Thermal hysteresis of antifreeze proteins considering Fragilariopsis cylindrus

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1 Algologial Studies 5/5 (06, p Pulished online June 06 Open Aess Artile Thermal hysteresis of antifreeze proteins onsidering Fragilariopsis ylindrus Bernd Kutshan, Silke Thoms & Maddalena Bayer-Giraldi FH Münster, Corrensstr. 5, 89 Münster, Germany Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Researh, Am Han-delshafen, 7570 Bremerhaven, Germany With 9 figures and tale Astrat: The diatom speies Fragilariopsis ylindrus produes antifreeze proteins (AFPs of moderate thermal hysteresis. Two onepts are often used in order to desrie a thermal hysteresis, on the one hand the irreversile nuleation growth desried y the Gis-Thomson relation and on the other hand a nonlinear adsorption kinetis of the ie-inding proteins. We refer to the Landau s phase transition theory and predit a saturation onentration of AFPs for the maximal freezing depression. The derived funtional relation etween temperature and AFP onentration is more omplex than a simple power law as suggested y thermal hysteresis experiments. With the parameter set otained for Fragilariopsis ylindrus the modelled urve is omparale with the often used square root law. Keywords: antifreeze proteins, thermal hysteresis, phase transition Introdution Diatoms represent an important fration of the sea-ie miroial ommunity (Brown & Bowman 00, whih onsideraly ontriutes to the produtivity of the polar oeans. They remain trapped in the ie during sea-ie formation, and survive and thrive within a porous system of rine hannels and rine pokets (Thomas & Diekmann 00. The diatom Fragilariopsis ylindrus is one dominant speies within seaie miroial assemlages (Bartsh 989, Günther & Diekmann 00, Lizotte 00, Thomas & Diekmann 00, Roerts et al As an adaptation to sea-ie onditions, this speies produes antifreeze-proteins (AFPs (Bayer-Giraldi et al. 00. These proteins have een shown to modify the mirostruture of ie rystals, i.e. their Corresponding author: Silke.Thoms@awi.de DOI: 0.7/algol_stud/06/05 06 The authors E. Shweizerart she Verlagsuhhandlung, Stuttgart, Germany,

2 70 Kutshan, Thoms & Bayer-Giraldi (A (B Fig.. Frozen samples of protein solutions oserved y refleted light mirosopy. (A Negative ontrol, frozen solution of ovine serum alumin (0.005 μm, (B Frozen solution of AFPs from F. ylindrus (0.0 μm (Reprinted with permission from Bayer-Giraldi et al. 0. size and shape (Bayer-Giraldi et al. 0, Fig.. Diatoms presumaly serete the proteins into the rine system to modify the surfae of ie, in order to maintain and shape their haitat onveniently. Antifreeze proteins have the harateristi property of shifting the freezing point of a solution elow the melting point, therefore ausing a thermal hysteresis (Barrett 00. The proteins attah to nasent ie rystals and inhiit further growth until a ritial temperature, defined as freezing point, is reahed (Kristiansen & Zahariassen 005. Ie growth is resumed at the freezing point. Antifreeze proteins are therefore also alled thermal hysteresis proteins, ie-inding or ie-struturing proteins, due to their effet on ie mirostruture. The interation etween the proteins and ie is ased on an intimate surfae-surfae omplementarity given y a spatial mathing

