Mechanical analysis of phase transition experiments of the bacterial flagellar filament

Transcription

1 Acta Mech Sin (2010) 26: DOI /s RESEARCH PAPER Mechanical analysis of phase transition experiments of the bacterial flagellar filament Xiao-Ling Wang Qing-Ping Sun Received: 20 May 2009 / Revised: 9 October 2009 / Accepted: 6 November 2009 / Published online: 8 July 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010 Abstract Bacterial flagellar filaments can undergo a polymorphic phase transition in both vitro and vivo environments. Each bacterial flagellar filament has 12 different helical forms which are macroscopically represented by different pitch lengths and helix radii. For external mechanical force induced filament phase transitions, there is so far only one experiment performed by Hotani in 1982, who showed a very beautiful cyclic phase transition phenomenon in his experiment on isolated flagellar filaments. In the present paper, we give a detailed mechanical analysis on Hotani s experiments. Through theoretical computations, we obtained a phase transition rule based on the phase transition mechanism. The theoretical analysis provides a foundation facilitating the establishment of phase transition theory for bacterial flagellar filaments. Keywords The polymorphic phase transition Bacterial flagellar filament Pitch lengths The cyclic phase transition 1 Introduction One of the characteristics of most living organisms is their ability to move, the motion of flagella on live bacteria was first The project was supported by the Hong Kong University of Science and Technology and the National Natural Science Foundation of China ( ). X.-L. Wang Department of Mechanical Engineering, University of Science and Technology Beijing, Beijing, China xiaoling@me.ustb.edu.cn Q.-P. Sun (B) Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Hong Kong, China meqpsun@ust.hk observed by Ehrenberg [1 in Flagella-based motility is a major mode of locomotion for bacteria. A bacterium has several flagella, the flagellum is mainly composed of three parts: a basal body consisting of a reversible motor imbedded in the cell wall, a short proximal hook which works as a flexible universal coupling joint and a long helical filament which is the propeller made up of only one kind of protein called flagellin [2,3. The filament is a hollow tube with an outer diameter of 20 nm and a length of µm macroscopically [4. When the bacterium swims, flagella form a rotating bundle as the motor in counter clockwise (CCW) rotation. When the motor changes into clockwise (CW) rotation, the bundle falls apart and the cell tumbles as Fig. 1 shows [5 7. At the same time when the motor rotation reverses a phase transition occurs in the flagellar filament [8. A flagellum has 12 different forms with different radii and pitch lengths [2, the phase transition can be induced by chemical change or by mechanical loading [9 15. The reason for the polymorphism of the bacterial flagellar filament is that its microstructure changes during the phase transition, a very slight difference of the protein subunit misfit, only 0.8Å, is responsible for the helix formation of flagellar filaments. The details of this part can be found in Refs. [2,3. The objective of this paper is to perform a mechanical analysis on Hotani s experiment which was conducted on a single bacteiral flagellar filament during the mechanical force-induced phase transition [16. 2 Mechanical analysis of the phase transition experiment of the bacterial flagellar filament Before the analysis, we shall first give a brief description of the experiments. The conventions used in the analysis are the same as those used in the paper of Macnab [17, as summarized in Fig. 2.

