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Calculations of magnetic susceptibility of magnetotactic bacteria culture
Journal of Magnetism and Magnetic Materials 234 (2001) 575–583
Calculations of magnetic susceptibility of magnetotactic bacteria culture V. Zablotskiia,*, V. Yurchenkoa, Y. Kamysab, M. Chelombetskayac a
Department of Magnetism, Institute of Physics, Czech Academy of Science, Na Slovance 2, 182 21 Prague, Czech Republic b Department of Physics, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine c Department of Biology, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine Received 22 September 2000; received in revised form 24 May 2001
Abstract Calculations of magnetic susceptibility of magnetotactic bacteria (MTB) culture are reported. The model was elaborated with regard to the effect of chemotaxis for two different geometrical compositions of the experiment. The results obtained allow one to determine concentration of MTB from magnetic measurements. It was shown that the characteristic parameters of chemotaxis can be extracted from the dependencies of the susceptibility on magnetic field strength. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetotactic bacteria; Magnetic susceptibility of magnetotactic bacteria; Magnetic moment of magnetotactic bacteria
1. Introduction The most prominent feature of magnetotactic bacteria containing intracellular Fe3O4 particles called magnetosomes is that they move by flagella driven motion in a direction imposed by the external magnetic field [1,2]. Recent studies [3–8] of these types of micro-organisms revealed a broad field of their practical applicability, e.g., for being engaged in the process of continuous radionuclide recovery, heavy metal adsorbtion etc. From the foregoing investigations of MTB it is well known that for an efficient, practical use of this kind of determination of concentration is required. Due to relatively short lifetimes of active magnetotactic bacteria and other accompanying effects, data on cultures’ concentrations are strongly affected. As it was shown previously, reliable and rapid data on this subject can be obtained from the investigations of magnetic properties of MTB cultures. In Ref. [9] authors reported about measurements of magnetic susceptibility of Magnetospirillum gryphiswaldense and Cocci cultures. The results provided by these experiments allowed MTB concentrations to be determined. Here we propose a theoretical background for the determination of mentioned concentrations by the mean of magnetic susceptibility measurements for two types of magnetotactic bacteria (with respect to the position of flagella).
*Corresponding author. E-mail address: [email protected] (V. Zablotskii). 0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 1 5 - 2
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To describe magnetotactic bacteria motion one should take into account two usually attendant effects: chemotaxis and magnetotaxis. The latter is a consequence of interaction between bacterial magnetosomes and external magnetic field. Whereas the former one is a result of bacteria culture’s sensitivity to the chemical environment. Counteraction between relevant energies predefines the favourable direction of bacteria’s motion. In what follows, a phenomenological model for calculation of MTB magnetic susceptibility is proposed taking into account both these effects. We believe that the bacteria are insensitive to chemotactants unless direct contact is made. The bacteria can then respond to the chemotactant by a series of random tumbling motions by means of flagella rotation. With regard to the type of specific chemicals, cells can be either attracted or repelled by them. It implies that bacteria migrate towards or away from the area with higher concentration of chemical stimuli. Since both cells and attractants are subjected to Brownian motion, the interaction between them has stochastic character. In respect of the distribution of chemotactants there are several ways to describe qualitatively, the evolution of the system. After an external magnetic field is applied all the magnetosomes tend to align themselves along the lines of magnetic induction, therefore stipulating the motion of bacteria in that direction. If the density of chemical attractants varies in space, i.e. a gradient of their concentration is present, preferable direction of culture’s flow appears. Thus, the medium becomes anisotropic. For the case of homogeneous distribution, chemotaxis can enter into the equation of bacteria motion as a viscous drag factor since it forces cells to deviate in random way from the direction of magnetically ordered motion. For both cases of non-homogeneous as well as homogeneous distribution of chemoattractants the energy of chemotaxis, Wch can be introduced as Wch ¼ Fvt;
ð1Þ
