MARTENSITE AND LIFE : DISPLACIVE TRANSFORMATIONS AS BIOLOGICAL PROCESSES

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TRANSFORMATIONS AS BIOLOGICAL PROCESSES

G. Olson, H. Hartman

To cite this version:

G. Olson, H. Hartman. MARTENSITE AND LIFE : DISPLACIVE TRANSFORMATIONS AS BIOLOGICAL PROCESSES. Journal de Physique Colloques, 1982, 43 (C4), pp.C4-855-C4-865.

�10.1051/jphyscol:19824140�. �jpa-00222128�

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JOURNAL DE PHYSIQUE

CoZZoque C4, suppZe'ment au n o 12, Tome 4 3 , dQcembre 1982 page C4-855

MARTENSITE AND L I F E

:

DISPLACIVE TRANSFORMATIONS AS BIOLOGICAL

PROCESSES

G . B. O l s o n a n d H. H a r t m a n MIT, Cambridge, MA, U.S.A.

( A c c e p t e d 9 A u g u s t 1 9 8 2 )

Abstract. - Martensi t i c transformations i n c y l i n d r i c a l p r o t e i n c r y s t a l s a r e found t o perform l i f e f u n c t i o n s i n p r i m i t i v e b i o l o g i c a l systems. T a i l - sheath c o n t r a c t i o n i n T-even bacteriophages can be described as an i r r e - v e r s i b l e s t r a i n - i n d u c e d m a r t e n s i t i c transformation, w h i l e polymorphic t r a n s - formations i n b a c t e r i a l f l a g e l l a appear t o be s t r e s s - a s s i s t e d mechanically r e v e r s i b l e martensi t i c transformations e x h i b i t i n g a shape memory e f f e c t . A v a i l a b l e i n f o r m a t i o n i n d i c a t e s t h a t t h e geometric, thermodynamic, and k i n e t i c aspects o f these transformations a r e c o n s i s t e n t w i t h m a r t e n s i t i c be- h a v i o r . S i m i l a r t r a n s f o r m a t i o n s i n v o l v i n g t h e motion o f p a r t i a l (coherency) d i s l o c a t i o n s under chemical forces may underly t h e mechanism o f motion i n h i g h e r organisms as w e l l .

I n t r o d u c t i o n . - Recent advances i n molecular b i o l o g y have revealed t h a t a remarkable number of b i o l o g i c a l s t r u c t u r e s i n v o l v e p e r i o d i c o r c r y s t a l l i n e a r r a y s o f p r o t e i n molecules. Even more remarkable, from t h e viewpoint o f a m a t e r i a l s s c i e n t i s t , i s t h e f a c t t h a t d i s p l a c i v e phase transformations between metastable s t a t e s f r e q u e n t l y occur and p r o v i d e t h e mechanism o f important l i f e processes. While c r y s t a l s t r u c - t u r e s o f importance t o m a t e r i a l s science, and m e t a l l u r g y i n p a r t i c u l a r , have been known f o r q u i t e some t i m e and considerable a t t e n t i o n has focussed on t h e mechanism and k i n e t i c s o f s o l i d - s t a t e transformations, the q u e s t i o n o f t r a n s f o r m a t i o n mechanisms i n the r e c e n t l y determined molecular c r y s t a l s t r u c t u r e s i s j u s t now being addressed i n t h e f i e l d o f b i o l o g y . These systems may p r o v i d e important t e s t s o f the g e n e r a l i t y o f fundamental concepts i n m a t e r i a l s science.

We here examine t h e relevance o f martensi t i c t r a n s f o r m a t i o n theory t o b i o l o g i - c a l systems adapting t h e t h e o r y t o reduced d i m e n s i o n a l i t y using two examples o f d i s p l a c i v e transformations i n c y l i n d r i c a l p r o t e i n c r y s t a l s performing 1 i f e f u n c t i o n s i n v i r u s e s and b a c t e r i a .

B a c t e r i a l - V i r u s Tai 1 -Sheath Contraction. - A t r a n s f o r m a t i o n f o r which t h e most pre- c i s e s t r u c t u r a l i n f o r m a t i o n i s a v a i l a b l e i s the case o f t a i l - s h e a t h c o n t r a c t i o n i n T-even bacteriophages, v i r u s e s t h a t i n f e c t E . c o l i b a c t e r i a . The c o n s t r u c t i o n o f a bacteriophage and t h e process o f c o n t r a c t i o n are i l l u s t r a t e d i n F i g u r e 1. The v i r u s c o n s i s t s o f a D N A - f i l l e d icosahedral head o r capsid attached t o a t a i l assembly composed o f a c y l i n d r i c a l sheath surrounding a narrow t u b u l a r core. A t t h e end o f t h e t a i l i s a hexagonal baseplate from which extend s i x long and s i x s h o r t t a i l f i b r e s . The v i r u s head i s a two-dimensional p r o t e i n c r y s t a l which closes on i t s e l f as a r e s u l t o f a p e r i o d i c a r r a y o f d i s c l i n a t i o n s . The t a i l sheath i s also a two- dimensional p r o t e i n c r y s t a l which closes t o form a c y l i n d e r . The l a t t e r c r y s t a l i s aP- p a r e n t l y i n a metastable s t a t e formed by " e p i t a x i a l " growth o f the sheath on t h e baseplate and core d u r i n g t h e i n i t i a l assembly process o f the v i r u s ( 2 ) . When t h e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19824140

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v i r u s t a i l contacts a bacterium, attachment o f t h e t a i l f i b r e s t o t h e b a c t e r i a l c e l l w a l l transforms t h e baseplate from t h e hexagonal t o a "star-shaped" form (3).

