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Molecular dynamics (MD) in homocysteine nanosystems – computer simulation
Biomolecular Engineering 24 (2007) 577–581 www.elsevier.com/locate/geneanabioeng
Molecular dynamics (MD) in homocysteine nanosystems – computer simulation Przemysław Raczyn´ski *, Aleksander Dawid, Zygmunt Gburski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland
Abstract Excessive of homocysteine in the human body is recently considered as a factor increasing the risk of the cardiovascular system diseases. The nanosystem composed of finite number of homocysteine molecules (n = 20, 50 and 80) have been studied by MD technique. Several physical quantities of homocysteine nanosystem have been calculated as a function of temperature and a number of molecules in homocysteine cluster. The total dipole moment autocorrelation function and dielectric loss of the cluster have been also obtained. # 2007 Elsevier B.V. All rights reserved. Keywords: Homocysteine; Homocysteine nanosystem; Cluster; Dielectric loss; Diffusion coefficient; Molecular dynamics (MD) simulation
1. Introduction The functions homocysteine (C4H9NO2S) plays in human body is the subject of current debate. For example, a high level of blood serum homocysteine is considered to be a marker of potential cardiovascular (risk factor for heart attack and stroke) disease. It is not clear yet whether high serum homocysteine itself is a problem or merely an indicator of existing problems (Selhub, 1999; Miller et al., 1994; Stehouwer and van Guldener, 2001; Zoungas et al., 2006). Elevated levels of homocysteine have been linked to increased fractures in elderly persons (van Meurs, 2004; McLean, 2004). Molecular level mechanism for the mentioned and other activities of C4H9NO2S are not fully understood. In this paper we used the molecular dynamics (MD) method to study the homocysteine nanosystems consisting finite number (n = 20, 50 and 80) molecules as well as a bulk sample of C4H9NO2S. Knowledge of the properties of pure homocysteine systems may help, when one considers the role its plays in the complex, biological environment. To our knowledge, the molecular dynamics simulations of homocysteine cluster have not been reported yet. As we show below, the homocysteine clusters studied (n < 100) vaporize below T 255 K so the pure nanosystem cannot exist at physiological temperature. However, homocysteine is component of umbilical cord blood.
* Corresponding author. E-mail address: [email protected] (P. Raczyn´ski). 1389-0344/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.bioeng.2007.08.011
The studies of the low temperature properties of homocysteine nanosystem can be related to the issue of cryopreserve stem cells and stem-cell-based therapies. The stem-cell-based therapy is one of the most promising issues of nowadays medicine (Schmidt, 2003; Hubel, 1997; Gao et al., 1998; Woo et al., 2000). 2. Simulation details We have used the standard Lennard–Jones (LJ) interaction potential V(rij) between the atoms (sites) of homocysteine (Allen and Tildesley, 1989). Namely, V(rij) = 4e[(a/rij)12 (s/ rij)6], where rij is the distance between the atoms ith and jth, e the minimum of potential at a distance 21/6s and kB is the Boltzmann constant. The LJ potentials parameters e and s are given in Table 1 (Daura et al., 1998; la Cour Jansen, 2002; Kuznetsova and Kvamme, 2002). The rigid-body homocysteine molecule contains 17 atomic sites. Moreover, we have included the dipole moment of homocysteine by putting the charges 0.376e on the oxide, 0.157e on the sulfur, 0.53e on the nitrogen and 0.376e, 0.157e, 0.0256e, 0.0256e on the bonded hydrogens, obtained from ArgusLab (see http://www.planaria-software.com/ and citations therein). The LJ potentials parameters between unlike atoms were calculated by the Lorentz–Berthelot rules sA–B = (sA + sB)/2 pffiffiffiffiffiffiffiffiffi and eAB ¼ eA eB (Frenkel and Smith, 2002; Rapaport, 1995), where A and B are different atoms. The classical Newton equations of motion were integrated by predictor–corrector
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
578 Table 1 Lennard–Jones potential parameters Atoms
e/kB (K)
˚) s (A
