Analysis of Rotating Flow Around a Growing Protein Crystal

Transcription

1 Available at ttp://pvamu.edu/aam Appl. Appl. Mat. ISSN: Vol. 4, Iue (June 9) pp. 3 5 (Previouly, Vol. 4, No. ) Application and Applied Matematic: An International Journal (AAM) Analyi of Rotating Flo Around a Groing Protein Crytal aniel N. Riai and Carle W. Obare epartment of Matematic Univerity of Texa-Pan American Edinburg, Texa USA driai@utpa.edu Received: Marc 3, 8; Accepted: ecember 8, 9 Abtract We conider te problem of teady flo around a groing protein crytal in a medium of it olution in a normal gravity environment. Te ole flo ytem i aumed to be rotating it a contant angular velocity about a vertical axi ic i anti-parallel to te gravity vector. Convective flo take place due to te olute depletion around te groing crytal ic lead to a buoyancy driven flo. Suc convective flo can produce inomogeneou olute concentration, ic ubequently generate non-uniformitie in te crytal tructure finalizing loer quality protein crytal. Uing caling analyi itin a diffuion boundary layer around te crytal, e etimate te magnitude of te convective flo velocity and oter dependent variable of te flo ytem for cae of eak, moderate and trong rotation and eiter trong or eak buoyancy. We find, in particular, tat for moderate rate of rotation, convective flo i eakened ignificantly a te rotation rate increae. Ti reult may be of interet for production of ig quality protein crytal ince te effect of te convective flo around te groing protein crytal i found to be eakened ignificantly by te rotational contraint. Keyord: Rotating Flo; Convective Flo; Flo Analyi; Convection; Nonlinear Flo MSC () No.: 76U5, 76R, 76R5. Introduction A large number of experiment on eart and in pace involving protein crytal grot ave been carried out in te lat to decade or o, including toe more recent one due to Cang et al. (3), Moreno et al. (7) and Heijna et al. (8). Some of te pace experiment under microgravity ave been able to produce ome type of crytal tat ere relatively larger and better ordered tan toe gron on eart under normal (eart) gravity condition [ay and McPeron (99), Long et al. (994)]. Te main reaon for groing protein crytal in pace 3

2 4 Riai and Obare a been to acieve diffuion controlled grot condition, ic a conidered to produce iger quality crytal. Hoever, tere ave been alo a number of fligt experiment tat yielded crytal it no improvement in te quality and internal order jut like toe lo quality crytal, ic ere gron under te normal gravity condition and in te abence of rotational contraint. One problem for te groing protein crytal in pace can be te undeirable effect of impulive diturbance from large truter firing due to uttle operation, ic can adverely affect te concentration field near te crytal face. Hoever, tere ave been ome experiment in centrifuge [Blagova et al. (997)] for groing ome type of protein crytal, like Uridine Poporylae (Up), ic indicated production of larger and iger quality crytal under te influence of a rotational contraint. Crytallization of larger ize of Up a become uitable for x-ray analyi. In addition, Up a been identified a te enzyme tat i alo reponible for te cleavage of ome pyrimidine nucleoide analog, poeing anti-tumor activity [Blagova et al. (997)]. Ramacandran et al. (995) carried out tudie of olutal convection in protein crytal grot under bot normal and lo gravity condition and in te abence of rotation. For teady flo, tey found, in particular, tat teir analytical olution baed on ome approximate metod, provided ueful information about te flo velocity and ma tranport in te vicinity of a groing cylindrical protein crytal. Lee and Cernov () tudied buoyancy driven axiymmetric convection during protein crytallization and in te abence of rotation. Tey conidered more realitic boundary condition for te protein olution at te crytal face and included te preence of impurity in te liquid. Tey carried out ome caling analyi [Barenblatt (996)] for te flo itin a boundary layer around a perical protein crytal and determined, in particular, ome etimation of te trengt of convection againt tat of diffuion. In addition, tey obtained reult for axiymmetric flo around a cylindrical protein crytal, ic ere baed on full-cale numerical imulation of te governing equation and ere independent from toe baed on te caling analyi, and found good agreement it te related reult baed on teir caling analyi of flo around te perical protein crytal. In te preent invetigation e carry out caling analyi [Barenblatt (996)] of te type ued before by Lee and Cernov () and by Worter (99) and Riai () in oter fluid flo problem. Suc analyi i a ueful tool often ued for implifying te governing equation ic can be lengty uc a toe encountered in te fluid mecanic. In uc analyi te leading term in te equation are identified and retained, and all te ret of te maller term in te equation are ignored and dropped from te equation. Ten, te leading order reult for te dependent variable and oter quantitie are determined, ic are baed on te leading term tat ere retained in te equation. For te axiymmetic flo around a groing perical protein crytal and in te abence of rotation, uc caling analyi a carried out by Lee and Cernov (), but ere e apply uc analyi for a more realitic cae of non-axiymmetric flo around uc groing protein crytal and in te preence of an external contraint of rotation. Te applied caling analyi ued ere can alo be tougt a an aymptotic analyi [Zeytounian (4)] in te limit of eiter ufficiently large axial flo (trong buoyancy) or ufficiently mall axial flo (eak buoyancy). We found intereting reult. In particular, e found tat preence