3 Thermal hysteresis of antifreeze proteins 7 etween the distane of the amino aids of the polypeptide hain and the lattie spaing of the rystal. The equilirium onfiguration of AFPs attahed to ie results from an entropy-driven optimization proess whih minimizes the Gis free energy. Two onepts are generally used when modeling thermal hysteresis due to AFPs. On the one hand there is the nuleation growth desried y the Gis-Thomson equation (Kaptay 0 and on the other hand a nonlinear adsorption kinetis of the ie-inding proteins. Li & Luo (99 used the first approah. They proposed a model for the thermal hysteresis ativity ased on the first order Gis-Thomson relation using hard sphere theory y Reiss et al. (959 and ompared the results with experimental data of the winter flounder (Psodeupleuronetes amerianus antifreeze protein type I (HPLC6. An example for the seond approah is given y Liu & Li (006. They applied a two-dimensional reversile adsorption kineti model for thermal hysteresis ativity on a hyperative antifreeze protein of the inset Tenerio molitor. Also the pinning or stones on a pillow model developed y Sander & Tkahenko (00 assumed an adsorption kinetis and an therefore e ounted among the seond shool. In the following we use a novel approah to justify the thermal hysteresis due to AFPs and apply Landau s phase transition theory. The two model approahes mentioned aove desrie a non-olligative ehavior of the freezing point depression as a funtion of AFP-onentration, ut do not always reflet the experimental values adequately. The well-estalished Landau theory of first order phase transitions an e used to desries hanges of the Gis free energy during the transition etween the ordinary hexagonal ie I modifiation and the superooled liquid. Therefore, a Landau model is est suited to desrie the thermal hysteresis aused y AFPs. In a reent work (Kutshan et al. 0 we applied Landau s phase transition theory to study the dynamial aspets of the nuleation growth and alulated the ritial size of the rystallization seed as a funtion of the AFP onentration. Here we use Landau s theory to desrie the impat of AFPs on the lower limit of the superooling region and alulate the AFP onentration with the maximal possile freezing point depression. The outline of the paper is as follows. In the next setion we introdue the Landau s approah to phase transitions, aording to whih a single variale, the order parameter, is introdued to keep trak of the phase hange. The phase transition is modified due to AFPs, whih we desrie y a seond variale, the AFP-onentration ρ. The desription of the experimental onditions and the determination of the model parameters are presented in the seond and third setion, leading to the Landau free energy desriing the thermal hysteresis due to the AFPs of Fragilariopsis ylindrus. The alulation of the AFP onentration with the maximal possile freezing point depression ompletes the fourth setion. In the fifths setion we summarize and onlude.

4 7 Kutshan, Thoms & Bayer-Giraldi Landau theory of phase transition General aspets A ruial point in Landau s theory plays the order parameter due to his duality of meanings. An order parameter ridges the gap etween a mirosopi point of view and a marosopi perspetive (Errington & Deenedetti 00, Medvedev & Naerukhin 987 and desries a measure of the degree of order of a system (Fig.. Suh a first order phase transition is assoiated with a symmetry hange during the transition from liquid water to hexagonal ie in the lassifiation of the point groups. It is assumed that the free energy density { an e desried y a power series expansion (Brokate & Sprekels 996, Kittel & Krömer 00. For a first order phase transition our approah enales a doule welled funtion whose two minima orrespond to two stale phases (Harrowell & Oxtoy 987. We ouple the order parameter with the funtion of the AFPs (t and otain the Gis free energy density { j { R W = K( 0(,,,,, j j j S - X S - X R W R W 6 ( with j j j KQ,,, V = R W 0 r = (- os S aros( d X 86 r $ S- os S aros( d XX ( and 6-8 d =- Q- V ( Fig.. 0 < < < degree of low order (Brownian random proess degree of high order.

5 Thermal hysteresis of antifreeze proteins 7 derived in Appendix Eq. (. The index j modifies only the onstant K in the potential funtion ( as shown in Fig. 9 in Appendix. We hoose j = eause this ase desries the superooling proess. However, the ase j = 0 haraterizes the superheating proess and the ase j = onnets oth ranhes. The oeffiient = p/c orresponds to the negative dimensionless AFP-onentration, wherey t designated the AFP-onentration whih eomes dimensionless y the onstant C. Furthermore, is a funtion of the temperature and vanishes for the ritial temperature T = T. The quadrati expression in Eq. (9 hange its sign at this point due = BT ( -T. ( Figure shows the asymmetri potential with a saddle point for t = 0 and T = T. On the other hand we have a temperature T = T that oinides with the atual first-order transition temperature that orresponds to a symmetri potential (Figs. and, i. e. the linear term in Eq. ( have to vanish 6 T - = 0 and one otains B = T Y /( T - T and for 8 T- T = T Y $ 8 T - T (5 (6 Finally, should only e dependent on the temperature T ut not on the (negative AFP onentration and set = 0 ( T = 8 T- T T - T T with the speial value( T = = 8. The reason for this is, that we understand as state variale dependent on the temperature only, whereas (t desries an external field dependent on the AFPs. Now, we onsider the symmetri potential for = 0 aording to Eq. ( in Appendix {( = S - X 076 (8 d ( and determine the extreme values { = ( - = 0. Therefore we d find the maximum for and the oth minima 0 (7