m (1025 kg)
C O N S H
58.2 88.7 53.22 209.76 12.4
3.851 2.95 3.494 3.6 2.81
0.199 0.266 0.116 0.531 0.017
Adams–Moulton algorithm (Allen and Tildesley, 1989). The integration time step was 0.4 fs which ensured total energy conservation within 0.01%. The total simulation time was 200 ps (1 106 time steps). The initial distribution of molecules was generated by the Monte-Carlo (MC) algorithm (Allen and Tildesley, 1989). Relating to the issue of cryopreserve of biomaterials we made our study for three temperatures of cryopreserving importance, namely the liquid nitrogen (T 80 K), dry ice (T 195 K). The third temperature chosen (T 240 K) was just below the evaporation of the smaller cluster studied (n = 20). 3. Results In this paper we made the standard MD simulations for the homocysteine clusters because to our knowledge, the molecular dynamics in these systems has not been studied before by computer simulation methods. Three clusters (C4H9NO2S)n (n = 20, 50 and 80) at one temperature T = 240 K and one cluster (C4H9NO2S)50 at three temperatures T = 80, 195 and 240 K have been studied. First quantity we discus is the mean square displacement (MSD) hjD~ rðtÞj2 i of the center of mass of ! homocysteine, where D~ rðtÞ ¼ ~ rðtÞ ~ rð0Þ and r is the position of centre of mass of a single molecule. Fig. 1a and b show MSD for one cluster (C4H9NO2S)50 at T = 80, 195, 240 K and three clusters (C4H9NO2S)n (n = 20, 50 and 80) at one temperature T = 240 K, respectively. The slope of MSD increases with the raising of temperature (Fig. 1a) and with reducing of the number of molecules (Fig. 1b). The bigger is the slope of hjD~ rðtÞj2 i, the higher is a mobility of molecule. The increasing of molecule mobility with heating of the cluster is natural. Explanation requires an increase of MSD with decreasing the number of molecules in a cluster (at the same temperature). This can be understood if one recognizes, that the less molecules compose the cluster, the given temperature causes more energetic/vigorous motions of molecules, i.e. the given temperature is ‘‘hotter’’ for the cluster which consists less molecules, than for the numerous one (Frenkel and Smith, 2002). One would like to know what the value of translational diffusion coefficient D of homocysteine in nanosystem is. This can be done because D is linked to MSD by the relation hjD~ rðtÞj2 i 6Dt (Hansen and Smith, 1986). The calculated diffusion coefficient, from the linear part of the slope of hjD~ rðtÞj2 i, in the cluster (C4H9NO2S)50 at T = 80, 195 and 240 K is: 1.0 106, 2.4 105, 3.6 105 cm2/s, respectively. The diffusion coefficient at T = 240 K for (C4H9NO2S)n clusters is: D = 4.2 105, 2.8 105, 2.4 105 cm2/s for
n = 20, 50 and 80, respectively. Note, that the diffusion coefficient at T = 80 K is one order of magnitude smaller than at T = 195 K or 240 K. It means the cluster is almost in the solid phase at T = 80 K and in liquid phase at higher temperatures studied. The linear velocity autocorrelation function C~v ðtÞ ¼ h~ vðtÞ~ vð0Þi h~ vð0Þ~ vð0Þi1 where~ vðtÞ is the translational velocity of homocysteine molecule is shown in Fig. 2. The initial, short time (t < 0.3 ps) and very fast decay (Fig. 2a) of C~v ðtÞ practically does not depend on temperature. At low temperature C~v ðtÞ exhibits perturbed dumped oscillations characteristic of the solid phase (Fig. 2a). The relaxation of C~v ðtÞ for the liquid phase behaves more regularly. The long time decay of C~v ðtÞin a given temperature is slower for the smaller cluster (Fig. 2b). Take a note, that for small cluster (in our case n = 20) almost all molecules form a ‘‘skin’’ of the cluster, i.e. the majority of homocysteines belong to the outer surface and only a few consist the internal core. The surface molecules have less the nearest neighbors comparing to the core molecules which are all over surrounded by others. As consequence of this, the core molecules (inside the cluster) statistically experience more intermolecular collisions. The bigger is the cluster, the percentage number of core molecules increases, the number of collisions increases too and the velocity correlation decays faster, comparing to the smaller cluster at the same temperature.