where F is the motive force of bacterium exerted by rotating flagella, v is the velocity of flagella-mediated bacterial motion and t is the mean time of flagella rotations. We should notice that we assume bacterial flagella to rotate invoking directed movement of a cell during a certain time after encountering a chemically active particle. The energy value (1) can be considered as the energy of chemotaxis for the given type of bacteria. The values v and t involved into the definition of the energy of chemotaxis are measurable parameters. The force F can be calculated from one of the following equations: mb v ¼ Ft mch vT FS t;
ð2Þ
mb v ¼ Ft þ mch vT FS t;
ð3Þ
or for attractive and repulsive chemotaxis, respectively. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHere mb and mch are the masses of bacterium and chemotactant particle, respectively. vT ¼ 8kB T=pmb is the average velocity of the Brownian motion of chemical particles. Velocity v entering these two formulas should be considered as the maximal velocity vm attained by bacteria immediately at the moment when flagella rotations are completed. Viscous drag force FS is given by the Stocks formula. For the time t cell moves for mean distance Dl ¼ vt: It was found that DlE38276 mm for mean tE122 s [10] of flagella rotations and vm E38 mm/s [7,11] of magnetotactic bacteria motion. Mean distance a between chemotactants, can be found from apparent expression: a ¼ n1=3 where n is the concentration of chemical stimuli dependent on coordinate. If Dloa then bacterial motion caused by chemotaxis gradient has to be considered as discrete, so then vovm : Otherwise, if Dl > a; movements of bacterium from one chemotactant to another may be specified as continuous and the average velocity is vEvm =2: To describe orientation action of chemotaxis we use the simplest angular dependence: Ean ðyÞ ¼ cch Wch cos y;
ð4Þ
where cch is a numerical constant of the order of unity which depends on the gradient of chemotactants concentration (it is equal to zero at zero gradient of chemical stimuli concentration) and y is the angle between bacterial magnetic moment and the chemotaxis gradient.
V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 575–583
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Fig. 1. Geometry of experiments with chemotactants placed on the top (left) and bottom (right) of the cylinder.
We consider two cases corresponding to different distributions of chemotactants. Throughout our consideration the cylindrical volume is selected. The external field direction coincides with the symmetry axle of the cylinder. The first considered case corresponds to the following geometry of experiment: MTB culture fills the volume, chemotactants are concentrated on the top of the volume (or on the bottom that reflects in the sign of the energy of chemotaxis) and penetrate into the volume as the result of diffusion. It implies that the gradient of chemical substance density is directed along the main axis and has only z-component (see Fig. 1). Further we will refer to this set of conditions as to case (i). We define the second case, call it case (ii), for transverse geometry: chemical stimuli are concentrated on the axis of cylinder (or its surface that stipulates the sign of the chemotaxis energy) diffusing inward to the volume (Fig. 2). In this way, the gradient of chemotactants concentration is radial. Since the magnetosomes can align themselves along the magnetic field direction and the magnetic interaction between different cells is negligibly small, a magnetotactic culture can be considered as an ideal gas of paramagnetic arrows. Thus, the susceptibility of the system can be obtained in the way similar to that commonly used for paramagnetic one. This approach is valid until the mean distance between bacteria is small compared to the characteristic size of the cells. As a rule this condition is fulfilled. Common size of a bacterium is about 0.6 mm (for cocci [9]). Appropriate concentrations of MTB to be used in practice is approximately 1014–1015 cells/m3 [9]. It makes the ratio of average distance between bacteria to their mean size be at least three orders. Let us now calculate magnetization M and susceptibility w ¼ qM=qH of the culture. The magnetization of MTB is equal to M ¼ C hPmz i; where Pmz is the z-component of the mean magnetic moment and C is the difference between the concentrations of bacteria with positively (North seeking bacteria) an negatively (South seeking bacteria) oriented magnetic moments with regard to the position of flagella: C ¼ ðNþ N Þ=V: Here Nþ and N are respectively, the numbers of North and South seeking bacteria, i.e. bacteria with the z-component of vector Pm parallel and antiparallel to the z-axis, respectively; V is the volume of the culture.
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Fig. 2. Geometry of the problem for the case of radial distribution of chemotactants.