6 7 Kutshan, Thoms & Bayer-Giraldi φ 0.0 Ρ 0, T T Ψ Ψ Ρ 79.0, T Ρ. Ρ 0, T T Fig.. The free energy { versus the order parameter for pure water (t = 0, phase transition at T = T, temperature at maximal superooling T =.5K and for the saturation onentration of AFPs t = t T T K Fig.. Dependene of the asolute minimum of the free energy (9 on temperature T for t = 0. The step at phase transition (T = T indiates the phase hange if one jumps from the first minimum to the asolute one in Figure. T- T = -! $ T - T (9 The parameter 0 is essential for a first order phase transition and determines the jump at T = T. The oeffiient haraterizes the jump for a first-order phase transition at the freezing point temperature T = T [see Eq. (7 and Fig. ].

7 Thermal hysteresis of antifreeze proteins 75 Experimental onditions Fragilariopsis ylindrus shows a moderate thermal hysteresis ativity (Bayer-Giraldi et al. 0. The thermal hysteresis experiments were performed at low salinity of S = 0 g/kg using a reominant protein (Bayer-Giraldi et al. 0 with a road onentration range. The atual physiologial onentration is unknown, sine no measurements have een done in sea ie. The freezing point depression DT f on the asis of the salinity an e alulated from DT f = K f S/M NaCl with the ryosopi onstant K f =.85 Kkg/mol for water and the molar mass for sodium hloride M NaCl = 58.5 g/mol and otain DT f = 0.6K. Hene, one finds the freezing temperature for the salinity T = T m DT f = 7.5K for S = 0 with T m = 7.5K. Parameter identifiation From the potential ( it an e found a nonlinear relation ( for the temperature depression T due to AFP onentration t t - 56 C d ( 9-8d r 9 D T = - oss aros( f X - (0 d 8d with 6 999d -78d (- 9 d ( 7-6d 8d f = ( 56 ( 9 8 ( 56 ( 9 8 d - d - d d - d d ( A nonlinear parameter fitting method provides always d = for our measured values, so that Eq. (0 is simplified to t r D T = oss aros( C f X - 8 ( with t t - 999T- Y 860 C C f = t T- 56 Y C 6 aording to Eq. (5 in Appendix. There are two unknown values in equation (, the parameter and a saling fator C for the AFP onentration. These parameters are determined for Fragilariopsis ylindrus using a nonlinear estimation method as

8 76 Kutshan, Thoms & Bayer-Giraldi T t in μm T s T Ρ Fig. 5. Parameter identifiation for Fragilariopsis ylindrus (6 kda aording to Eq. (: C = 99.8, =.6. shown in Figure 5. The model reflets the measured values exept for a onstant shift onditioned y T~ T. The Gis free energy density ( an e expressed as {( = K(, - ( 7 for = 8 with the translation K(, K (, os aros 68 r = 7776 U U- ( Z Z $ U os aros 68 r - U U- Z ( ZZ An asolute potential annot e determined ut only the differene etween two potentials. An additional ondition suh as a doule root in Eq. (0 an e used in order to determine this aritrary onstant K(,. If one hooses K = 0 { (, K = 0 = - 7 ( one sees the shift ased on K etween {(K(, and {(K = 0 (ompare Fig. 6.

9 Thermal hysteresis of antifreeze proteins 77 φ K Ρ 0 φ Ρ K Β, Β Ψ Ρ 5 Ψ Ρ Ρ Ρ Ρ Ρ 79.0 Fig. 6. The free energy { versus the order parameter for various AFP onentrations t at onstant temperature T = T (left: { for the aritrary onstant K = 0, right: { renormalized to zero at the first minimum. In the right figure, the asolute minimum indiates the ontriution of the AFPs to {. Limit of maximal AFP onentration The extreme values of the potential { (0 or ( are determined y yields S - X - = d{ ( d = 0 and (5 if we identify with in Eq. (0. The index an e omitted eause the index of the multiple root of multipliity is hosen aritrarily. With = 8 and t =- one otains the loops of a istale system C t = C S - 6 X (6 or rather y means of Eq. (7 j 68 t r = os aros j 6 C ( U U Z Z (7 with j = 0,, (see Fig. 7 resp. Fig. 8. The funtion (6 has a maximum and a minimum! dt =! =! deduile from the ondition 0. = The extreme d values deompose the profile into two stale ranhes for - or - and