Fig. 1. The mean square displacement of the center of mass of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
Fig. 2. The linear velocity autocorrelation function of the center of mass of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
The angular velocity autocorrelation function C~v ðtÞ ¼ h~ vðtÞ~ vð0Þi h~ vð0Þ~ vð0Þi1 , where ~ vðtÞ is the angular velocity of homocysteine is shown in Fig. 3a for several temperatures. The lowest temperature plot exhibits a pronounced dip (at t = 2.8 ps) and a small pulsation—the distinguish mark of the solid phase of the cluster. As a temperature grows up, C~v ðtÞ becomes smoother and featureless with the almost exponential decay (liquid phase). The plot of correlation function C~v ðtÞ for n = 20, 50, 80 at the same temperature T = 240 K (Fig. 3b) once again stress the role of ‘‘skin’’ effects on dynamics of molecule in a small clusters. The molecules belonging to the skin of the cluster interacts with the fewer number of neighbors. The reorientational relaxation, similarly to translational one (see Fig. 2b) is the slowest for the cluster with n = 20 where almost all homocysteines belong to the outer surface. We have also calculated for (C4H9NO2S)n at T = 240 K the second-rank orientational order parameter P2 = 0.21, 0.12 and 0.09, where n = 20, 50 and 80, respectively (de Gennes, 1974; Allen, 1995). In the case of an isotropic liquid the parameter, P2, should be zero, if the system is infinite. Indeed, the value of P2 decreases from P2 = 0.21 for extremely sparse nanosystem (n = 20) to P2 = 0.09 for much bigger cluster. An example of the normalized total dipole moment ˆ ¼ hM ~ MðtÞi=h ~ ~ 2 i, where M ~¼ correlation function CðtÞ Mð0Þ
579
Fig. 3. The angular velocity autocorrelation function of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
Pn
mi is the dipole moment of ith homocysteine, is mi and ~ i¼1 ~ ˆ decays almost presented in Fig. 4. The correlation function CðtÞ exponentially (typical Debye relaxation) with the correlation time t 159 ps. In a dielectric experiment one measures the frequency dependence of the dielectric loss e00 (n), which is the imaginary part of complex dielectric permittivity e*(n) = e0 (n) ie00 (n) (Bo¨ttcher and Bordewijk, 1995; Raczyn´ski
Fig. 4. The total dipole moment autocorrelation function in (C4H9NO2S)20 at T = 195 K.
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
580
4. Conclusions We have shown, that at the liquid nitrogen temperature the homocysteine cluster is in solid state and liquid for dry ice temperature. All clusters studied vaporize at the lower temperature than that of typical refrigerator (freezing compartment, T 255 K). The translational diffusion coefficient of homocysteine and the second-rank orientational order parameter P2 were calculated for the liquid phase of the cluster. The total dipole moment correlation function decays exponentially (typical Debye relaxation). The maximum dielectric absorption is predicted in the frequency range 25– 30 GHz. Our results may serve as preliminary report on the molecular dynamics in the small homocysteine clusters and can be linked to the issue of safe cryopreserving of selected blood’s biocomponents.
Fig. 5. The normalized dielectric loss in (C4H9NO2S)20 at T = 195 K.
Acknowledgement This work was supported in part by Ministry of Education and Science, Grant No. 1 P03B 002 30. References
Fig. 6. The normalized absorption coefficient in (C4H9NO2S)20 at T = 195 K.
pffiffiffiffiffiffiffi et al., 2005), i ¼ 1. In case of pure dipolar absorption and in the classical limit ðih ! 0Þ e00 (n) is related to the cosine Fourier ~ MðtÞi: ~ transform of CðtÞ ¼ hMðtÞ e00 ðnÞ n
Z
1
dt CðtÞ cosð2pntÞ:
(1)
0 00 Fig. R 1 5 shows the normalized dielectric loss en ðnÞ ¼ ˆ cosð2pntÞ of studied cluster. n 0 dt CðtÞ Neglecting the frequency dependence of the refractive index, the dielectric loss e00 (n) can be connected with the absorption coefficient a(n) (Bo¨ttcher and Bordewijk, 1995), a(n) / ne00 (n). The normalized absorption coefficient an(n), i.e. a(n) divided by its maximum value, is presented in Fig. 6 for (C4H9NO2S)20 at T = 195 K. Figs. 5 and 6 tell that one should expect the maximum dielectric absorption around 25–30 GHz. As far as we know the dielectric absorption of homocysteine clusters has not been announced yet.