3 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ] 5 of an intermediate rotation rate i beneficial in te ene tat te flo velocity near te crytal face i eakened a te magnitude of te rotation rate increae.. Governing Sytem We conider te problem of a protein crytal groing in a cylindrical camber containing te olution. Suc crytal ill obviouly gro until no more protein i available in te olution. Hoever, ince te protein crytal i muc maller in ize tan te volume of te olution, a teady olution of te governing equation can be attained to te leading order term after an initial pae of flo adjutment around te groing crytal. We conider a perical protein crytal of radiu, in te central part of te container of it olution under a normal (Eart) gravity environment (Figure ). We aume te ole flo ytem i rotating about te vertical axi z, ic i anti-parallel to te force of gravity, it a uniform rotation rate, and te governing equation for te flo are aumed in a coordinate ytem ic i rotating it te flo at uc rotation rate and direction. Te governing equation are te continuity equation for incompreible fluid flo, te momentum equation under Bouineq approximation [Candraekar (96)] for te flo velocity vector and te concentration equation for te protein and impurity. It turn out to be ufficient to conider te governing equation in cylindrical coordinate, ere te Laplacian operator i given by r r r r r z, ic i in cylindrical frame ( r,, z). Here r and are te radial and azimutal variable, repectively, and z i te axial variable oe axi coincide it te vertical axi. Te continuity equation i v ( ru). (a) r r r z Te radial, azimutal and axial component of te momentum equation are u u v u v u p u v u u v r t r r r z r r r v v v v uv v p v u u v t r r r z r r r ( ) u [ ] rco in, [ ] co, (b) (c)

4 6 Riai and Obare ] [ g z p z r v r u t, (d) ere it i aumed tat tere i only ligt amount of contamination due to te impurity concentration, o tat te buoyancy term due to te impurity i neglected in (d). Te concentration equation for protein and impurity are z r v r u t (e) and z r v r u t. (f) Here u, v and are te radial, azimutal and axial component of te flo velocity vector u, repectively, i te kinematic vicoity, p i te modified preure (preure divided by ), i te protein olutal expanion coefficient, g i te acceleration due to gravity, and are te coefficient of te olutal diffuivity for te protein and impurity, repectively, and i te initial uniform concentration of te crytallizing protein. Te boundary condition at te crytal urface are [Lee and Cernov ()] c c V n,, V K n C K, v u, (a-c) ere (cm/) i te kinetic coefficient, ic caracterize te rate at ic molecule are incorporated by te groing protein crytal, V i defined in (a), and n i te normal ditance along te axi pointing into te liquid from te crytal face. Te boundary condition at te container urface are n n v u. (d) It ould be noted tat te main matematical difference beteen te preent model and tat due to Lee and Cernov () i tat te preent model treat non-axiymmetric ytem ubjected to an external contraint of rotation, ile te model of Lee and Cernov () a for te impler cae of non-rotating axiymmetric ytem.