This i n t u r n t r i g g e r s a transformation i n t h e t a i l sheath i n v o l v i n g a c o n t r a c t i o n which d r i v e s t h e r i g i d core through t h e b a c t e r i a l c e l l w a l l , i n j e c t i n g t h e v i r u s

DNA

i n t o t h e bacterium.

Figure 1. T a i l - s h e a t h c o n t r a c t i o n The c r y s t a l l a t t i c e s t r u c t u r e o f t h e t a i l sheath has been p r e c i s e l y determined by e l e c t r o n microscopy and X-ray d i f f r a c t i o n f o r t h e case o f t h e T4 bacteriophage. The c y l i n d r i c a l c r y s t a l can be described as a s t a c k i n g o f 24 annular r i n g s o f s i x g l o b u l a r gP18 p r o t e i n sub- u n i t s each. I n t h e "extended" form the hexagonal r i n g s a r e spaced 4nm and each i s r o t a t e d about t h e c y l i n d e r a x i s r e l a - t i v e t o i t s neighbor by 17"

.

"Unwrap- p i n g " t h e c y l i n d e r t o form a plane l a t - t i c e gives t h e s t r u c t u r e d e p i c t e d i n F i g u r e 2a. As discussed by H a r r i s and Scriven (4) the s t r u c t u r e o f a c y l i n d r i - c a l c r y s t a l can be s p e c i f i e d by combin- i n g a two dimensional l a t t i c e o f p r i m i - t i v e vectors a l , and a2 w i t h a charact- e r i s t i c v e c t o r C which f o l l o w s t h e c i r - cumference o f t h e c y l i n d e r . The vectors a1 and a2 i n F i g u r e 2a were chosen along d i r e c t i o n s o f c l o s e s t packing. On t h e c l o s e d c y l i n d e r , each o f t h e d i r e c t i o n s d e f i n e s s i x h e l i c e s , l e f t - h a n d e d and right-handed, r e s p e c t i v e l y .

Upon t a i l - s h e a t h c o n t r a c t i o n , the spacing o f t h e a n n u l i decreases t o 1.8nm, t h e annular r o t a t i o n between them increa- ses t o 3 2 O , and t h e c y l i n d e r o u t e r d i a -

process i n T4-bacteriophage (1)

Figure 2. C r y s t a l l a t t i c e s t r u c t u r e of T4-bacteriophage t a i l - s h e a t h i n (a) extended and ( b ) contracted forms, showing corresponding l a t t i c e vectors.

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meter expands from 24 t o 34

nm,

giving t h e l a t t i c e s t r u c t u r e depicted i n Fig. 2b.

Electron microscopy observation of tail-sheaths in an a r t i f i c a l ly-induced s t a t e of p a r t i a l transformation not only indicates t h a t the transformation i s displacive (and diffusionless) in nature, but allows a d i r e c t determination of the operative l a t t i c e correspondence ( 5 ) . This correspondence i s depicted by the corresponding vectors a i , a?, and C' i n Figure 2b.

The d i s t o r t i o n s of corresponding vectors in Figure 2 define the l a t t i c e deformation accompanying t h i s transformation. Inasmuch as the conversion of re1 a t i v e rotation o r t w i s t about the cylinder axis t o shear s t r a i n i n the two dimensional l a t t i c e i s dependent

on

the cylinder diameter, the l a t t i c e s of Figure 2 were con- s t r u c t e d using an e f f e c t i v e i n i t i a l diameter of 13 nm based on positions of the protein subunit "centers" estimated from electron density contours generated by electron microscopy ( 6 ) . Adopting an orthogonal coordinate system with xi

[I

^{C and}

x2 along the cylinder a x i s , the t o t a l l a t t i c e deformation

S

accompanying tail-sheath contraction i s expressed by:

b J

The deformation shear component of r), 0.6050 corresponds t o a large twist about the cylinder axis by which t h e virus

I

^ead⁼ (and t a i l core) makes one corriplete revolution. While tail-sheath contraction i s often likened t o the operation of a hypodermic needle, the analogy of a d r i l l press, whereby the t a i l core "bores"

through the bacterial c e l l wall, i s more appropriate.

The t o t a l deformation can be factored into a rigid-body rotation

I?

and a pure deformation ij expressed as:

Ihe rotation i s a clockwise rotation of 17.8O. Expressed i n principal coordinates X I ,

x2

(defined by a counter clockwise rotation of 24.1°), the pure l a t t i c e deformation i s :

1.5872

= [ 0

0.3920

.

o I

⁽³⁾

This i s a r e l a t i v e l y large l a t t i c e deformation, amounting t o a -60% expansion along one principal axis and a -60% contraction along the other. I t i s , however, the smallest principal d i s t o r t i o n r e l a t i n g the two s t r u c t u r e s of Figure 2. The l a t t i c e d i s t o r t i o n i s predominately deviatoric with the t o t a l shear s t r a i n exceeding 100%.

Such a large shear-dominant l a t t i c e d i s t o r t i o n would suggest t h a t t h i s diffusionless displ acive transformati on be classed as martensitic ( 7 ) .