The average magnetic moment of the culture can be calculated from the following expression: R Pmz dn ; hPmz i ¼ R dn Taking into account the Boltzman distribution of MTB we rewrite Eq. (5) in the form: R 2p R p Pmz eW=kB T sin y dy dj hPmz i ¼ 0 R 2p0 R p ; W=kB T sin y dy dj 0 0 e
ð5Þ
ð6Þ
where W is the energy of bacterium, Pmz ¼ Pm cos y; y and j are corresponding angles in the spherical coordinate frame and kB is the Boltzman constant. The energy W is the sum of the magnetic energy Wm ; i.e. the energy of interaction of bacterial magnetic moment with the external magnetic field: !! Wm ¼ m0 Pm H ; ð7Þ where m0 is the permeability of free space, and the anisotropy energy caused by chemotaxis: Ean ¼ cch Wch cos y for the case (i) or for the case of radial distribution of chemotactants (ii): Ean ¼ cch Wch sin y: Substituting relevant expressions for the energy of a bacterium into Eq. (6) we obtain the expression for the average value of z-component of magnetic moment: R 2p R p cch Wch cos y m0 Pm H cos y cos y sin y dy dj 0 0 Pm exp kB T ð8Þ hPmz i ¼ R 2p R p cch Wch cos y m0 Pm H cos y sin y dy dj 0 0 exp kB T Integration of (8) for the given geometry gives us the Langevin function with shifted argument: " # m0 Pm H cch Wch m0 Pm H cch Wch 1 ; hPmz i ¼ Pm cth kB T kB T
ð9Þ
V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 575–583
Thus, from Eq. (9) we obtain for the susceptibility: " 2 # m m P H c W m P H c W m ch ch m ch ch 0 w ¼ CP2m 0 sh2 0 ; þ kB T kB T kB T
579
ð10Þ
For case (ii) an expression for the z-component of the mean magnetic moment can be inferred in a similar way to the one used above. Hence, it has a form: R 2p R p cch Wch sin y sin j m0 Pm H cos y P exp cos y sin y dy dj m 0 0 kB T : ð11Þ hPmz i ¼ R 2p R p cch Wch sin y sin j m0 Pm H cos y exp sin y dy dj 0 0 kB T Integral (11) can be calculated numerically. In Fig. 3 the dependence of the mean magnetic moment on external magnetic field strength is shown for the case (i). Different curves correspond to various energies of chemotaxis: Wch1 > Wch2 > Wch3 > Wch4 : Here and further on the energy of chemotaxis is expressed in units kB Troom and denoted as Z ¼ Wch =kB Troom : For calculations we used the next numerical values: p ¼ 2:7 1016 Am2 [9], T ¼ 293 K, C ¼ 6 1014 cells/m3. From the plot one can see that with the increase of chemotactic energy the initial value of the mean magnetic moment grows. This is quite reasonable since the growing gradient of chemotaxis leads to higher ordering of the culture in specific direction, so, cells and obviously magnetosomes align in this direction. In Fig. 4 the magnetic susceptibility of the culture is plotted as a function of H for different values of the energy of chemotaxis. The culture’s magnetic susceptibility decreases with increasing energy of chemotaxis (see Fig. 4). This decreasing can be explained as the following. It is clear that wp/Pmz S=H because wpM=H and Mp/Pmz S: Magnetic field strength H increases in a much faster rate than /Pmz S and so the magnetic susceptibility decreases with increasing applied magnetic field. Since the rate of the mean magnetic moment changes drops as Wch grows (see Fig. 3, here /Pmz S is close to the saturation for large values of the chemotaxis energy), the susceptibility
Fig. 3. Dependence of the mean magnetic moment on magnetic field strength for different values of the energy of chemotaxis Z:
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Fig. 4. Dependence of magnetic susceptibility on magnetic field strength for different values of chemotaxis energy Z:
Fig. 5. Magnetic susceptibility versus the energy of chemotaxis for the case of chemotactants concentration gradient parallel to the field direction for different values of field strength H:
decreases. The magnetic susceptibility reveals similar behaviour being plotted versus the chemotactic energy for different values of the magnetic field strength (see Fig. 5). If the chemical stimuli are concentrated on the bottom, i.e. the gradient of chemotactant concentration is antiparallel to the field direction, initial magnetization of the culture will be different from zero and negative. Fig. 6 represents the mean magnetic moment dependency on the magnetic field strength for different values of the chemotaxis energy. Different signs of magnetic field and gradient of chemotaxis lead to the presence of a peak on the plot of field dependence of susceptibility. As the energy of chemotaxis grows the position of the peak shifts to the range of higher field values as it is shown in Fig. 7.