5 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ] 7 3. Analyi and Reult Conider a perical protein crytal of radiu in te medium of it olution under a normal S gravity condition. At initial time t =, te uniform denity of olution i, and te uniform concentration of te crytallizing protein and impurity are denoted by and, repectively. For t, te concentration are non-omogeneou and non-uniform in general and are denoted by and. Crytal-olution equilibrium i reaced en = e. Inide te protein crytal, te denitie for te protein and impurity are deignated by c and c. We conider te fully nonlinear regime but for eakly non-axiymmetric cae, ere te flo dependence are aumed to vary eakly it repect to te azimutal variable. For trong buoyancy driven flo, ic implie trong convection, te Peclet number P r =/, ic i a non-dimenional parameter repreenting te axial flo, i large (P r >>), ile for eak buoyancy driven flo P r i mall (P r <<) [Lee and Cernov ()]. Te preent caling analyi can be tougt a an aymptotic analyi in te limit of eiter very large P r (trong buoyancy) or very mall P r (eak buoyancy). 3.. Convection Veru iffuion Folloing Lee and Cernov (), e firt focu on convection veru diffuion apect. We compare four velocitie involved in te ytem. Tee are te flo rate (in te axial direction), diffuion rate /, kinetic coefficient and te crytal grot rate V. Te flo rate can be large or mall depending on te order of magnitude of te buoyant force. For ufficiently lo gravity cae, diffuion dominate and o it i reaonable to put aide. Protein crytal grot i lo, o tat at te crytal-olution interface i typically loer tan te diffuion rate /, at ic molecule are tranported troug te olution to te urface. Since i typically mall, e e are intereted ere in ig uper-aturation, ere. c c c No, from (a), e ave V / / at te interface becaue. Tu, e ave / V. If V i o mall tat v V, ten te problem can be treated a a purely diffuion one becaue V i of order 8 cm / and can ardly tir up convection. Problem i about a crytal groing in a finite iolated body of olution. So crytal ill gro until no more protein i available. Since crytal <<volume of olution, a teady tate i attained after an initial pae of adjutment of olution around crytal long before protein olute i ignificantly depleted. 3.. Te Cae of Weak Rotation and Strong Buoyancy We conider te cae ere convection occur due to te olute depletion around te groing crytal leading to a decreae in denity by an amount denoted by and a reulting buoyant force of g. In an aumed leading order teady tate, te buoyant force and te oppoing vicou force convert te depleted region into a diffuion boundary layer of tickne. For te momentum equation and itin ti boundary layer, diffuion of vorticity, repreented by

6 8 Riai and Obare vicou term, i important and dominate to leading term over te nonlinear inertial term in te aumed teady tate. For ufficiently mall rotation rate, te centrifugal force, ic contain a factor of, i mall in comparion to te Corioli force, ic contain a factor of. Comparing te Corioli force it te leading term of te vicou force in (b) or (c), e obtain te condition <</, (3) ic i valid en rotation i eak. Tu, te rotational contraint ere a at mot minor effect and ill not enter te leading order balance of force ic determine te flo rate. In ti paper, en rotation i eak, e obtain te leading expreion for te axial velocity, ic i te main component of te velocity vector, from te axial component (d) of te momentum equation. Te equation (d) ten yield te folloing leading balance g, (4a) ere n p n. (4b) Here,, and p are te volume, denity of ater and denity of a protein molecule, repectively, n and n are te number of protein molecule per unit volume in te ambient and at interface repectively, i evaluated at interface, and /( ) p p. Te expreion in te rigt-and ide in (4b) i due to te gravity term in (d) and te ydrotatic approximation of te axial-derivative of te preure term in (d). From (a) e find, e ic ten yield e, (5) becaue / (on te bai of / and ) and 3 and e 5.35 for Ferritin, a an example, o tat, ic e aumed initially. Tu, (5) follo.

7 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ] 9 No, by diffuion alone, te boundary layer tickne gro like t.ince balancing diffuion term it temporal derivative term in eiter (e) or (f) implie uc reult. Wen tere i enoug convection, reace it limiting value en flo reace a teady tate. Te axial flo rate i aumed to be te dominant flo component tat togeter it radiu of te protein, determine te relevant t in t. Tu, t. (6) No, uing (4b) and (5)-(6) into (4a), e ave Tu, 5 3 g 3 g e g e or e g e 5 3 / 5 5 e Since, (6) become to te leading term g 5 3 / (7). (8) Te reult given in (8) i obtained for te cae of eak rotation and trong buoyancy driven flo Cae of Weak Rotation and Weak Buoyancy We no analyze te cae ere convection i dominated by protein diffuion ince buoyancy i aumed to be ufficiently eak. Ten protein concentration i mainly determined by diffuion. Tu in (e) e neglect convective term and e ave t In teady tate, (9) ten become. (9). ()

8 Riai and Obare Here, tere i no diffuion boundary layer, o tat te lengt cale naturally replace in (4a) and (5) leading to te reult e g,. (a-b) Uing (4b) and (b) in (a), e find g, ic yield g 3. () Te reult given in () i obtained for te cae of eak rotation and eak buoyancy driven flo Cae of Moderate Rotation and Strong Buoyancy We no conider te regime deignated a moderate rotation cae, ere te Corioli force dominate over te centrifugal force. Ti regime correpond to << (u + v ) / /, (3) ic i determined by an order of magnitude comparion beteen tee to force given in (b)-(c). We, tu, keep te Corioli force plu te oppoing vicou force in te non-axial component of te momentum equation (b)-(c) and find u v v, u. (4a-b) For te axial component of te momentum equation (d), e keep te buoyant force and te oppoing vicou force and find g g e. (5) From (4a-b), e find u implying v