As w i l l occur f o r any two-dimensional deformation in which the principal s t r a i n s are of opposite sign, the l a t t i c e deformation leaves two vectors unchanged in length. The orientation of two undistorted l i n e s ^Rjand 2.2 r e l a t i v e t o the parent- crystal and deformation coordinate systems a r e depicted in Figure 3; El i s 29.2' from xl with R 12.6' from x2. The undistorted l i n e s a r e f a i r l y close t o the [l 01 and [O 13 crystal close-packed directions, consistent with a r e l a t i v e l y small change in near-neighbor separations during transformation. ( I t i s i n t e r e s t i n g t o note, however, t h a t the [I T] direction, nearly p a r a l l e l t o k2, undergoes such a large contraction t h a t i t becomes more close-packed than the original close-packed directions, see Figure 2 ) . The s t r a i n energy associated with a martensitic i n t e r - face in a planar two-dimensional crystal would be minimized i f t h e i n t e r f a c e l i e s

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Figure 3. Orientation of undistorted Figure 4. Dislocation model of coherent l i n e s R1, R r e l a t i v e t o parent-crystal m r t e n s i t i c i n t e r f a c e i n T4 bacteriophage and deformaZion coordinate systems. t a i l - s h e a t h . Six dislocations each move Defopation axes x l , x2; principal axes, along [TO] and [01] helical l a t t i c e rows.

x2; crystal directions [lo] and [01]

parallel t o a1 and a2.

along an undistorted l i n e , a1 o r 9,

.

Meeting of the s t r u c t u r e s along t h i s l i n e would leave the l i n e unrotated s u c i t h a t the l a t t i c e deformation would then be an

invariant-1 ine s t r a i n , the two-dimensional equivalent of an invariant-plane s t r a i n . Thus, provided principal l a t t i c e s t r a i n s a r e of opposite sign such t h a t an undistorted 1 ine e x i s t s , no additional deformation (e.g. l a t t i c e - i n v a r i a n t shear) i s required t o achieve s t r e s s - f r e e matching a t a martensitic i n t e r f a c e in a planar two-dimensional c r y s t a l . The s i t u a t i o n i s more complicated f o r the case of a cylindrical c r y s t a l . The periodic boundary conditions imposed by the c h a r a c t e r i s t i c vector C d i c t a t e t h a t the only " s t r a i g h t " l i n e which closes on i t s e l f , and therefore the only orientation of a " f l a t " interface, i s parallel t o C. Imposed meeting of the two s t r u c t u r e s along C guarantees t h a t t h i s vector i s unrotated across the i n t e r - face and t h e parent-product orientation relation i s thus specified by t h e condition:

consistent with the observed behavior of p a r t i a l l y transformed t a i l sheaths ( 5 ) . The s t r a i n energy associated with an interface constrained t o l i e a l o n g C could be minimized i f the i n t e r f a c e i s made semicoherent by the introduction of a l a t t i c e - invariant deformation t o make C a "macroscopically" invariant l i n e . In view of the small f i n i t e dimension of C, however, exact matching would be d i f f i c u l t t o obtain through (anti coherency) i n t e r f a r i a l dislocations with l a t t i c e Burgers vectors.

Whether the i n t e r f a c e i s f u l l y o r p a r t i a l l y coherent, the orientation relation will always be specified by equation 4.

Observation of p a r t i a l l y transformed t a i l sheaths indicates t h a t the interface remains f u l l y coherent ( 5 ) . E l a s t i c d i s t o r t i o n in the coherent i n t e r f a c e will make the i n t e r f a c e l e s s sharp than would be the case f o r an invariant l i n e interface, and t h i s i s observed ( 5 ) . Despite the more diffuse character of the interface, the nature of i t s long-range displacements and i t s underlying discrete-crystal s t e p

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s t r u c t u r e a l l o w i t s d e s c r i p t i o n v i a i n t e r f a c i a l d i s l o c a t i o n a r r a y s . Resolving t h e t o t a l l a t t i c e deformation

S

i n t o i n v a r i a n t l i n e s t r a i n s on t h e [I 01 and [0 1 1 close-packed d i r e c t i o n s , t h e f u l l y coherent i n t e r f a c e can be modelled by two s e t s o f coherency ( o r t r a n s f o r m a t i o n ) d i s l o c a t i o n s (8,9) as d e p i c t e d i n F i g u r e 4. The h e l i c a l n a t u r e o f t h e close-packed d i r e c t i o n s i n t h e c y l i n d r i c a l c r y s t a l provides a b u i l t - i n "pole mechanism" whereby o n l y s i x d i s l o c a t i o n s o f each type need s p i r a l up t h e

[f

01 and [0 1 1 d i r e c t i o n s (as i n d i c a t e d by t h e arrows i n F i g u r e 4) t o completely t r a n s f o r m the c r y s t a l . Diffuseness o f t h e coherent i n t e r f a c e i s equiva- l e n t t o a widening o f t h e d i s l o c a t i o n cores.