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Fig. 6. Dependence of the mean magnetic moment on external field strength for different energies of chemotaxis Z: Magnetic field direction is antiparallel to the gradient of chemotactants concentration.
Fig. 7. Susceptibility as function of external magnetic field for different values of chemotaxis energy Z: Magnetic field is directed antiparallel to the gradient of chemotactans concentration.
In the case (ii) the initial value of system’s magnetization was found to be equal to zero due to the radial symmetry of chemotactants’ distribution. In Fig. 8 z-component of the mean magnetic moment was plotted versus external magnetic field strength for different energies of chemotaxis. Steeper curves correspond to lower values of the energy. Field dependence of the susceptibility for given geometry is shown in Fig. 9. A comparison between Figs. 3 and 8 shows a reduction in the mean magnetic moment of the culture for the geometry with the radial gradient of chemotactants concentration. This is to be expected because the bacteria become aligned radially perpendicular to the z-axis.
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Fig. 8. Mean magnetic moment versus the magnetic field strength for different energies of chemotaxis Z: Field direction is perpendicular to the gradient of chemotactants concentration.
Fig. 9. Field dependence of magnetic susceptibility for different values of chemotaxis energy Z: Magnetic field is perpendicular to the gradient of chemotactants concentration.
2. Conclusions and discussion The proposed model allowed us to calculate the key magnetic characteristics of a magnetotactic bacteria colony (the mean magnetic moment and magnetic susceptibility) as functions of the external magnetic field strength and the chemotaxis energy. It was shown that geometry of the problem (relative orientation of the chemotactants concentration gradient and the applied magnetic field) has an essential influence on both the magnetization curve shape and magnetic susceptibility. If the gradient of chemotactants concentration is parallel to the field direction, the field dependence of the mean magnetic moment is the Langevin function of H (Eq. (9)) with shifted argument (see Fig. 3). This result coincides with the one previously referred to in Ref. [8]. For the gradient of chemotactants concentration antiparallel to the field direction magnetization depends on H as it is shown in Fig. 6. In this case the magnetization at H ¼ 0 can be negative (call it
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polarization by chemotaxis) and the magnetic susceptibility of bacterial culture has a maximum when the energy of chemotaxis is equal to the energy of interaction between the bacteria magnetic moment and magnetic field (Fig. 7). The shift of the maximum position field ðHmax Þ with increasing chemotaxis energy is described by expression Hmax ¼ Wch =mm0 Pm On this basis one can propose an experimental method for the determination of the chemotaxis energy of MTB from the magnetic susceptibility measurements at the given geometry. Another substantial for experiment and practical applicability detail is that the mean magnetic moment of bacteria is several orders larger than the magnetic moment of a paramagnetic atom. Thus, it is possible to observe saturation in the magnetization curves of MTB in reachable range of magnetic field strength values, Hs E102 A/m (Figs. 3, 6 and 8). ~ gradients of chemotactants concentration we have found a For both the radial and that coincided with H decrease of the magnetic susceptibility with increasing energy of chemotaxis.
References [1] A.J. Kalmin, R.P. Blackmore, in: K. Schmidt-Koening, W.T. Keeton (Eds.), Animal Migration, Navigation and Homing, Springer, New York, 1978, p. 354. [2] R.B. Frankel, R.P. Blackmore, J. Magn. Magn. Mater. 15–18 (1980) 1562. [3] T. Matsunaga, S. Kamiya, Appl. Microbiol. Biotechnol. 26 (1987) 328. [4] M. Funaki, H. Sakai, T. Matsunaga, J. Geomag. Geoelectr. 41 (1989) 77. [5] K. Futchik, H. Pfu. tzner, A. Doblander, P. Scho. nhuber, T. Donebeck, N. Petersen, H. Vali, Phys. Scr. 40 (1989) 518. [6] P. Scho. nhuber, H. Pfu. tzner, G. Harasko, T. Klinger, K. Futchik, J. Magn. Magn. Mater. 112 (1992) 349. [7] A.S. Bahaj, I.W. Croudace, P.A.B. James, F.D. Moeschler, P.E. Warwick, J. Magn. Magn. Mater. 184 (1998) 241. [8] J.H.P. Watson, B.A. Cressey, A.P. Roberts, D.C. Ellwood, J.M. Charnock, A.K. Sober, J. Magn. Magn. Mater. 214 (2000) 13. [9] G. Harasko, H. Pfu. tzner, E. Rapp, K. Futchik, D. Schu. ler, Jpn. J. Appl. Phys. 32 (1993) 252. [10] R.M. Atlas, Principles of Microbiology, Mosby-Year Book Inc., St. Louis, MO, 1994, p. 888. [11] A.S. Bahaj, P.A.B. James, F.D. Moeschler, J. Magn. Magn. Mater. 177–181 (1998) 1453.