2 778 X.-L. Wang, Q.-P. Sun a resistance to reactions δn = V N C N δs,δt = V T C T δs, where C N and C T are the frictional constants in the appropriate directions; θ is the pitch angle of the helix. Now we define Fig. 1 Illustration of movements of E. coli, when its motor rotates CCW, flagella are in a bundle and the cell swims; when the motor reverses direction, flagella fall apart and the cell tumbles. The filament undergoes polymorphic transition when the cell tumbles In Fig. 2a, helical sense is defined as left-handed (LH) or right-handed (RH) by the usual convention that the phase vector rotates CCW or CW, respectively, as the locus moves away from an observer looking along the helical axis. In Fig. 2b, sense of flagellar rotation is defined as CCW or CW from the viewpoint of an observer looking along the helical axis of the flagellum into its point of attachment to the cell. In Fig. 2c, the torque applied by the motor to the flagellum combines with viscous resistance to rotation to give a twisting moment, or torsion, which is LH if the motor is rotating CCW and RH if the motor is rotating CW. 2.1 Experiment of bacterial filament in vivo The experiment was to investigate the natural flagellar filament attached to an energized cell body [17. During cell swimming, Macnab observed normal to curly form transition on both wide and mutant bacteria filament. Figure 3 shows the motion of a filament of radius η and wavelength (pitch) λ moving at a constant velocity through a viscous medium. Let V N and V T be the velocities normal and tangential to an element of the filament of length δs. The medium offers δf = (δn sin θ δt cos θ)δs, δc = (δn cos θ + δt sin θ)δsη. Each element of a filament moving in the x direction (Fig. 3) has two velocity components: u is in the axial direction and ηω is perpendicular to it. The helix is assumed to be rotating at an angular frequency of ω, thus tan θ = 2πη λ, V T = u cos θ + ηϖ sin θ, (2) V N = usin θ + ηϖ cos θ. The total propulsive force and the torque exerting on the helix axis are, respectively F = mλ sec θ(c N V N sin θ C T V T cos θ), C = mλ sec θ(v N C N cos θ + V T C T sin θ), where m is the wave number along the filament, δs =δx sec θ. Substituting sin θ = 2πη/ and cos θ = λ/ into Eq. (3), and also based on Ref. [18, C N = 2C T, C T = 2πμ/[ln(d/2λ) + 1/2, we finally get the generated hydrodynamic forces, i.e., the axial force and the torque on the moving filament as follows F = C = C T m [4 f π 2 η 2 λ (8π 2 η 2 + λ 2 )u 2πη2 C T m [ f (4π 2 η 2 + 2λ 2 ) λu, where f is the rotation frequency of the filament and u is the fluid flow speed. We can see that near the cell body the cell is subjected to a propulsive force to propel it to swim in the fluid. However the filament undergoes compression at the cross section, which is different from Hotani s experiment where the filament was subjected to a tension force when, (1) (3) (4) Fig. 2 Conventions used in this paper: a helicity, b rotation, c torsion

3 Mechanical analysis of phase transition experiments of the bacterial flagellar filament 779 Fig. 3 The mechanical forces acting on a filament when cell swims in viscous fluid viscous fluid flowed along the filament from the fixed end to the free end. Figure 4 shows the fundamental mechanism underlying the mechanically generated normal-to-curly polymorphic transitions. The normal left-handed helical forms, when subjected to a right-handed torsion during CW rotation, are converted to the curly right-handed form. The curly form, when subjected to a left-handed torsion during CCW rotation, is converted to the normal form. Here we note that both cell swimming and filament transition need filament rotation to generate a propulsive axial force and a torque to induce the filament transition. Twenty years later, Berg [4 conducted a similar experiment on the flagellar filament transition during cell swimming under better experiment conditions, giving more clear images, as schematically shown in Fig. 5. From these two experiments, we arrive at the following conclusions: (1) The right handed curly form is produced when the flagella are under right-handed torsion and is initiated near the basal body and propagates to the end of the filament. (2) Dynamic transition occurs in the rotating flagella only, and it is the consequence of flagella motility. The torque provided by the motor at the base is balanced by viscous forces from the medium, giving a distributed torque to the filament. 2.2 Experiment of isolated bacterial filament in viscous fluid Hotani [16 showed a cyclic phase transition phenomenon in his experiment on isolated bacterial flagellar filaments, as illustrated in Fig. 6. The normal and the semi-coiled forms are always nucleated at the stuck end, and then it grows until it pervades the whole filament. Each zone of the filament also has rotational movement as observed from the experiment. The left handed normal form rotates CCW, and the right handed semi-coiled form rotates CW. Cyclic transition can occur between the left handed normal form and right handed semi-coiled form, or normal and right handed curly form, depending on how the filament was attached to the glass. Simple mechanical analyses were performed on the experiment results, and the experimental data are summarized as shown in Fig. 7. Figure 7a changes of zone lengths in a filament undergoing cyclic transformation between the normal and the semi-coiled forms. The thin or thick portion of each line represents the end-to-end length of each zone (zone length) for normal or semi-coiled forms, respectively. The broken line represents the flow speed of the fluid. The rotational frequency (1/s) of each zone is given by the number beside the zone. The top end of each line corresponds to the stuck end of the filament. Mechanical forces (Torque and tension) are generated on a filament undergoing cyclic transformation between the normal and the semi-coiled forms (see Fig. 7b). The scales of torque and tension are given on the left-hand and right-hand ordinates, respectively. ( ) torque at the stuck end. Dash line is torque at the transition point where the transformation from normal to semi-coiled occurs. Dot-dash line is torque at the transition point where the transformation from semi-coiled to normal occurs, ( ) tension at the stuck end. Each datum point has a one-to-one correspondence to each line in Fig. 7a. It was concluded that nucleation of the new forms at the stuck end corresponded to the peak value of the torque; this is reasonable from the transition theory. However, we still need to answer the following questions regarding this experiment:

4 780 X.-L. Wang, Q.-P. Sun Fig. 4 Fundamental mechanisms underlying the mechanically generated normal-to-curly polymorphic transitions Fig. 5 E. coli cell with one flagellar filament undergoing polymorphic transformation among the normal, semi-coiled and curly forms (1) In the viscous flow, the filament is subjected to several kinds of forces, such as torque, bending, tension and shear forces, which force dominates the transition process of the filament, such as nucleation and the propagation of the new phases? (2) After the nucleation of the new phase, what kinds of criteria make the interface to propagate in this experiment? In the following analysis, we will answer the above two questions. 2.3 Mechanical analysis of the cyclic transition of filament in the experiment of Hotani Force components calculation Figure 8 shows the force components at the filament cross-section, where M 1 and M 2, the bending moments, are, respectively, along and perpendicular to the direction of the curvature of the filament, T is the torque, Q 1 and Q 2 are the shearing forces, respectively, and P is the tension in the filament. The shear forces are neglected in the analysis. As to viscosity, the frictional constants in the normal and the tangent direction are different. We analyzed force conditions for the left handed normal form and the right handed semi-coiled form separately. As shown in Fig. 9, an axial force δf and torque δc are exerting on the helical filament element, and the total force on the filament is the summation along the filament length. A filament of radius η and wavelength (pitch) λ moves at a constant velocity through a viscous medium, let Fig. 6 Illustration of the cyclic transformation between the normal and the semi-coiled forms. A flagellar filament is attached to a glass surface at its top portion and viscous fluid flow from top to bottom. Time interval is 5/9 s V N and V T be the velocity components normal and tangential to an element of the filament of length δs. The medium offers a resistance to reactions δn = V N C N δs,δt = V T C T δs, where C N and C T are the frictional constants in the appropriate directions; θ is the pitch angle of the helix, δf = (δn sin θ δt cos θ)δs, δc = (δn cos θ + δt sin θ)δsr, (5) δm = 1 2 δn(δs)2, where r is the radius of the helix, and δs = δx sec θ. Each element of a filament moving in the x direction has two velocity components: u in the axial direction and ηω perpendicular