Allen, M.P., 1995. Observation, Prediction and Simulation of Phase Transitions in Complex Fluids. Kluwer, Dordrecht (and the references therein). Allen, M.P., Tildesley, D.J., 1989. Computer Simulation of Liquids. Oxford University Press, Oxford. Bo¨ttcher, C.L.F., Bordewijk, P., 1995. Theory of Dielectric Polarization. Elsevier, Amsterdam. Daura, X., Mark, A.E., van Gunsteren, W.F., 1998. Parametrization of aliphatic CHn united atoms of GROMOS96 force field. J. Comput. Chem. 19, 535– 547. de Gennes, P.G., 1974. The Physics of Liquid Crystals. Clarendon, Oxford. Frenkel, D., Smith, B., 2002. Understanding Molecular Simulation. Academic Press, New York. Gao, D.Y., Chang, Q., Liu, C., Farris, K., Harvey, K., McGann, L.E., English, D., Jansen, J., Critser, J.K., 1998. Fundamental cryobiology of human hematopoietic progenitor cells. I. Osmotic characteristics and volume distribution. Cryobiology 36, 40–48. Hansen, J.P., Mcdonald, 1986. Theory of Simple Liquid. Academic Press, London. Hubel, A., 1997. Parameters of cell freezing: implications for the cryopreservation of stem cells. Transfusion Med. Rev. 11, 224–233. Kuznetsova, T., Kvamme, B., 2002. Atomistic computer simulations for thermodynamic properties of carbon dioxide at low temperatures. Energy Conver. Manage. 43, 2601–2623. la Cour Jansen, T., 2002. Theoretical simulation of nonlinear spectroscopy in the liquid phase. Ph.D. Thesis. McLean, R.R., 2004. Homocysteine as a predictive factor for hip fracture in older persons. N. Engl. J. Med. 350, 2042–2049. Miller, J.W., Nadeau, M.R., Smith, D., Selhub, J., 1994. Vitamin B-6 deficiency vs folate deficiency: comparison of responses to methionine loading in rats. Am. J. Clin. Nutr. 59, 1033–1039. Raczyn´ski, P., Dawid, A., Dendzik, Z., Gburski, Z., 2005. Dielectric relaxation in water–cholesterol mixture cluster: molecular dynamics simulation. J. Mol. Struct. 750, 17–21. Rapaport, D.C., 1995. The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge. Schmidt, P.J., 2003. Frozen blood. Lancet 361, 2084. Selhub, J., 1999. Homocysteine metabolism. Annu. Rev. Nutr. 19, 217–246. Stehouwer, C., van Guldener, C., 2001. Homocysteine-lowering treatment: an overview. Expert. Opin. Pharmacother. 2 (9), 1449–1460.
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581 van Meurs, J.B., 2004. Homocysteine levels and the risk of osteoporotic fracture. N. Engl. J. Med. 350, 2033–2041. Woo, D.K., Dudrick, S.J., Sumpio, B.E., 2000. Homocysteine stimulates MAP kinase in bovine aortic smooth muscle cells. Surgery 128, 59– 66.
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Zoungas, S., McGrath, B.P., Branley, P., Kerr, P.G., Muske, C., Wolfe, R., Atkins, R.C., Nicholls, K., Fraenkel, M., Hutchison, B.G., Walker, R., McNeil, J.J., 2006. Cardiovascular morbidity and mortality in the atherosclerosis and folic acid supplementation trial (ASFAST) in chronic renal failure: a multicenter, randomized, controlled trial. J. Am. Coll. Cardiol. 47 (6), 1108–1116.