Calculations of magnetic susceptibility of magnetotactic bacteria culture V. Zablotskiia,*, V. Yurchenkoa, Y. Kamysab, M. Chelombetskayac a
Department of Magnetism, Institute of Physics, Czech Academy of Science, Na Slovance 2, 182 21 Prague, Czech Republic b Department of Physics, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine c Department of Biology, Donetsk National University, Universitetskaya str. 24, 83055 Donetsk, Ukraine Received 22 September 2000; received in revised form 24 May 2001
Abstract Calculations of magnetic susceptibility of magnetotactic bacteria (MTB) culture are reported. The model was elaborated with regard to the effect of chemotaxis for two different geometrical compositions of the experiment. The results obtained allow one to determine concentration of MTB from magnetic measurements. It was shown that the characteristic parameters of chemotaxis can be extracted from the dependencies of the susceptibility on magnetic field strength. r 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetotactic bacteria; Magnetic susceptibility of magnetotactic bacteria; Magnetic moment of magnetotactic bacteria
1. Introduction The most prominent feature of magnetotactic bacteria containing intracellular Fe3O4 particles called magnetosomes is that they move by flagella driven motion in a direction imposed by the external magnetic field [1,2]. Recent studies [3–8] of these types of micro-organisms revealed a broad field of their practical applicability, e.g., for being engaged in the process of continuous radionuclide recovery, heavy metal adsorbtion etc. From the foregoing investigations of MTB it is well known that for an efficient, practical use of this kind of determination of concentration is required. Due to relatively short lifetimes of active magnetotactic bacteria and other accompanying effects, data on cultures’ concentrations are strongly affected. As it was shown previously, reliable and rapid data on this subject can be obtained from the investigations of magnetic properties of MTB cultures. In Ref. [9] authors reported about measurements of magnetic susceptibility of Magnetospirillum gryphiswaldense and Cocci cultures. The results provided by these experiments allowed MTB concentrations to be determined. Here we propose a theoretical background for the determination of mentioned concentrations by the mean of magnetic susceptibility measurements for two types of magnetotactic bacteria (with respect to the position of flagella).
