Fermi surface by putting a cut-off nmax O9D=kB Trc(2nmax + 1 ) .

(3.5) 155

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Thus 4yco~ ,4=ge~rx2o(O) In r&BT'

7= the Euler constant.

(3.6)

The coefficient of I0c 12 terms F[ G, 9" ] is then /(72

4 •0) o "~

1 --geffO'(O ) In 7rkBTJ O*G .

(3.7)

The expression within the brackets gives the critical temperature as To_ 4yah) exp ( rrk8

1 ) geff~r(0) .

(3.8)

Note that the terms appearing before the logarithm in (3.7) appear in the exponent while terms which are arguments for the log appear as ADo) in the formula for To. Since the Higgs mechanism is the relativistic version of the above, one may write the entire calculation of oneloop diagrams as well as the calculation of Tc in the form of corrections to the calculation of mass of the Higgs field. We shall adopt this procedure with slight variations. 4. The point we wish to make is summarized by (3.8) which has the form AO)D exp [ -2/gecfa(O) ]. In order that the exponential factor does not present too much of a restriction 1/geffa(O) should be of the order of unity. This gives Tc'~ (709D/r&B) exp ( -- 2). Before proceeding, we shall make some remarks about the amino acids. The crystalline structure of amino acids is characterized by the graph shown in fig. 1. Apart from the N terms, here R~ to R4 are the residues which specify the amino acid concerned. The peptide bond which gives rise to proteins is formed, for example, where the molecules of water are consistently expelled and the lattice structure is reduced to the simple form

f

polypeptid~ !~i__i'_ m.,n0.a,n ;

I

-N--

I H C--C~N--

0

H

0

C--C--~

II O--

I

H

I

(~ I

Amino acid residue Ra

c o n t i n u o u s and unbranched polymer

I

+1

I

+ -,

Fig. 1. This diagram is taken from ref. [4]. 156

O--H

L

I Amino acid residue I Amino acid I RI I residue R 2 --

H--}- N - - C - - C - -

I

I . . . . . . . . .

I

/

H H 0

M22 R group

T

Free amino acid

R4 monomer -

I

I [

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H H O I I II --N--C--C--. I R

The hydrogens give up their loose electrons and act as metallic hydrogens. (Superconductivity has been established for a similar case by Carr [ 5 ]. )

5. Helicity and chirality of the matter fields ~1. Let us review the definition of chirality, which is the eigenvalue of Y5, with Ys= + 1 corresponding to right-handedness, and 75 = - 1 to left-handedness: 7sR=R, 1~7s=-1{,

75L=-L,

/275=/5,

(5.1)

where R, L are Dirac spinors with only two independent components. They may be obtained from a four-component Dirac spinor q/by the following projections: R=½(l+Ts)9/,

/~=½~(1-75),

L=½(1-75)q/,

/S=½~(l+7s).

(5.2)

For later applications, it is important to note that

v2~u=I~R+ I{L, v27u~=ff.,TUL+ l~TUR ,

(5.3)

written as

(a.p+flm)~=E~,

E = (p2+m2) 1/2

(5.4)

Using the identity a = 75a, and the fact that 75 and a commute, we can rewrite this in the form

o.,OR= E R - m---ilL, p= IPl, P

~r.pt= - E-L- mflR.

P

P

(5.5)

P

These equations become decoupled if m = 0:

cr.~R=R, a . p L = - L .

(5.6)

Therefore, for massless Dirac particles, chirality is the same as helicity, for antiparticles, chirality is the opposite of helicity. [An antiparticle has the same chirality as the particle, by definition; but it has the opposite helicity due to a change in the sign of E in (5.4). ] A conventional mechanical mass term in the lagrangian density cannot be invariant under SU (2), because it is proportional to q?~v=LR + I~L. Therefore, in this theory the electron mass can arise only by virtue of a spontaneous breakdown of SU (2). A convenient way to implement this is to introduce a doublet Higgs field

Oo '

0=

where the subscripts refer to the electric charges. The mass term is ....