Molecular dynamics (MD) in homocysteine nanosystems – computer simulation Przemysław Raczyn´ski *, Aleksander Dawid, Zygmunt Gburski Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland
Abstract Excessive of homocysteine in the human body is recently considered as a factor increasing the risk of the cardiovascular system diseases. The nanosystem composed of finite number of homocysteine molecules (n = 20, 50 and 80) have been studied by MD technique. Several physical quantities of homocysteine nanosystem have been calculated as a function of temperature and a number of molecules in homocysteine cluster. The total dipole moment autocorrelation function and dielectric loss of the cluster have been also obtained. # 2007 Elsevier B.V. All rights reserved. Keywords: Homocysteine; Homocysteine nanosystem; Cluster; Dielectric loss; Diffusion coefficient; Molecular dynamics (MD) simulation
1. Introduction The functions homocysteine (C4H9NO2S) plays in human body is the subject of current debate. For example, a high level of blood serum homocysteine is considered to be a marker of potential cardiovascular (risk factor for heart attack and stroke) disease. It is not clear yet whether high serum homocysteine itself is a problem or merely an indicator of existing problems (Selhub, 1999; Miller et al., 1994; Stehouwer and van Guldener, 2001; Zoungas et al., 2006). Elevated levels of homocysteine have been linked to increased fractures in elderly persons (van Meurs, 2004; McLean, 2004). Molecular level mechanism for the mentioned and other activities of C4H9NO2S are not fully understood. In this paper we used the molecular dynamics (MD) method to study the homocysteine nanosystems consisting finite number (n = 20, 50 and 80) molecules as well as a bulk sample of C4H9NO2S. Knowledge of the properties of pure homocysteine systems may help, when one considers the role its plays in the complex, biological environment. To our knowledge, the molecular dynamics simulations of homocysteine cluster have not been reported yet. As we show below, the homocysteine clusters studied (n < 100) vaporize below T 255 K so the pure nanosystem cannot exist at physiological temperature. However, homocysteine is component of umbilical cord blood.
* Corresponding author. E-mail address: [email protected] (P. Raczyn´ski). 1389-0344/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.bioeng.2007.08.011
The studies of the low temperature properties of homocysteine nanosystem can be related to the issue of cryopreserve stem cells and stem-cell-based therapies. The stem-cell-based therapy is one of the most promising issues of nowadays medicine (Schmidt, 2003; Hubel, 1997; Gao et al., 1998; Woo et al., 2000). 2. Simulation details We have used the standard Lennard–Jones (LJ) interaction potential V(rij) between the atoms (sites) of homocysteine (Allen and Tildesley, 1989). Namely, V(rij) = 4e[(a/rij)12 (s/ rij)6], where rij is the distance between the atoms ith and jth, e the minimum of potential at a distance 21/6s and kB is the Boltzmann constant. The LJ potentials parameters e and s are given in Table 1 (Daura et al., 1998; la Cour Jansen, 2002; Kuznetsova and Kvamme, 2002). The rigid-body homocysteine molecule contains 17 atomic sites. Moreover, we have included the dipole moment of homocysteine by putting the charges 0.376e on the oxide, 0.157e on the sulfur, 0.53e on the nitrogen and 0.376e, 0.157e, 0.0256e, 0.0256e on the bonded hydrogens, obtained from ArgusLab (see http://www.planaria-software.com/ and citations therein). The LJ potentials parameters between unlike atoms were calculated by the Lorentz–Berthelot rules sA–B = (sA + sB)/2 pffiffiffiffiffiffiffiffiffi and eAB ¼ eA eB (Frenkel and Smith, 2002; Rapaport, 1995), where A and B are different atoms. The classical Newton equations of motion were integrated by predictor–corrector
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
578 Table 1 Lennard–Jones potential parameters Atoms
e/kB (K)
˚) s (A
m (1025 kg)
C O N S H
58.2 88.7 53.22 209.76 12.4
3.851 2.95 3.494 3.6 2.81
0.199 0.266 0.116 0.531 0.017