10 78 Kutshan, Thoms & Bayer-Giraldi an unstale region for - haraterized y the dotted line in Figure 7. C The appropriate values of the funtion are t (! =! t =! R W. Besides 68 (- t = = there is ( t = = f for the ranh j 0 6 (Fig. 8. Aording to the phase jump in Eq. (9 for T = T in Fig. one find Z =- r for j = 0 j ] ( t = 0 = [ 0 for j = (8 ]- = r \ for j = for a pure water/ie system without AFPs. The dashed lines in Figures 7 and 8 orrespond to the measured loop for the AFP onentration of 5 μm and the lines to the onentration of t =79.0 μm that indue a maximal freezing point depression of T =. K as a maximal theoreti limit. Aordingly, the minima of the energy density hange their shape and position, depending on the AFP onentration at the freezing temperature T in Figure 6. The minima are muh smaller than the minimum for the superooling temperature T of pure water without AFPs in Figure. The line of the value t desries preisely the limit for the infletion points. The limit lies not so far away from the measured values (Bayer-Giraldi et al. 0. AFP-onentrations up to 50 μm was measured and seems to onverge to a saturation value, t = 75 μm orresponding to delta T = 0.9 K. The instaility region of the loop shown in Figure 7 an also e understood as Maxwell onstrution. The line at t = 0 shown in Figure 8 disonnets the superooled and the superheated region. It should e mentioned here that oth Maxwell s phase transition theory of real gases and Landau s phase transition theory ase on a polynomial approah. Summary Until now we onsidered an isotropi order parameter. This simplifiation was suffiient in order to derive a suitale non olligative relation etween temperature depression and AFP-onentration. The Landau approah an reprodue the experimental values exept for a onstant shift. Our funtion is more omplex than a simple power law for the AFP onentration as suggested y thermal hysteresis experiments. However, with the parameter set derived for Fragilariopsis ylindrus the modelled urve is omparale with the often used square root law (Raymond & DeVries 977. Moreover, we determined a theoretial limit for the maximal effetive AFP onentration. It is reommended to onfirm our theoretial limit y means of appropriate measurements. It is useful to inlude an anisotropi order parameter with the aim to investigate the hange of the symmetry elements of the point group of the ie rystal.

11 Thermal hysteresis of antifreeze proteins 79 Ρ 00 Ψ f, Ρ 00 Ψ 5, Ψ Ψ, Ρ Fig. 7. Relationship etween the AFP onentration t and the two minima of the free energy { at T = T (solid lines. The maximal AFP onentration t is defined y the limiting infletion point in Figure 6. Ψ ,Ψ Ρ,Ψ r f Ψ j 0 Ρ,Ψ Ψ j Ρ j Ψ 5,Ψ j 5 Fig. 8. The inverse of the funtion in Figure 7. Here, the maximal AFP onentration t is defined y the termination of the solution ranh for the first minimum in the Figure 6.

12 80 Kutshan, Thoms & Bayer-Giraldi Aknowledgements This work is ased on the results of the DFG Priority Program SPP 58 Antarti Researh. Appendix The potential { an e expressed y a polynomial { ( = 0 (9 with = a- t. The shift = - yields {( = S X S X (0 6 = ( - ( - ( - with =. The requirement = for the fatorization using the fundamental theorem of algera in Eq. (0 {( = ( - ( - ( - ( - = ( - ( - ( - redues the numer of independent oeffiients from four down to three. The method of equating the oeffiients allows us to find the fators 0- - = ( - = - ( 6 ( - = ( ( 0 =- ( ( =. (5 The requirement = yields to a doule root and the four dimensional parameter spae { 0,,, redues to a three dimensional parameter spae {,, with 0 = 0 (,, aording to Eq. (. The sustitution of and in the system of equation ( to ( yields - S - X = 0 (6