=O(EOR+I~o,LP),

(5.8)

where/20 is an SU (2) singlet, and a Dirac spinor. If 0 has non-zero vacuum value, then for low excitations and

P=Po, (5.8) is indistinguishable from a conventional mass term. The Weinberg-Salam model is obtained by gauging SU (2) × U ( 1 ), generated by weak hypercharge. ~t This chapter is taken essentially from the book of Huang [6]. 157

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W i t h these, t h e e o v a r i a n t d e r i v a t i v e c a n b e w r i t t e n i n t h e f o r m

W~t~ + W~t2) + ieQAU+ieQ'Z~,.

D~'=~'+ig(

(5.9)

E l e c t r i c c h a r g e = eQ, Q= r3 + Zo w h e r e t h e n e u t r a l c h a r g e m a t r i x Q ' is d e f i n e d b y

eQ' = r3 cot 0w - ~o t a n 0w.

(5.10)

H e r e ro is t h e singlet m a t r i x a m o n g ~o, z~, r2, r3, w h i l e Q = r3 + to. To s t u d y t h e m a s s e s o f t h e gauge fields, it is c o n v e n i e n t to go to u n i t a r y gauge #~, i n w h i c h

O=

IPo

'

w h e r e P/Po is a real field. I n t e r m s o f fields in t h e u n i t a r y gauge, t h e l a g r a n g i a n d e n s i t y is

~ - - - - l ( a . a + H . ~ ) + ~ g ~ p ~ w~ + W~ + co--p~ ~ +op.op-~(p~-pg) ~.

(5.12)

R e w r i t e i n t h e u n i t a r y gauge, w h e r e t / i s t h e real Higgs f e l d in u n i t a r y gauge, i n w h i c h O(x) h a s t h e f o r m and

(5.12)

p(x) =Po + r/(x).

( 5.13 )

6. T o c o m p u t e a n y f u r t h e r , we m u s t g u a r a n t e e t h e gerf i n t h e e f f e c t i v e l a g r a n g i a n is a p o s i t i v e n u m b e r corres p o n d i n g t o a t t r a c t i o n . F u r t h e r , t h e p r o t o n left b e h i n d m i g r a t e s t o t h e n i t r o g e n so t h a t t h e Z ° m e s o n b e f o r e b e i n g a n n i h i l a t e d gives rise to a n i n t e r a c t i o n b e t w e e n t h e q u a r k s c o n t a i n e d i n s i d e t h e p r o t o n a n d i n s i d e t h e n i t r o g e n . ( T h a t it h a s to b e n i t r o g e n r a t h e r t h a n t h e c a r b o n ( n e x t d o o r ) is e m p i r i c a l l y g u a r a n t e e d b y t h e " r i g h t " c o n f i g u r a t i o n w h i c h is i m p a r t e d to t h e c o r r e s p o n d i n g m o l e c u l e w h i c h c o n t a i n s s u g a r s ~3 a n d n o n i t r o g e n s ~4. ) T o c o n s i d e r Z ° c o n t a i n i n g p a r t o f t h e l a g r a n g i a n , w r i t e it as ,2 We should in fact have a triplet of fields