Adams–Moulton algorithm (Allen and Tildesley, 1989). The integration time step was 0.4 fs which ensured total energy conservation within 0.01%. The total simulation time was 200 ps (1 106 time steps). The initial distribution of molecules was generated by the Monte-Carlo (MC) algorithm (Allen and Tildesley, 1989). Relating to the issue of cryopreserve of biomaterials we made our study for three temperatures of cryopreserving importance, namely the liquid nitrogen (T 80 K), dry ice (T 195 K). The third temperature chosen (T 240 K) was just below the evaporation of the smaller cluster studied (n = 20). 3. Results In this paper we made the standard MD simulations for the homocysteine clusters because to our knowledge, the molecular dynamics in these systems has not been studied before by computer simulation methods. Three clusters (C4H9NO2S)n (n = 20, 50 and 80) at one temperature T = 240 K and one cluster (C4H9NO2S)50 at three temperatures T = 80, 195 and 240 K have been studied. First quantity we discus is the mean square displacement (MSD) hjD~ rðtÞj2 i of the center of mass of ! homocysteine, where D~ rðtÞ ¼ ~ rðtÞ ~ rð0Þ and r is the position of centre of mass of a single molecule. Fig. 1a and b show MSD for one cluster (C4H9NO2S)50 at T = 80, 195, 240 K and three clusters (C4H9NO2S)n (n = 20, 50 and 80) at one temperature T = 240 K, respectively. The slope of MSD increases with the raising of temperature (Fig. 1a) and with reducing of the number of molecules (Fig. 1b). The bigger is the slope of hjD~ rðtÞj2 i, the higher is a mobility of molecule. The increasing of molecule mobility with heating of the cluster is natural. Explanation requires an increase of MSD with decreasing the number of molecules in a cluster (at the same temperature). This can be understood if one recognizes, that the less molecules compose the cluster, the given temperature causes more energetic/vigorous motions of molecules, i.e. the given temperature is ‘‘hotter’’ for the cluster which consists less molecules, than for the numerous one (Frenkel and Smith, 2002). One would like to know what the value of translational diffusion coefficient D of homocysteine in nanosystem is. This can be done because D is linked to MSD by the relation hjD~ rðtÞj2 i 6Dt (Hansen and Smith, 1986). The calculated diffusion coefficient, from the linear part of the slope of hjD~ rðtÞj2 i, in the cluster (C4H9NO2S)50 at T = 80, 195 and 240 K is: 1.0 106, 2.4 105, 3.6 105 cm2/s, respectively. The diffusion coefficient at T = 240 K for (C4H9NO2S)n clusters is: D = 4.2 105, 2.8 105, 2.4 105 cm2/s for
n = 20, 50 and 80, respectively. Note, that the diffusion coefficient at T = 80 K is one order of magnitude smaller than at T = 195 K or 240 K. It means the cluster is almost in the solid phase at T = 80 K and in liquid phase at higher temperatures studied. The linear velocity autocorrelation function C~v ðtÞ ¼ h~ vðtÞ~ vð0Þi h~ vð0Þ~ vð0Þi1 where~ vðtÞ is the translational velocity of homocysteine molecule is shown in Fig. 2. The initial, short time (t < 0.3 ps) and very fast decay (Fig. 2a) of C~v ðtÞ practically does not depend on temperature. At low temperature C~v ðtÞ exhibits perturbed dumped oscillations characteristic of the solid phase (Fig. 2a). The relaxation of C~v ðtÞ for the liquid phase behaves more regularly. The long time decay of C~v ðtÞin a given temperature is slower for the smaller cluster (Fig. 2b). Take a note, that for small cluster (in our case n = 20) almost all molecules form a ‘‘skin’’ of the cluster, i.e. the majority of homocysteines belong to the outer surface and only a few consist the internal core. The surface molecules have less the nearest neighbors comparing to the core molecules which are all over surrounded by others. As consequence of this, the core molecules (inside the cluster) statistically experience more intermolecular collisions. The bigger is the cluster, the percentage number of core molecules increases, the number of collisions increases too and the velocity correlation decays faster, comparing to the smaller cluster at the same temperature.