*Corresponding author. E-mail address: [email protected] (V. Zablotskii). 0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 1 5 - 2
576
V. Zablotskii et al. / Journal of Magnetism and Magnetic Materials 234 (2001) 575–583
To describe magnetotactic bacteria motion one should take into account two usually attendant effects: chemotaxis and magnetotaxis. The latter is a consequence of interaction between bacterial magnetosomes and external magnetic field. Whereas the former one is a result of bacteria culture’s sensitivity to the chemical environment. Counteraction between relevant energies predefines the favourable direction of bacteria’s motion. In what follows, a phenomenological model for calculation of MTB magnetic susceptibility is proposed taking into account both these effects. We believe that the bacteria are insensitive to chemotactants unless direct contact is made. The bacteria can then respond to the chemotactant by a series of random tumbling motions by means of flagella rotation. With regard to the type of specific chemicals, cells can be either attracted or repelled by them. It implies that bacteria migrate towards or away from the area with higher concentration of chemical stimuli. Since both cells and attractants are subjected to Brownian motion, the interaction between them has stochastic character. In respect of the distribution of chemotactants there are several ways to describe qualitatively, the evolution of the system. After an external magnetic field is applied all the magnetosomes tend to align themselves along the lines of magnetic induction, therefore stipulating the motion of bacteria in that direction. If the density of chemical attractants varies in space, i.e. a gradient of their concentration is present, preferable direction of culture’s flow appears. Thus, the medium becomes anisotropic. For the case of homogeneous distribution, chemotaxis can enter into the equation of bacteria motion as a viscous drag factor since it forces cells to deviate in random way from the direction of magnetically ordered motion. For both cases of non-homogeneous as well as homogeneous distribution of chemoattractants the energy of chemotaxis, Wch can be introduced as Wch ¼ Fvt;
ð1Þ
where F is the motive force of bacterium exerted by rotating flagella, v is the velocity of flagella-mediated bacterial motion and t is the mean time of flagella rotations. We should notice that we assume bacterial flagella to rotate invoking directed movement of a cell during a certain time after encountering a chemically active particle. The energy value (1) can be considered as the energy of chemotaxis for the given type of bacteria. The values v and t involved into the definition of the energy of chemotaxis are measurable parameters. The force F can be calculated from one of the following equations: mb v ¼ Ft mch vT FS t;
ð2Þ
mb v ¼ Ft þ mch vT FS t;
ð3Þ
or for attractive and repulsive chemotaxis, respectively. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHere mb and mch are the masses of bacterium and chemotactant particle, respectively. vT ¼ 8kB T=pmb is the average velocity of the Brownian motion of chemical particles. Velocity v entering these two formulas should be considered as the maximal velocity vm attained by bacteria immediately at the moment when flagella rotations are completed. Viscous drag force FS is given by the Stocks formula. For the time t cell moves for mean distance Dl ¼ vt: It was found that DlE38276 mm for mean tE122 s [10] of flagella rotations and vm E38 mm/s [7,11] of magnetotactic bacteria motion. Mean distance a between chemotactants, can be found from apparent expression: a ¼ n1=3 where n is the concentration of chemical stimuli dependent on coordinate. If Dloa then bacterial motion caused by chemotaxis gradient has to be considered as discrete, so then vovm : Otherwise, if Dl > a; movements of bacterium from one chemotactant to another may be specified as continuous and the average velocity is vEvm =2: To describe orientation action of chemotaxis we use the simplest angular dependence: Ean ðyÞ ¼ cch Wch cos y;
ð4Þ
where cch is a numerical constant of the order of unity which depends on the gradient of chemotactants concentration (it is equal to zero at zero gradient of chemical stimuli concentration) and y is the angle between bacterial magnetic moment and the chemotaxis gradient.
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Fig. 1. Geometry of experiments with chemotactants placed on the top (left) and bottom (right) of the cylinder.
We consider two cases corresponding to different distributions of chemotactants. Throughout our consideration the cylindrical volume is selected. The external field direction coincides with the symmetry axle of the cylinder. The first considered case corresponds to the following geometry of experiment: MTB culture fills the volume, chemotactants are concentrated on the top of the volume (or on the bottom that reflects in the sign of the energy of chemotaxis) and penetrate into the volume as the result of diffusion. It implies that the gradient of chemical substance density is directed along the main axis and has only z-component (see Fig. 1). Further we will refer to this set of conditions as to case (i). We define the second case, call it case (ii), for transverse geometry: chemical stimuli are concentrated on the axis of cylinder (or its surface that stipulates the sign of the chemotaxis energy) diffusing inward to the volume (Fig. 2). In this way, the gradient of chemotactants concentration is radial. Since the magnetosomes can align themselves along the magnetic field direction and the magnetic interaction between different cells is negligibly small, a magnetotactic culture can be considered as an ideal gas of paramagnetic arrows. Thus, the susceptibility of the system can be obtained in the way similar to that commonly used for paramagnetic one. This approach is valid until the mean distance between bacteria is small compared to the characteristic size of the cells. As a rule this condition is fulfilled. Common size of a bacterium is about 0.6 mm (for cocci [9]). Appropriate concentrations of MTB to be used in practice is approximately 1014–1015 cells/m3 [9]. It makes the ratio of average distance between bacteria to their mean size be at least three orders. Let us now calculate magnetization M and susceptibility w ¼ qM=qH of the culture. The magnetization of MTB is equal to M ¼ C hPmz i; where Pmz is the z-component of the mean magnetic moment and C is the difference between the concentrations of bacteria with positively (North seeking bacteria) an negatively (South seeking bacteria) oriented magnetic moments with regard to the position of flagella: C ¼ ðNþ N Þ=V: Here Nþ and N are respectively, the numbers of North and South seeking bacteria, i.e. bacteria with the z-component of vector Pm parallel and antiparallel to the z-axis, respectively; V is the volume of the culture.