13 Thermal hysteresis of antifreeze proteins 8 with the real solutions r = - oss aros( d jx 6 (7 for j = 0,, and 6-8 d =- (- (8 for the asus irreduiilis. We onsider the solution for j = 0,, j r = - oss aros( d j X 6 (9 and otain the other zeros in onsideration of ( and ( j =- j! -( j - (0 and =- " -( j j j - ( Thus, the onstant in the potential (0 is K = ( j j j = r = S- X os aros( j S d X 86 ( r $ S- os S aros( d j XX and the potential funtion j { Q V r = S- X os aros( j S d X 86 r $ S- os S aros( d j XX j S- X 6 j j S - Q Q. X V V (

14 8 Kutshan, Thoms & Bayer-Giraldi φ Ψ j 0 φ Ψ j Ψ j j Fig. 9. Potential { ( Eq. ( for j = 0,, with =.6, =, 8 = 0 (straight line and = 0.0 (dashed line. The potential is symmetri if - = 0 & = and the 6 8 equation ( is simplified to φ (- j - ( { Q V = 076 j ( for j = 0, and j j j { Q V = U 6 - Z ( ( 7 (5 for j =. The potential (0 depends on the zeros (7 and on d aordingly. Therefore we solve the equation X 6 8 -S- d = -. (6 We otain 9 U - Z U d d 8 d = 0 d d 78 d 78 Z (7 move = - U - Z and get 6 d

15 Thermal hysteresis of antifreeze proteins 8 9 U d d 96 d Z = d d 56 d d 8 d 69 d (8 The solution is 56d ( 9-8d r = Sos aros( f j X (9 d resp. 56d ( 9-8d r d 6 d os aros( j = S f X - (0 999 d -78 d (- 9 d 6 ( 7-6d 8d with f = ( 56 ( 9 8 ( ( d - d - d d - d d for j = 0,,. The oeffiient (T implies the measured temperature T, the ritial temperature T, and the atual freezing temperature T T- T T- T T T - = = = 8 T - T 8 T - T T T DT = - ( 8 T T - T Y = 8 T T - T Y with T = D 8 T - T and =. Comining the equations (0 and ( we get 8 DT T - T C = 9-8d t ( - 56 d 9-8 d C r oss aros( f j X d ( for = t/c. The refletion aross the x-axis and the dimensionless saling fator T T C = - yields K

16 8 Kutshan, Thoms & Bayer-Giraldi DT C T - T C = - t ( - 56 d d C d r 9 $ oss aros( f j X - 8d ( or for j = D T = - t ( - 56 d 9-8 d C r 9 oss aros( f X -. d 8d ( The maximal superooling is otained for d =. For this ase equation ( is simplified to t r D T = os aros( S f X - C 8 (5 t t -999 T- Y 860 C C with f = t T- 56 Y C 6. Referenes Barrett, J. (00: Thermal hysteresis proteins. IJBCB : Bartsh, A. (989: Die Eisalgenflora des Weddelmeeres (Antarktis: Artenzusammensetzung und Biomasse sowie Ökophysiologie ausgewählter Arten (Sea ie algae of the Weddel Sea (Antartia: Speies omposition, iomass, and eophysiology of seleted speies. Ber. Polarforshung 6: 0 p. Bayer-Giraldi, M., Weikusat, I., Besir, H. & Diekmann, G. (0: Charaterization of an antifreeze protein from the polar diatom Fragilariopsis ylindrus and its relevane in sea ie. Cryoiology 6: 0 9. Bayer-Giraldi, M., Uhlig, C., John, U., Mok, T. & Valentin, K. (00: Antifreeze proteins in polar sea ie diatoms: diversity and gene expression in the genus fragilariopsis. Environmental Miroiology : Brokate, M. & Sprekels, J. (996: Hysteresis and Phase Transitions. Springer-Verlag, New York, Berlin, Heidelerg, 57 p. Brown, M. & Bowman, J. (00: A moleular phylogeneti survey of sea-ie miroial ommunities (SIMCO. FEMS, Miroiology, Eology 5: Errington, J. & Deenedetti, P. (00: Relationship etween strutural order and the anomalies of liquid water. Nature 09: 8. Günther, S. & Diekmann, S. (00: Vertial zonation and ommunity transition of sea-ie diatoms in fast ie and platelet layer, Weddell Sea, Antartia. Annals of Glaiology :

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