Fig. 1. The mean square displacement of the center of mass of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
Fig. 2. The linear velocity autocorrelation function of the center of mass of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
The angular velocity autocorrelation function C~v ðtÞ ¼ h~ vðtÞ~ vð0Þi h~ vð0Þ~ vð0Þi1 , where ~ vðtÞ is the angular velocity of homocysteine is shown in Fig. 3a for several temperatures. The lowest temperature plot exhibits a pronounced dip (at t = 2.8 ps) and a small pulsation—the distinguish mark of the solid phase of the cluster. As a temperature grows up, C~v ðtÞ becomes smoother and featureless with the almost exponential decay (liquid phase). The plot of correlation function C~v ðtÞ for n = 20, 50, 80 at the same temperature T = 240 K (Fig. 3b) once again stress the role of ‘‘skin’’ effects on dynamics of molecule in a small clusters. The molecules belonging to the skin of the cluster interacts with the fewer number of neighbors. The reorientational relaxation, similarly to translational one (see Fig. 2b) is the slowest for the cluster with n = 20 where almost all homocysteines belong to the outer surface. We have also calculated for (C4H9NO2S)n at T = 240 K the second-rank orientational order parameter P2 = 0.21, 0.12 and 0.09, where n = 20, 50 and 80, respectively (de Gennes, 1974; Allen, 1995). In the case of an isotropic liquid the parameter, P2, should be zero, if the system is infinite. Indeed, the value of P2 decreases from P2 = 0.21 for extremely sparse nanosystem (n = 20) to P2 = 0.09 for much bigger cluster. An example of the normalized total dipole moment ˆ ¼ hM ~ MðtÞi=h ~ ~ 2 i, where M ~¼ correlation function CðtÞ Mð0Þ
579
Fig. 3. The angular velocity autocorrelation function of homocysteine: (a) in (C4H9NO2S)50 at several temperatures and (b) in (C4H9NO2S)n clusters at T = 240 K.
Pn
mi is the dipole moment of ith homocysteine, is mi and ~ i¼1 ~ ˆ decays almost presented in Fig. 4. The correlation function CðtÞ exponentially (typical Debye relaxation) with the correlation time t 159 ps. In a dielectric experiment one measures the frequency dependence of the dielectric loss e00 (n), which is the imaginary part of complex dielectric permittivity e*(n) = e0 (n) ie00 (n) (Bo¨ttcher and Bordewijk, 1995; Raczyn´ski
Fig. 4. The total dipole moment autocorrelation function in (C4H9NO2S)20 at T = 195 K.
P. Raczyn´ski et al. / Biomolecular Engineering 24 (2007) 577–581
580
4. Conclusions We have shown, that at the liquid nitrogen temperature the homocysteine cluster is in solid state and liquid for dry ice temperature. All clusters studied vaporize at the lower temperature than that of typical refrigerator (freezing compartment, T 255 K). The translational diffusion coefficient of homocysteine and the second-rank orientational order parameter P2 were calculated for the liquid phase of the cluster. The total dipole moment correlation function decays exponentially (typical Debye relaxation). The maximum dielectric absorption is predicted in the frequency range 25– 30 GHz. Our results may serve as preliminary report on the molecular dynamics in the small homocysteine clusters and can be linked to the issue of safe cryopreserving of selected blood’s biocomponents.
Fig. 5. The normalized dielectric loss in (C4H9NO2S)20 at T = 195 K.
Acknowledgement This work was supported in part by Ministry of Education and Science, Grant No. 1 P03B 002 30. References
Fig. 6. The normalized absorption coefficient in (C4H9NO2S)20 at T = 195 K.
pffiffiffiffiffiffiffi et al., 2005), i ¼ 1. In case of pure dipolar absorption and in the classical limit ðih ! 0Þ e00 (n) is related to the cosine Fourier ~ MðtÞi: ~ transform of CðtÞ ¼ hMðtÞ e00 ðnÞ n
Z
1
dt CðtÞ cosð2pntÞ:
(1)
0 00 Fig. R 1 5 shows the normalized dielectric loss en ðnÞ ¼ ˆ cosð2pntÞ of studied cluster. n 0 dt CðtÞ Neglecting the frequency dependence of the refractive index, the dielectric loss e00 (n) can be connected with the absorption coefficient a(n) (Bo¨ttcher and Bordewijk, 1995), a(n) / ne00 (n). The normalized absorption coefficient an(n), i.e. a(n) divided by its maximum value, is presented in Fig. 6 for (C4H9NO2S)20 at T = 195 K. Figs. 5 and 6 tell that one should expect the maximum dielectric absorption around 25–30 GHz. As far as we know the dielectric absorption of homocysteine clusters has not been announced yet.
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