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Fig. 2. Geometry of the problem for the case of radial distribution of chemotactants.
The average magnetic moment of the culture can be calculated from the following expression: R Pmz dn ; hPmz i ¼ R dn Taking into account the Boltzman distribution of MTB we rewrite Eq. (5) in the form: R 2p R p Pmz eW=kB T sin y dy dj hPmz i ¼ 0 R 2p0 R p ; W=kB T sin y dy dj 0 0 e
ð5Þ
ð6Þ
where W is the energy of bacterium, Pmz ¼ Pm cos y; y and j are corresponding angles in the spherical coordinate frame and kB is the Boltzman constant. The energy W is the sum of the magnetic energy Wm ; i.e. the energy of interaction of bacterial magnetic moment with the external magnetic field: !! Wm ¼ m0 Pm H ; ð7Þ where m0 is the permeability of free space, and the anisotropy energy caused by chemotaxis: Ean ¼ cch Wch cos y for the case (i) or for the case of radial distribution of chemotactants (ii): Ean ¼ cch Wch sin y: Substituting relevant expressions for the energy of a bacterium into Eq. (6) we obtain the expression for the average value of z-component of magnetic moment: R 2p R p cch Wch cos y m0 Pm H cos y cos y sin y dy dj 0 0 Pm exp kB T ð8Þ hPmz i ¼ R 2p R p cch Wch cos y m0 Pm H cos y sin y dy dj 0 0 exp kB T Integration of (8) for the given geometry gives us the Langevin function with shifted argument: " # m0 Pm H cch Wch m0 Pm H cch Wch 1 ; hPmz i ¼ Pm cth kB T kB T
ð9Þ
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Thus, from Eq. (9) we obtain for the susceptibility: " 2 # m m P H c W m P H c W m ch ch m ch ch 0 w ¼ CP2m 0 sh2 0 ; þ kB T kB T kB T
579
ð10Þ
For case (ii) an expression for the z-component of the mean magnetic moment can be inferred in a similar way to the one used above. Hence, it has a form: R 2p R p cch Wch sin y sin j m0 Pm H cos y P exp cos y sin y dy dj m 0 0 kB T : ð11Þ hPmz i ¼ R 2p R p cch Wch sin y sin j m0 Pm H cos y exp sin y dy dj 0 0 kB T Integral (11) can be calculated numerically. In Fig. 3 the dependence of the mean magnetic moment on external magnetic field strength is shown for the case (i). Different curves correspond to various energies of chemotaxis: Wch1 > Wch2 > Wch3 > Wch4 : Here and further on the energy of chemotaxis is expressed in units kB Troom and denoted as Z ¼ Wch =kB Troom : For calculations we used the next numerical values: p ¼ 2:7 1016 Am2 [9], T ¼ 293 K, C ¼ 6 1014 cells/m3. From the plot one can see that with the increase of chemotactic energy the initial value of the mean magnetic moment grows. This is quite reasonable since the growing gradient of chemotaxis leads to higher ordering of the culture in specific direction, so, cells and obviously magnetosomes align in this direction. In Fig. 4 the magnetic susceptibility of the culture is plotted as a function of H for different values of the energy of chemotaxis. The culture’s magnetic susceptibility decreases with increasing energy of chemotaxis (see Fig. 4). This decreasing can be explained as the following. It is clear that wp/Pmz S=H because wpM=H and Mp/Pmz S: Magnetic field strength H increases in a much faster rate than /Pmz S and so the magnetic susceptibility decreases with increasing applied magnetic field. Since the rate of the mean magnetic moment changes drops as Wch grows (see Fig. 3, here /Pmz S is close to the saturation for large values of the chemotaxis energy), the susceptibility
Fig. 3. Dependence of the mean magnetic moment on magnetic field strength for different values of the energy of chemotaxis Z:
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Fig. 4. Dependence of magnetic susceptibility on magnetic field strength for different values of chemotaxis energy Z:
Fig. 5. Magnetic susceptibility versus the energy of chemotaxis for the case of chemotactants concentration gradient parallel to the field direction for different values of field strength H:
decreases. The magnetic susceptibility reveals similar behaviour being plotted versus the chemotactic energy for different values of the magnetic field strength (see Fig. 5). If the chemical stimuli are concentrated on the bottom, i.e. the gradient of chemotactant concentration is antiparallel to the field direction, initial magnetization of the culture will be different from zero and negative. Fig. 6 represents the mean magnetic moment dependency on the magnetic field strength for different values of the chemotaxis energy. Different signs of magnetic field and gradient of chemotaxis lead to the presence of a peak on the plot of field dependence of susceptibility. As the energy of chemotaxis grows the position of the peak shifts to the range of higher field values as it is shown in Fig. 7.