9 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ], u v. (6a-b) In addition to (3), (6a) i a econd condition for te moderate rotation cae. Uing (6a) in (5), e find g, o tat 3 g. (7) Te reult given in (7) o tat te axial velocity reduce a te rotation rate increae Cae of Moderate Rotation and Weak Buoyancy Similar to te cae of eak rotation and eak buoyancy preented in te ub-ection 3.3, e ould apply te lengt cale in place of. Tu, ere e need to replace in (5) by, ic lead to g 3. (8) Ti reult i eentially te ame a te one given in (), and te reult given in (6a) i replaced by. (9) Te reult given in (6b) i till valid in ti cae Cae of Strong Rotation and Strong Buoyancy Here e conider te cae deignated a trong rotation regime, ere te centrifugal force dominate over te Corioli force. It correpond to te regime oppoite to tat given in () ere >> (u + v ) / /, () ic i determined by an order of comparion beteen tee to force given in (b)- (c). Tu, for trong rotation cae, e keep te centrifugal force plu buoyant force and te oppoing

10 Riai and Obare vicou force in te momentum equation (b)-(d) and find te folloing etimation for te non-axial component of te flo velocity: u v r co, r co in From (4a-b), (5) and trong buoyancy cae, e ave v g. (a-c),. () Uing () in (), e find u r co, v r co in, v g. (3a-c) Uing (6) for te order of magnitude of te boundary layer tickne, e ave. (4) From (4) and (3), e find g g 5, (5a), 5 g 3 5 u, g 3 5 v. (5b-d) g Te reult given in (5a-d) appear to be reaonable. A vicou effect increae, te boundary layer tickne increae; velocity component u, v and decreae it increaing vicoity, and te non-axial velocity component u and v increae it te rotation rate 3.7. Cae of Strong Rotation and Weak Buoyancy Again imilar to te eak buoyancy cae preented in te ub-ection 3.3 and 3.5, e need to ue lengt cale intead of. Tu, ere e replace in ()-() and (3c) by and find

11 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ] 3 g 3, (6a) v u tan, u ( r co )/( ). (6b-c) 4. Concluding Remark We invetigated te problem of convective flo around a groing protein crytal and in te preence of an externally impoed rotational contraint. Uing caling analyi for te flo itin a tin boundary layer adjacent to te crytal face, e found te folloing main reult tat are given a follo. (i) For eak rotation, ere (3) old, and trong buoyant force cae, te relevant lengt cale i te boundary layer tickne, and te leading order vicou force balance tat due to buoyancy in te axial component of te momentum equation leading to (4a). In addition, te leading order diffuion term in te concentration equation, for eiter protein or impurity, balance te correponding temporal derivative term leading to a reult tat combined it (4a) yield (8). In ti cae rotational effect are inignificant. (ii) For eak rotation and eak buoyant force cae, te relevant lengt cale i te radiu of te crytal, and te imilar type of balance of te force a in te cae (i) lead to (). In ti cae rotational effect are again inignificant. (iii) For moderate rotation and trong buoyant force, te relevant lengt cale i te boundary layer tickne, and te Corioli force in te non-axial momentum equation i ignificant and balance te correponding leading order vicou force leading to (4a-b). Te leading order vicou force balance te buoyant force in te axial component of te momentum equation, ic togeter it (4a-b) lead to te reult (7). It can be een from (7) tat axial convective flo can decreae in magnitude if te rotational effect i moderate in te ene tat te Corioli force dominate over te centrifugal force. (iv) For moderate rotation and eak buoyancy cae, te relevant lengt cale i te radiu of te crytal intead of te boundary layer tickne, and te imilar type of te force balance a in te cae (iii) lead to (8)-(9), ic imply tat te axial flo decreae it increaing te rotation rate. Te reult for bot cae of (iii) and (iv) are valid in te range ere (3) and (6a) old for te rotation rate. (v) For trong rotation and trong buoyant force, te relevant lengt cale i te boundary layer tickne, and te centrifugal force in te non-axial momentum equation i ignificant and balance te correponding leading order vicou force leading to (a-b). Te leading order vicou force balance it te buoyant force in te axial component of te momentum equation, ic lead to (c). Te condition of trong buoyancy and te order of magnitude of te boundary layer tickne togeter it te reult given in (a-c) ten yield (5b-d), ic imply, in particular, tat nonaxial convective flo can be intenified if te rotational effect i trong in te ene tat te centrifugal force dominate over te Corioli force. (vi) For trong rotation and eak buoyancy cae, te relevant lengt cale i te radiu of te protein crytal, and te imilar type of force balance a in te cae (v) lead to (6a-c), ic imply, in particular, tat te non-axial flo increae it te rotation rate.