The p o s s i b l e relevance o f d i s l o c a t i o n s moving along close-packed h e l i c e s i n t a i l - s h e a t h c o n t r a c t i o n was f i r s t proposed by H a r r i s and Scriven ( 4 ) . However, t h e i r s p e c i f i c model amounts t o deformation by s l i p , whereby d i s l o c a t i o n s w i t h l a t - t i c e Burgers v e c t o r s deform t h e c r y s t a l v i a a l a t t i c e - i n v a r i a n t deformation. I n c o n t r a s t t o t h e coherency d i s l o c a t i o n a r r a y o f F i g u r e 4, such a deformation i n v o l v e s severe bond-breaking d i s t o r t i o n s a t t h e d i s l o c a t i o n cores, does n o t transform t h e c r y s t a l t o a new s t r u c t u r e , and must be d r i v e n by e x t e r n a l l y - a p p l i e d mechanical forces. By coherently t r a n s f o r m i n g t h e c r y s t a l t o a new s t r u c t u r e o f d i f f e r i n g f r e e energy t h e coherency d i s l o c a t i o n s propagate t h e l a t t i c e - d e f o r m a t i o n i n response t o " i n t e r n a l " chemical forces, thereby p r o v i d i n g an e f f e c t i v e means o f c o n v e r t i n g chemical energy t o mechanical work. The l a c k o f d i s r u p t i o n o f near-neighbor bonds a l s o provides t h e i n t r i n s i c p o t e n t i a l f o r s t r u c t u r a l r e v e r s i b i l i t y .

A r e c e n t l y proposed d e f i n i t i o n o f m a r t e n s i t i c transformations as a subset o f d i f f u s i o n l e s s / d i s p l a c i v e t r a n s f o r m a t i o n s r e q u i r e s t h a t l a t t i c e - d i s t o r t i v e shear displacements be s u f f i c i e n t l y l a r g e t h a t t h e t r a n s f o r m a t i o n i s dominated by s t r a i n energy (7); a c o r o l l a r y o f t h i s d e f i n i t i o n i s t h a t m a r t e n s i t i c transformations a r e f i r s t - o r d e r i n nature. While t h e t r a n s f o r m a t i o n s t r a i n s i n v o l v e d i n t a i l - s h e a t h c o n t r a c t i o n a r e q u i t e large, t h e a c t u a l magnitude o f e l a s t i c s t r a i n energy i n v o l v e d depends on t h e e l a s t i c moduli o f the c r y s t a l . As w i l l be discussed l a t e r , t h e shear modulus IJ o f c y l i n d r i c a l p r o t e i n c r y s t a l s i s estimated t o be two orders of magnitude s m a l l e r than t h a t o f metals. Since t h e e f f e c t i v e t r a n s f o r m a t i o n shear y i s an o r d e r o f magnitude h i g h e r than f o r t y p i c a l m a r t e n s i t i c transformations i n metals, t h e s t r a i n energy product py2 w i l l be o f comparable magnitude. The f i r s t o r d e r n a t u r e o f t h e t r a n s f o r m a t i o n i s e v i d e n t from c a l o r i m e t r i c measurements which reveal a l a r g e l a t e n t heat o f AH=-190 kJ/mole (10).

Another f e a t u r e o f t a i l - s h e a t h c o n t r a c t i o n which i s c h a r a c t e r i s t i c o f a s t r a i n - energy-dominated m a r t e n s i t i c t r a n s f o r m a t i o n i s t h e apparent d i f f i c u l t y o f nucleation.

Judging from t h e r e l a t i v e l y h i g h amount o f work r e q u i r e d t o r u p t u r e the b a c t e r i a l c e l l w a l l d u r i n g c o n t r a c t i o n , t h e t r a n s f o r m a t i o n must i n v o l v e a s u b s t a n t i a l f r e e - energy change. M a i n t a i n i n g t h e extended sheath i n a h i g h l y metastable s t a t e r e q u i r e s a l a r g e b a r r i e r t o m a r t e n s i t i c n u c l e a t i o n . This i s c o n s i s t e n t w i t h t h e need f o r s t r a i n - i n d u c e d n u c l e a t i o n i n which the t a i l base-plate becomes a heterogeneous n u c l e a t i o n s i t e as a r e s u l t o f i t s d i s t o r t i o n v i a displacement o f t h e t a i l f i b r e s . The i n t e r f a c i a l d i s l o c a t i o n s o f Figure 4 would then be generated from displacements i n the baseplate.

Polymorphic Transformations i n B a c t e r i a l F l a g e l l a . - Another important d i s p l a c i v e t r a n s f o r m a t i o n f o r which d e t a i l e d s t r u c t u r a l and k i n e t i c i n f o r m a t i o n i s a v a i l a b l e i n v o l v e s t h e mechanism o f motion o f many b a c t e r i a i n c l u d i n g Salmonel l a , E . c o l i

,

and B.subti1 i s . These b a c t e r i a propel themselves by r i g i d r o t a t i o n o f " f l a g e l l a " , which are he1 i c a l f i l a m e n t s c o n s i s t i n g o f d i s t o r t e d c y l i n d r i c a l c r y s t a l s o f t h e p r o t e i n f l a g e l l i n . A bacterium c o n t r o l s i t s t r a v e l by a l t e r n a t i n g between two modes of motion d e p i c t e d i n F i g u r e 5. I n normal swinuning, t h e f l a g e l l a are i n a l e f t handed h e l i c a l form, termed t h e "normal" form, and a r e r o t a t e d counter-clockwise (viewed l o o k i n g toward t h e b a c t e r i a l c e l l ) t o a c t as screw-type p r o p e l l e r s . Although t h e i n d i v i d u a l f l a g e l l a a r e l o c a t e d randomly on t h e c e l l w a l l , motion causes them t o group t o g e t h e r t o form a coordinated h e l i c a l bundle as shown i n F i g u r e 5a. The d i r e c t i o n o f t r a v e l o f t h e bacterium can be changed i n a random manner by changing t o a "tumbling" motion v i a r e v e r s a l o f t h e d i r e c t i o n o f