W+=Wj - ~ , likewise W- = W~+ IW2/~f2 belonging to a vector representation of the internal group SU (2), while the Higgs panicles are a complex representation of the same group SU (2). We have neglected W + and W - throughout this paper because their contribution is small at low momenta. ,3 We may consider 2, 8, 20, 28, 50, 82 and 126 as "magic numbers" for nuclei, provided spin-orbit coupling is taken into account. (See ref. [ 7 ]. ) Nuclei containing 2, 8, 20, 28, 50, 82 or 126 neutrons or protons are particularly stable. The detailed evidence supporting this point of view is discussed in ref. [ 8 ] with the fact of 20 coming out naturally. Goeppert Mayer goes on to consider in detail the protons and neutrons and the spin-orbit couplings in terms of a potential energy which has a shape somewhat between that of a square well and a three-dimensional isotropic oscillator. (See table 1. ) ,4 The discussion from now on is not exclusively relevant to the rest of this paper. The problem for the usual quark model is to see if the Pauli principle holds for quark assignments. This means that in the final analysis the quark model which substitutes quarks for nucleons will have quark spheres of influence reduced depending on the number of colours. This implies that certain discrepancies in the nucleon model are removed when quarks are taken into account. These discrepancies, for example, concern the spin of l~Na j2. The magnetic moment of this nucleus would indicate P3/2 rather than d3/2 orbit. That these considerations have something new to tell us is an important point in its favour. (We would urge very strongly that parity violation among the proteins used by this mechanism should be further examined in order that the Z ° and its decay are properly considered. ) 158

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Table 1 Oscillator number

Square well

Spectral term

Spin term

Number of states

0

ls

ls

ls~/2 lp~/2

l

lp 1d

2p 3d

lp3/2 I d3/2

2s

2s

ld3/2 2sj/2

2 4 2 6 4 2

2

Shells

Total number

2

2

6

8

12 20

eZ °

Lint ( Z ° ) - sin 0 c o s 0 [ (T3c-sin2OJem L+R) ] •

(6.1)

H e r e Jem is the electromagnetic current, T3L is the left-handed third c o m p o n e n t of the weak isospin consisting

o f the ( a n o m a l y - f r e e ) c o m b i n a t i o n o f the p r o t o n a n d neutron (p, n ) , (or the quarks inside the proton or the n e u t r o n ) . Z ° here requires a mass because o f the spontaneous s y m m e t r y breaking i m p l i e d by the last equation. L . . . . =geff[ T3 - (4 sin20Jem L+R) ]Lint ~o •

(6.2)

Here geff is p r o p o r t i o n a l to (e 2/ 16 sin20 cos20) rh,, / (k 2 - m ~) where t/u, = 1, - 1, - 1, - 1. This is positive prov i d e d m ~ >> k 2 a n d we consider the three space-rotation invariant part o f the expression above. We shall take ( 1 - 4 sin20) ~ ~3 with the present empirical value of the p a r a m e t e r sin20g 0.231. We shall not take this quantity to equal zero as has been done by authors o f refs. [8,9 ] in the hope that the renormalization group will give the " e x a c t " value as 1 (personal c o m m u n i c a t i o n ) . We do not believe this will ever happen. F o r the mass term we take the ansatz, as the s y m m e t r y breaking which is given by eq. (2.4), which follows on from (2.5) and (2.7) where x is a constant with d i m e n s i o n o f mass. We shall introduce a source for the 0 field in the form o f eq. (2.8). In terms o f the quantity ( (~0) o = 250 GeV ) the electron mass Me turns out to be a very tiny n u m b e r ~ 2 X 10 - 6. Such a n u m b e r is large only for the top quark if its mass is in excess of 100 GeV. Thus the term which gives rise to this, looks likef~0/-c tR wheref>~ 2. Some physicists (like Y. N a m b u ) take this as the defining property o f the field ~0, i.e. ~0is considered as a t t composite. According to Lee and Drell, a proposal is considered according to which the masses o f the fermions in the standard m o d e l are d e t e r m i n e d by d y n a m i c a l s y m m e t r y breaking rather than being introduced as arbitrary p a r a m e t e r s in the lagrangian, they are d e t e r m i n e d self-consistently by the requirement that the proper selfenergy vanish the fermion mass shell. They find that in the one-loop a p p r o x i m a t i o n it is possible to generate a heavy top quark mass d y n a m i c a l l y while the other fcrmions r e m a i n massless [ 10 ]. We find that non-zero solutions for m, do exist and they are always greater than 70 GeV for all values of cutoffs A = 5 m~, 10 m~, 20 mz and is the only Higgs mass explored in the problem. One m a y thus be led to conclude that, while a heavy top quark mass can be generated dynamically, the observed masses o f the light fermion cannot be generated in this approach. Recently, Ruiz-Altaba, Gonzalez and Vargas [ 11 ] extended this to a two-loop calculation predicting mt = 124 GeV, mH = 234 G e V and the weak mixing angle sin20w-~ 0.24. Our physical m o t i v a t i o n is to r e m o v e the fermion masses arbitrary p a r a m e t e r s o f the standard model and treat t h e m as p a r a m e t e r s d e t e r m i n e d by the d y n a m i c s in the sub-TeV region. Rather than cancel the divergences, they impose and interpret the s t a n d a r d m o d e l 5° as an effective lagrangian in order to d e t e r m i n e the physical masses. Thus using Sakita's formulation, we get the result (~0) exp[ Tc=~