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Fig. 6. Dependence of the mean magnetic moment on external field strength for different energies of chemotaxis Z: Magnetic field direction is antiparallel to the gradient of chemotactants concentration.
Fig. 7. Susceptibility as function of external magnetic field for different values of chemotaxis energy Z: Magnetic field is directed antiparallel to the gradient of chemotactans concentration.
In the case (ii) the initial value of system’s magnetization was found to be equal to zero due to the radial symmetry of chemotactants’ distribution. In Fig. 8 z-component of the mean magnetic moment was plotted versus external magnetic field strength for different energies of chemotaxis. Steeper curves correspond to lower values of the energy. Field dependence of the susceptibility for given geometry is shown in Fig. 9. A comparison between Figs. 3 and 8 shows a reduction in the mean magnetic moment of the culture for the geometry with the radial gradient of chemotactants concentration. This is to be expected because the bacteria become aligned radially perpendicular to the z-axis.
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Fig. 8. Mean magnetic moment versus the magnetic field strength for different energies of chemotaxis Z: Field direction is perpendicular to the gradient of chemotactants concentration.
Fig. 9. Field dependence of magnetic susceptibility for different values of chemotaxis energy Z: Magnetic field is perpendicular to the gradient of chemotactants concentration.
2. Conclusions and discussion The proposed model allowed us to calculate the key magnetic characteristics of a magnetotactic bacteria colony (the mean magnetic moment and magnetic susceptibility) as functions of the external magnetic field strength and the chemotaxis energy. It was shown that geometry of the problem (relative orientation of the chemotactants concentration gradient and the applied magnetic field) has an essential influence on both the magnetization curve shape and magnetic susceptibility. If the gradient of chemotactants concentration is parallel to the field direction, the field dependence of the mean magnetic moment is the Langevin function of H (Eq. (9)) with shifted argument (see Fig. 3). This result coincides with the one previously referred to in Ref. [8]. For the gradient of chemotactants concentration antiparallel to the field direction magnetization depends on H as it is shown in Fig. 6. In this case the magnetization at H ¼ 0 can be negative (call it
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polarization by chemotaxis) and the magnetic susceptibility of bacterial culture has a maximum when the energy of chemotaxis is equal to the energy of interaction between the bacteria magnetic moment and magnetic field (Fig. 7). The shift of the maximum position field ðHmax Þ with increasing chemotaxis energy is described by expression Hmax ¼ Wch =mm0 Pm On this basis one can propose an experimental method for the determination of the chemotaxis energy of MTB from the magnetic susceptibility measurements at the given geometry. Another substantial for experiment and practical applicability detail is that the mean magnetic moment of bacteria is several orders larger than the magnetic moment of a paramagnetic atom. Thus, it is possible to observe saturation in the magnetization curves of MTB in reachable range of magnetic field strength values, Hs E102 A/m (Figs. 3, 6 and 8). ~ gradients of chemotactants concentration we have found a For both the radial and that coincided with H decrease of the magnetic susceptibility with increasing energy of chemotaxis.
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