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Figure 5. Appearance o f bacterium and f l a g e l l a i n (a) forward swimming and (b) tumbl i n g mode. White arrows i n d i c a t e d i r e c t i o n o f b a c t e r i a l motion, dark arrows apparent propagation o f r o t a t i n g he1 i c a l f l a g e l l a (1 1 ).

r o t a t i o n o f the f l a g e l l a . Viscous drag o f t h e surrounding f l u i d then i n i t i a t e s a t r a n s f o r m a t i o n t o a right-handed h e l i c a l form w i t h a d i f f e r e n t p i t c h angle, termed t h e " c u r l y " form. A sharp change i n d i r e c t i o n o f t h e h e l i c a l a x i s across t h e t r a n s f o r m a t i o n i n t e r f a c e i n p a r t i a l l y transformed f l a g e l l a causes t h e f l a g e l l a r bundle t o f l y a p a r t and renders t h e i n d i v i d u a l f l a g e l l a i n e f f e c t i v e as p r o p e l l e r s , r e s u l t i n g i n a random c h a o t i c motion as depicted i n Figure 5b. Reversal back t o t h e o r i g i n a l sense o f r o t a t i o n induces r e v e r s i o n o f t h e f l a g e l l a t o t h e o r i g i n a l left-handed s t a t e and n e t motion ensues i n a new d i r e c t i o n o f t r a v e l . By a d j u s t i n g t h e t i m e spent i n each mode, a bacterium can s t e e r i t s e l f t o a d e s i r e d environment.

Studies o f i s o l a t e d f l a g e l l a reveal t h a t , i n a d d i t i o n t o t h e normal and c u r l y forms important i n b a c t e r i a l motion, a wide v a r i e t y o f metastable h e l i c a l forms e x i s t s , described as polymorphism, and r e v e r s i b l e transformations between these forms can be induced by a l t e r a t i o n o f t h e thermodynamic environment. For t h e case o f Salmonella and E. c o l i f l a g e l l a , phase diagrams have been constructed mapping t h e s t a b i l i t y o f the various forms as a f u n c t i o n o f pH, i o n i c concentration, and temp- e r a t u r e (12-14).

Among t h e observed f l a g e l l a r forms i s a p e r f e c t l y s t r a i g h t form which permits d i r e c t determination o f i t s c r y s t a l s t r u c t u r e by e l e c t r o n microscopy; t h e s t r u c t u r e f o r s t r a i g h t f l a g e l l a o f Salmonella i s d e p i c t e d i n Figure 6. Although t h e corresponding p r e c i s e s t r u c t u r a l i n f o r m a t i o n i s n o t a v a i l a b l e f o r the h e l i c a l f l age1 l a r forms, i t i s g e n e r a l l y accepted t h a t t h e various forms a r e associated w i t h s l i g h t changes ( d i f f u s i o n l e s s and d i s p l a c i v e ) o f t h e p r o t e i n c r y s t a l s t r u c t u r e , and s p e c i f i c s t r u c t u r a l models have been proposed c o n s i s t e n t w i t h observed c o r r e l a t i o n s between t w i s t and c u r v a t u r e o f the h e l i c a l forms (16). The s t r a i g h t c y l i n d r i c a l c r y s t a l o f Figure 6 can be converted t o a macroscopically h e l i c a l form by inhomogeneous normal s t r a i n s along t h e n e a r l y l o n g i t u d i n a l ( a x i a l ) l a t t i c e rows. I f these rows were e x a c t l y l o n g i t u d i n a l , expansion on one s i d e o f t h e c y l i n d e r and con- t r a c t i o n on t h e o t h e r would i m p a r t a simple c u r v a t u r e (bending) t o t h e c y l i n d e r . By f o l l o w i n g n e a r l y l o n g i t u d i n a l rows which are h e l i c a l i n nature, these normal s t r a i n s convert t h e c y l i n d e r i n t o t h e form o f a macroscopic h e l i x . If t h e normal s t r a i n s remain d i r e c t e d along these l a t t i c e rows, t w i s t deformation about t h e c y l i n d e r a x i s can c o n v e r t t h e p i t c h and handedness o f t h e h e l i c a l form. Referred t o an i n i t i a l l y r e c t a n g u l a r u n i t c e l l , the l a t t i c e deformation accompanying d i s - p l a c i v e t r a n s f o r m a t i o n s between t h e h e l i c a l forms i s then o f t h e type depicted i n Figure 7. While t h e p r e c i s e nature o f the i n t r a - c e l l s h u f f l i n g accompanying the near1 y-axi a1 i nhomogeneous displacements i s n o t known, t h e shear deformation which

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Figure 6. Crystal l a t t i c e s t r u c t u r e of Figure 7. Deformation of i n i t i a l l y Salmonell a f l agel 1 a , cyl inder axis v e r t i - rectangular unit cell during f l agel l a r cal

.

Lines depict 11 nearly 1 ongi tudinal polymorphic transformation, showing he1 i cal l a t t i c e rows, one short-pi tch inhomogeneous axial displacements (near circumferential) helical l a t t i c e and uniform shear o r twist about

row (15). vertical cylinder axis.

constitutes the twist about the cylinder axis can be precisely specified. The important role of t h i s shear identifies' these transformations as martensitic.