5 Mechanical analysis of phase transition experiments of the bacterial flagellar filament 781 Fig. 7 The magnitude of the accumulated tension at the stuck end and the accumulated torque along the filament to it, and the helix is assumed to be rotating at an angular frequency of ω, thus for the right handed and left handed form, the velocity components are tan θ = 2πη λ, (6) v TL = u cos θ rϖ sin θ, v NL = u sin θ + rϖ cos θ, v TR = u cos θ + rϖ sin θ, v NR = u sin θ + rϖ cos θ. The total forces exerting in the direction of the helix are, respectively F = mλ sec θ(c N V N sin θ C T V T cos θ), C = mλ sec θ(v N C N cos θ + V T C T sin θ), (7) M = 1 2 V NC N m 2 λ 2 sec 2 θ, where m is pitch number along the filament, Substituting into Eq. (3), and based on Ref. [19 sin θ = 2πη 4π 2 η 2 + λ, cos θ = λ 2 4π 2 η 2 + λ, 2 C N = 2C T, C T = 2πμ ln(d/2λ) + 1/2, Fig. 8 All force components on the helical filament. Bending moment M 1, M 2 ; torque T ; shear force Q 1, Q 2 ;tensionp 1 we finally get the following generated hydrodynamic forces on the moving helical cross section for the right handed form C T m F = [ 4 f π 2 η 2 λ + (8π 2 η 2 + λ 2 )u, C = 2πη2 C T m [ f (4π 2 η 2 + 2λ 2 ) λu, (8) M = 2πηC T m 2 (u f λ). The forces for the left handed form are C T m F = [4 f π 2 η 2 λ + (8π 2 η 2 + λ 2 )u, C = 2πη2 C T m [ f (4π 2 η 2 + 2λ 2 ) + λu, (9) M = 2πηC T m 2 (u + f λ). There is a little difference between the two forms in different charity in the force component expressions on the signs of velocity and frequency Results of theoretical analysis From these analytical force expressions and by using the parameters given in the experiment [16, we get rotation frequencies for each filament part, as shown in Fig. 10. The torque T (= C cos θ), bending moments M 1 (= C sin θ), M 2

6 782 X.-L. Wang, Q.-P. Sun Fig. 9 Geometric parameters and force components of a filament s element for describing a left handed or a right handed helical filament in a viscous fluid Fig. 10 Rotation frequencies of each filament zone Fig. 12 Bending moment M 1 /(N m) Fig. 11 Torque/(N m) and tension F at the stuck end and at interfaces of the filament cross section are shown in Figs. 11, 12, 13 and 14, respectively. In Figs. 11, 12, 13 and 14, the black lines with black dots are torque and tension at the stuck end, respectively; the color lines with black dots are torque and tension at the forward transition (normal to semi-coiled) points; the black lines with color dots are torque and tension at the reverse transition (semi-coiled to normal) points. Time interval 5/9 s. We can see that at both the forward (normal-semi-coiled form) and the reverse (semi-coiled-normal form) transition interface, all torque T (Fig. 11) and bending moment M 1 (Fig. 12) are zero at the cross section of the filament. The maximum T and M 1 at the stuck end correspond to the nucleation of the right handed semi-coiled form, and the minimum values at the stuck end correspond to the nucleation of the left handed normal form nucleation.