12 4 Riai and Obare Te preent reult about poible beneficial effect of an externally impoed rotational contraint on eakening te convective flo around a groing protein crytal and te intereting recent experimental reult for te protein crytallization ytem and production of iger quality protein crytal in a ig gravity environment [Blagova et al. (997)] point at future ueful tudie of protein crytal grot on a centrifuge. It i oped tat uc tudie can be carried out teoretically and computationally by te firt autor in te near future. Finally it ould be noted tat effect of a magnetic field on te grot rate of certain protein crytal, uc a lyozyme and oter protein crytal, a been tudied experimentally in te pat [Yanagiva et al. (), Moreno et al. (7)], and it a been found tat uc effect to be notable for trong field cae. Suc experimental reult togeter it te preent teoretical reult for te cae of convective flo during protein crytallization and toe in Riai () for te cae of alloy crytallization ugget tat application of bot a magnetic field and a rotation vector on te flo during protein crytal grot could lead to a more effective procedure to control te trengt of te flo around te groing protein crytal and tereby producing iger quality protein crytal. Tu, it i alo recommended future tudie of convection during protein crytallization and in te preence of externally impoed rotation and magnetic field in order to detect poible relevant parameter regime ere te flo adjacent to te crytal face can be eakened a muc a poible. Acknoledgement Ti reearc a upported by a grant from UTPA-FRC. REFERENCES Barenblatt, G. I. (996). Scaling, Self-Similarity, and Intermediate Aymptotic. Cambridge Univerity Pre, UK. Blagova, E., Mogunova, E., Smimova, A., Mikailov, S., Armtrong, C., Mao, C. and Ealick, S. (997). Centrifugal Material Proceing, edited by L. L. Regel and W.R. Wilcox, 3. Cang, H-X, Wang, Y-P, Han, Y., Zou, J-X and Bi, R-C (3). Te pace experiment of protein crytallization abroad te Cinee pacecraft SZ-3. Microgravity-Science and Tecnology, 4(3), 3-6. Candraekar, S. (96). Hydrodynamic and Hydromagnetic Stability. Oxford Univerity Pre, UK. ay, J. and McPeron, A. (99). Macromolecular crytal grot experiment on international microgravity laboratory. Protein Science,, Heijna, M. C. R., Van Enckevort, W. J. P. (8). Grot inibition of protein crytal: A tudy of Lyozyme polymorp. Crytal Grot Reearc, 8(), Lee, C. P. and Cernov, A. A. (). Solutal convection around groing protein crytal and diffuional purification in pace. Journal of Crytal Grot, 4, Long, M. M., eluca, L. J., Smit, C., Caron, M., Moore, K., Harrington, M.., Pillion,.J., Biop, S.P., Roenblum, W.M., Naumann, R.J., Cait, A., Pral, J. and Bugg, C.E. (994).

13 AAM: Intern. J., Vol. 4, Iue (June 9) [Previouly, Vol. 4, No. ] 5 Protein crytal grot in microgravity-temperature induced large cale crytallization of inulin, Microgravity Science and Tecnology, 7, 96-. Moreno, A., Quiroz-Garcia, B., Yokaiciya, F., Stojanoff V., and Rudolp, P. (7). Protein crytal grot in gel and tationary magnetic field, Crytal Grot and Tecnology, 4(3), Ramacandran, N., Bauger, C.R. and Naumann R.J. (995). Modeling flo and tranport in protein crytal grot, Microgravity Science and Tecnology, 8, Riai,. N. (). Effect of centrifugal and Corioli force on a ydro-magnetic cimney convection in a muy layer, J. Crytal Grot, 6, Worter, M. G. (99). Natural convection in a muy layer, J. Fluid Mec., 4, Yanagiya, S., Sazaki, G., urbin, S.., Miyaita, S., Nakajima, K., Komatu, H., Watanabe, K. and Motokaa, M. (). Effect of a magnetic field on te grot rate of tetragonal lyozyme crytal. J. Crytal Grot, 8, Zeytounian, R.K. (4). Aymptotic Modeling of Fluid Flo Penomena, Springer Neterland. Figure. Ti i a diagram for te pyical ytem.