The occurence of a macroscopic shape s t r a i n across the transformation i n t e r - face, an important c h a r a c t e r i s t i c of martensitic transformations, plays an essent- i a l role i n bacterial tumbling. Studies of the geometry of p a r t i a l l y transformed f l a g e l l a show t h a t the angle r e l a t i n g the helix axis of the two helical forms across the i n t e r f a c e i s always equal t o the difference between the two helical pitch angles (17). Equivalent t o the usual geometric features of martensitic shape s t r a i n s , t h i s r u l e has been explained using the orientation r e l a t i o n of equation 4 (unrotated i n t e r f a c e l i n e ) with the additional constraint t h a t l a t t i c e rows of maximum and minimum axial d i s t o r t i o n are continuous across t h e interface; the

l a t t e r constraint i s rationalized on the basis of minimum e l a s t i c energy of i n t e r - f a c i a l dislocation arrays (16). There i s no evidence t h a t the transformation i n t e r - face i s other than f u l l y coherent. I f t h e transformation displacements a r e only of the type depicted in Figure 7, the l a t t i c e deformation i s an invariant l i n e s t r a i n . Circumferential normal s t r a i n s t h a t would cause a deviation from an invariant l i n e condition have not been detected so f a r .

The e l a s t i c moduli necessary f o r a quantitative determination of the s t r a i n energies involved in the f l a g e l l a r polymorphic transformations can be estimated from

a

measured bending modulus f o r Salmonella f l a g e l l a of 3x10-24 ~ . (18). m ~ Treating the crystal as a thick-wall cylin e r of 19nm O.D. and 4nm I.D. (15)

.!!

gives a Young's modulus of E = 4 . 7 ~ 1 0 8 N/m

.

This modulus compares well w i t h t h a t estimated f o r F a c t i n , a two-stranded helical f i r e of a protein s i m i l a r t o f l a g e l - l i n ;

a

measured bending modulus of 5.3~10-26 N - m (19 gives a Young's modulus of

3

E = 4 . 5 ~ 1 0 8 ~ / m ~ . A Young's modulus of E = 5x108 N/m has also been reported f o r DNA (20). Assuming a Poisson's r a t i o of v = 1/3, these Young's moduli correspond t o an isotropic shear modulus of p = 2x108 ~/m2, approximately two orders of magnitude lower than f o r metals and ceramics which undergo martensitic transformations.

In s p i t e of the r e l a t i v e l y s o f t e l a s t i c behavior, a s i g n i f i c a n t transformation nucleation b a r r i e r e x i s t s as indicated by a hysteresis accompanying fotward and reverse transformation. Stress-assisted transformation under forces induced by f l u i d flow has been studied i n isolated f l a g e l l a of Salmonella SJ25 with one end

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attached t o a glass s l i d e , and the forces acting were quantitatively analyzed (21).

The r e s u l t s support t h e martensitic character of the transformations i n t h a t axial loads exert l i t t l e influence, but the transformation i n t e r a c t s strongly with torsional (shear) loads. The positive torque required f o r forward transformation and a smaller negative torque required f o r reversion were both measured, identifying the t o t a l transformation hysteresis. A sharp load drop following i n i t i a t i o n of single-interface transformation indicated t h a t a s u b s t a n t i a l l y higher driving force i s required f o r nucleation than f o r growth. Transformations from the normal (N) t o curly (C) forms and t o a "semi-coiled" (SC) form were studied. The net torque necessary t o drive the nucleation process can be estimated from one-half t h e t o t a l torque hysteresis, giving 7.5

x

^10-19N.m f o r the N+SC transformation and 6

x

10-19 N.m f o r N-tC. Taking an average cylinder radius of 5.8 nm, the twist increments f o r these transformations (16) correspond t o transformation shear s t r a i n s of y N+SC = 0.023 and y N+C = 0.032. Converting torques t o maximum shear s t r e s s T in a thick wall cylinder, c r i t i c a l driving forces AGc f o r nucleation can be determined from the work term -TY (22). With a protein molar volume of 7.8 x m3, t h i s gives c r i t i c a l values of AGc N+SC = -1 .O1 kJ/mole and AGc N+C = -1 . l l kJ/mole. As molar q u a n t i t i e s , these values a r e s i m i l a r t o the c r i t i c a l driving forces f o r FCC-SCC transformation i n s t e e l s ; however, the comparison i s misleading due t o the much l a r g e r molar volume of the protein. Expressed as a f r e e energy change per unit volume, Ag, and normalized t o the estimated shear modulus, c r i t i c a l driving forces of Agc N+SC=-7 . 4 x 1 0 - ~ ~ and Agc N+C = - 8 . 2 ~ 1 0 - 5 ~ a r e comparable t o those of metallic systems involving s i m i l a r transformation shears of a few percent. Direct observation of these transformations revealed t h a t they nucleated heterogeneously a t the point of attachment to the glass side, and whether the C o r SC form nucleated depended on the detailed geometry of the d i s t o r t i o n imposed by the attachment. I t i s i n t e r - e s t i n g t o note t h a t , because of the low-pi tch (nearly circumferential ) he1 i c a l l a t t i c e row in the s t r u c t u r e of Figure 6 ( a similar feature e x i s t s in the struc- t u r e of E.Co1i (23) f l a g e l l a ) , accomplishing the transformation shear requires the formation of only a single coherency dislocation which can s p i r a l up t h i s helix t o completely transform the c r y s t a l .