2/geff~ ( 1 - 4 sin20) ] ~ 2 . 5 × 102 K .

(6.3)

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The exponential factor gives exp ( - 2 6 ) ~ 10-1 o. Assuming that a ( 0 ) ~ rn ~ we obtain gefra ( 0 ) ~ 1. 7. N o w we go on to the supersymmetry case. The symmetry resolves the dilemma of hierarchy. In a theory possessing fermion-boson symmetry because of the presence o f both fermionic and bosonic radiative loops, the radiative shifts for scalar fields are not quadratically divergent. In the limit o f broken supersymmetry, the shifts in v 2 only depend logarithmically on the cutoffA. To guarantee that geffa(0) ~ 1, we must take some mass in the theory which is large enough. The simplest mass is obtained by taking supersymmetry such that 0o--*~o + a [ AM2 ] In ~o-o'

(7.1)

where A M 2 is the mass splitting between supersymmetry fermions and is of order 1 TeV [ 12 ] which gives ~ k F / 9 2 m z2 ~ 1 and thus geffCr(0 ) of the order o f unity once again. This paper started with the possibility of defining a science of life based on atomic physics. What we have shown is that if the model is Zo it is possible to have the quark instead o f the nucleons. The mass o f Zo is ~ 100 GeV, while the mass of the quarks is ~ 200 GeV so that ifZo loops are considered the tt quarks go through these loops. The gefr which is proportional to m ~ 2 then is o f the right magnitude to be able to let these quarks make composites which are manifested through their passage through these loops.

References [ 1 ] B. Sakita, Quantum theory of many-variable systems and fields (World Scientific, Singapore, 1985) pp. 42-51. [2] V.L. Ginzberg and L.D. Landau, Soy. Phys. JETP 20 (1950) 1064. [3] L.P. Gorkov, Sov. Phys. JETP Z. (1958) 505; JETP 9 (1959) 1364. [4 ] A.R. Rees and M.J.E. Sternberg, From cells to atoms - An illustrated introduction to molecular biology (Blackwell, Oxford, 1984) p. 13. [5] W.J. Cart Jr., Phys. Rev. B 33 (1986) 1585. [ 6 ] K. Huang, Quarks, leptons and gauge fields (World Scientific, Singapore, 1982 ). [7] M. Goeppert Mayer, Phys. Rev. 78 (1950) 16. [ 8 ] G.E. Tranter and A.J. MacDermott, Croatica Chem. Acta 62 (2A) ( 1989 ) 165. [9] S.F. Mason and G.E. Tranter, Molec. Phys. 53 (1984) 1091. [ 10] I.-H. Lee and S.D. Drell, in: Fermion masses in the standard model, M.A.B. Brg Memorial Volume, eds. A. Alt and P. Hoodbhoy (World Scientific, Singapore, 1991 ) p. 13. [ 11 ] M. Ruiz-Altaba, B. Gonzalez and M. Vargas, preprint CERN-TH 5558/89, UGVA-DPT 89/11-638 (unpublished); see also discussion in I. Jack and D.R.T. Jones, Phys. Lett. B 234 (1990) 321. [ 12 ] U. Amaldi, Scientific American ( 1991 ), and CERN preprint.

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