Given the large number of polymorphic forms observed i n isolated f l a g e l l a , a problem of i n t e r e s t t o biologists has been the question of why bacterial tumbling involves d i s c r e t e transformation between the N and C s t a t e s without involving s t r u c t u r e s such as t h e SC s t a t e which can be considered s t r u c t u r a l l y intermediate between N and C. One explanation i s t h a t these intermediate s t a t e s involve a large axial contraction which i s opposed by t h e viscous drag of the surroundings (16). The c h a r a c t e r i s t i c s of martensitic transformations suggest some additional factors. As has been well demonstrated f o r the case of the competing metastable

B ' and y ' martensitic phases in B CuAlNi alloys (23), the thermodynamics of trans-

formation under s t r e s s can favor t h e transformation with the l a r g e r transformation s t r a i n , even when the lower s t r a i n transformation gives the nearest metastable s t a t e ( i n f r e e energy) under s t r e s s - f r e e conditions. The " s i t e s p e c i f i c i t y "

aspect of heterogeneous martensitic nucleation observed in the experiments j u s t discussed i s another c h a r a c t e r i s t i c t h a t warrants attention. Consistent with dislocation-dissociation theory of martensi t i c nucleation (24), d e t a i l s of the

"hook" assembly where the flagellum i s attached t o the bacterium can control which coherency dislocations can be most e a s i l y derived from t h i s nucleation s i t e . Perhaps most important, the nature of crystal 1 ine bonding i s such t h a t r e l a t i v e free energies can not be simply inferred from s t r u c t u r e . That the SC s t a t e can be regarded as structurallyintermediate between N and C s t a t e s does not imply t h a t i t i s energetically intermediate. Calculating f r e e energies from the mid- points of the experimental torque hysteresis measurements ( 2 1 ) , t h e free-energy difference between the N and C s t a t e s under s t r e s s - f r e e conditions i s AG N+C =

+3.71x10Z J/mole, while the SC s t a t e has a higher f r e e energy with AG N 4 C = a4.71 x 1 0 ~ J/mol e .

Fol lowing a procedure analogous t o t h e Clausius-Clapeyron r e l a t i o n , the foregoing r e l a t i v e free-energy estimates can be combined with the available phase

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diagram i n f o r m a t i o n t o d e r i v e f u r t h e r thermodynamic i n f o r m a t i o n concerning t h e f l a g e l l a r polymorphic transformations. Denoting c h l o r i d e i o n c o n c e n t r a t i o n as C, t h e phase diagram (12) f o r t h e Salmonella SJ25 f l a g e l l a ( a t an ambient temp- e r a t u r e of t h e microscope experiments o f -30°C) i n d i c a t e s an NSC e q u i l i b r i u m 1 in e o f s l o p e dC/dpH = -0.06M passing through C=O a t pH=5. The s h i f t o f t h e e q u i l i b r i u m 1 in e w i t h temperature i s described by dpH/dT=-0.037 OK-1. For t h e c o n d i t i o n s o f t h e s t r e s s - a s s i s t e d t r a n s f o r m a t i o n experiments (21) (C=0.15M, pH=7), t h e d e v i a t i o n from the NPSC e q u i l i b r i u m temperature T can then be estimated as T-To=1230K g i v i n g To=1800K. The AG N+SC value o f 8.71~102 J/mole estimated from the torque h y s t e r e s i s m i d p o i n t then defines a t r a n s f o r m a t i o n entropy change of AS=-aAG/aT *

-

AG/(T-To) = -3.8 J/mole-OK. The t r a n s f o r m a t i o n enthalpy change

i s then estimated as AH = TOAS = -670 J/mole. This i s two orders o f magnitude s m a l l e r than t h e enthalpy change o f t h e l a r g e - s t r a i n m a r t e n s i t i c transformation i n t h e case o f v i r u s t a i l - s h e a t h c o n t r a c t i o n .

I n c o n t r a s t t o t h e v i r u s t a i l - s h e a t h where t h e f u n c t i o n i s a one-time per- formance o f a l a r g e amount o f work r e q u i r i n g a l a r g e d e v i a t i o n form e q u i l i b r i u m , t h e N+C f l a g e l l a r t r a n s f o r m a t i o n d u r i n g b a c t e r i a l tumbling need o n l y produce a mechanically r e v e r s i b l e change o f f l a g e l l a r shape, i .e. "shape memory e f f e c t " , r e q u i r i n g a r e l a t i v e l y small amount o f work. Mechanical r e v e r s i b i l i t y i s i n s u r e d by a small t r a n s f o r m a t i o n s t r a i n promoting a r e l a t i v e l y small c r i t i c a l d r i v i n g f o r c e f o r n u c l e a t i o n such t h a t c o n d i t i o n s f o r forward and reverse t r a n s f o r m a t i o n can be met by modest changes i n thermodynamic c o n d i t i o n s t h a t do n o t destroy t h e c r y s t a l . The behavior i s analogous t o t h e r m o e l a s t i c m a r t e n s i t i c transformations.

The incomplete n a t u r e o f t h e N-tC t r a n s f o r m a t i o n i n tumbling i s a l s o suggestive o f a t h e r m o e l a s t i c growth f o r c e balance, b u t t h e o r i g i n o f t h e growth r e s t r a i n i n g f o r c e i s i n t h i s case y e t unclear.

Discussion.- Thus f a r o u r treatment o f b i o l o g i c a l processes as d i s p l a c i v e phase transformations has n o t considered t h e n a t u r e o f p r o t e i n bonding as i t d i f f e r s from the bonding o f metals and ceramics. D e t a i l s o f t h e short-range bonding i n t e r a c t i o n s between p r o t e i n subunits w i l l be i m p o r t a n t t o t h e s t r u c t u r e and energy o f i n t e r f a c i a l d i s l o c a t i o n cores, t h e i n t r i n s i c l a t t i c e r e s i s t a n c e t o d i s l o c a t i o n motion, and t o t h e physics u n d e r l y i n g the observed thermodynamic q u a n t i t i e s . However, t h e long-range c o n s t r a i n t s imposed by c r y s t a l l i n e period- i c i t y a l l o w t h a t many aspects o f s t r a i n - e n e r g y dominated, f i r s t - o r d e r martensi t i c transformations can be p r e d i c t e d on t h e basis o f c r y s t a l geometry, l i n e a r e l a s t i - c i t y , b a s i c d i s l o c a t i o n theory, and macroscopi c thermodynamics, w i t h o u t s e n s i t i v i t y t o l o c a l bonding d e t a i l s ; t h i s forms t h e s t r e n g t h o f t h e t h e o r y o f m a r t e n s i t i c transformations as i t c u r r e n t l y stands. The numerous para1 l e l s i n t h e behavior o f martensi t i c transformations i n three-dimensional metal 1 i c c r y s t a l s and t h e c y l i n d r i c a l p r o t e i n c r y s t a l s examined here, i n s p i t e o f t h e bonding d i f f e r e n c e s , a t t e s t s t o t h e usefulness o f t h e approach.

I n a d d i t i o n t o t h e examples o f c o n t r a c t i l e v i r u s t a i l sheaths and b a c t e r i a l f l a g e l l a , o t h e r examples such as p i 1 i, tubules, and microtubules show t h a t c y l i n d r i c a l c r y s t a l s a r e u b i q u i t o u s i n b i o l o g i c a l s t r u c t u r e s . Other forms o f two dimensional c r y s t a l s a r e a l s o common, and may undergo m a r t e n s i t i c transforma- t i o n s . A twinned s t r u c t u r e representing t h e i n t e r n a l s u b s t r u c t u r e o f a martensi- t i c t r a n s f o r m a t i o n product i s observed i n p h o s p h o l i p i d membranes (25) b u t i t i s n o t c l e a r whether t h e s t r u c t u r e a r i s e s from a m a r t e n s i t i c mechanism. A two- dimensional d i l a t a t i o n a l d i s p l acive t r a n s f o r m a t i o n apparently occurs i n some icosahedrel v i r u s capsids. Arrays o f c y l i n d r i c a l c r y s t a l s w i t h one dimensional p e r i o d i c i t y along t h e i r l i n e s o f attachment a r e important t o t h e mechanism o f motion i n h i g h e r organisms. Nabarro (26) has discussed t h e r e l a t i v e motion o f muscle f i b r e s i n terms of d i s l o c a t i o n s moving along such one dimensional arrays.

I t appears t h a t i n many o f these cases t h e displacements i n v o l v e d a r e p a r t i a l l a t t i c e v e c t o r s and t h e r e f o r e produce a change i n s t r u c t u r e ( l a t t i c e deformation) w i t h an associated chemical f r e e energy change. An i m p o r t a n t p r o p e r t y o f p a r t i a l d i s l o c a t i o n s compared t o 1 a t t i c e d i s l o c a t i o n s i s t h e i r response t o chemical r a t h e r than p u r e l y mechanical forces. Mechanical r e v e r s i b i l it y i s a l s o promoted by

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t h e preservation of near neighbor bonds a t p a r t i a l dislocations i n contrast t o l a t t i c e dislocations. I t i s l i k e l y t h a t a central feature of the mechanisms of motion and shape change in periodic biological s t r u c t u r e s of a l l dimension- a l i t i e s i s the movement of p a r t i a l (coherency) dislocatiors in response t o chemical forces.

As a f i n a l note, man i s credited w i t h making use of martensitic transfornations i n metal1 i c systems f o r about three thousand years and the shape memory e f f e c t f o r a few decades. I t now appears from the behavior of the viruses and bacteria examined here t h a t the e a r l i e s t application of these phenomena dates back 3.5 b i l l i o n years. Me may be unwittingly exploiting even more sophisticated applications as we l i v e .

Conclusions

.-

Displacive phase transformations in virus tail-sheaths and bacterial flagel l a a r e i d e n t i f i e d as martensitic. A1 lowing f o r special constraints of cylindrical symmetry, the geometric, thermodynamic, and k i n e t i c aspects of these transformations a r e consistent with strain-energy-dominated martensitic behavior, as dictated by the long-range constraints of c r y s t a l l i n e periodicity. The transformations a r e describable by p a r t i a l dislocation mechanisms which may be of broader relevance t o t h e mechanisms of motion and shape change of periodic biological s t r u c t u r e s i n general.

Acknowledgements

.-

Research on martensi t i c transformation a t M. I .T. i s sponsored by the National Science Foundation under Grant #DMR-79-15196. The authors a r e grateful f o r helpful discussions with Dr. D. Goldenburg, Prof. J . King, Mr. M. Grujicic, and Prof. M. Cohen of M.I.T.

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