7 Mechanical analysis of phase transition experiments of the bacterial flagellar filament 783 From Figs. 11, 12, 13 and 14, we can also see that the interface between two filament forms is different from the phase interface in normal metals, such as Shape Memory Alloys. During phase transition in a filament, a new phase needs rotation to propagate, with zero torque and zero bending moment at the phase interface. From this point, at the interface between two phases, only some fluctuations can make transition continue, the torque and bending moment is entirely used to cause rotation of the filament below the interface. In the experiments of Hotani, this special loading condition can make each zone of a filament have the proper rotation to let the transition occur. M 2 can make the filament straight and transform from the left handed to right handed. There is a tension force at the stuck end, where it reaches the maximum value as shown in Fig. 14. When the phase interface propagates along the filament from the stuck end to the free end, the axial force will decrease almost linearly and reach zero at the free end. From Fig. 14 we find some points where the axial force at the interfaces are greater than that at the stuck end; this results from metrical difference between the normal and the semi-coiled forms. The axial force produce only linear elastic deformation in the filament. In short, the nucleation of a new phase of filament needs some critical torque and bending moment at the stuck end, Fig. 13 Bending moment M 2 Fig. 14 Drag force the torque and the bending moment is accumulated to maximums at this end. During the new phase propagation, the filament below the phase interface needs to rotate; also the rotation of the filament is a geometrically necessary condition for converting the handedness of a helix; the rotation frequency depends on the flow velocity and increases with it. The frequency is constant when the torque at helix interfaces is zero Discussions This section includes two parts: (1) In the experiment on the attached filament under viscous fluid, the cross section of the filament is subjected to a tension force which is at the stuck end. This is different from the analysis of Hotani shown in Fig. 7b. The difference may come from the following factors: Firstly we consider the inertia force: the unit mass of filament per 1 µm is about kg, we see the interface between the non-rotating semi-coiled part and the normal part. Its rotation rate at the interface changes from 0 to 1.5 at one time interval, so we estimated the torque generated from the velocity change to be about N m. Compared to the viscous torque at the stuck end of about Nm, the inertia torque is no doubt negligible. Secondly we considered the torque arising from rigid deformation of the filament per unit length: the shear modulus of the filament is about Pa and the helix radius is about 0.2 µm, we estimated the induced torque to be about N m. It can also be neglected. From the above analysis we found that this filament transition is dominated by viscosity only, and all other effects on the force components of the filament can be neglected as they are relatively small. As we can see from Eqs. (8) and (9), the two key parameters determining the torque and tension are the viscous velocity and the filament rotation frequency. The passive rotation movement is induced by the viscous drag generated from the viscosity, and thus the friction torque caused by the rotation movement should be balanced by the torque generated from the viscous fluid on the filament rotating at a constant rotation angular velocity. After checking each rotational zone of the filament undergoing transition process, we found that nearly all the friction torque induced by the passive rotation of the filament is larger than the viscous torque which is generated by the viscous fluid on the filament and makes the filament rotate, and this is impossible. From the above analysis we suspect that there might be some problems in measuring either the viscous velocity or the filament rotation frequency. Here we use a new method to estimate the torque at the interface during transition. Firstly we found the maximum angular velocity ω 0 of the zones below the transition point through a balance between the viscous fluid induced torque T

8 784 X.-L. Wang, Q.-P. Sun and the friction torque induced rotation. The rotation velocity can reach this value when the torque at the interface is zero. We defined the torque at the interface as T i during the transition, the part below the transition point rotates at an angular velocity ω, and the friction torque due to ω is T f. We assume power balance between the active torque and passive torque, i.e. T ω 0 = T f ω + T i v, and we have T i = T T f from torque balance and v = ωλ/2π, Delta = 22π/λ from geometric relation. It was found that ω = ω 0 = [2πλ/(4π 2 r 2 + λ 2 )u, here we can see that the torque at the interface is zero, the interface transition velocity is v = ωλ/2π =[λ 2 /(4π 2 r 2 + λ 2 )u. The torque depends on the fluid velocity and geometry of the filament only. (2) The torque of the helix cross section reaches its peak value when new forms are nucleated at the stuck end and decreases to zero at the interface when the interface propagates, there is an acceleration process for the new form propagation, we try to estimate the acceleration time. First we calculate the acceleration time of the normal form. When it nucleates at the stuck end, the torque reaches a negative peak value at the 2, 8, 15 time intervals in Fig. 7, then its propagation results in the semi-coiled part of the filament below the transition point to rotate right handedly. This is because the torque on the semi-coiled filament is right handed, the angular velocity of this part accelerates from zero to the balanced value which is determined by the balance between the active torque generated by the viscous fluid flow and the frictional torque due to rotation movement. For simplicity, we consider a unit length of filament in viscous fluid flow of velocity u = 7 µm/s. The torque acting on the semi-coiled filament is T = 2πη 2 C T su, here η is the radius of the helical filament and λ is the pitch length, C T = 2πμ/[ln(d/2λ) + 1/2 is the viscous friction coefficient parallel to the filament, μ is the viscosity constant kg/ms, these parameters were chosen from Ref. [18, the radius η and pitch λ of the semi-coiled filament are m and m, respectively. The torque T at the cross section is about N m. When the filament rotates at a constant angular frequency 1.6, the tangent velocity is v = 2π f η, and the mass of the filament per unit µm is about kg. T/η = m(v/t) 2, finally the acceleration time is about s which is too short to be observed from the experiment. This means when the normal form is nucleated at the stuck end, the torque at the transition interface drops quickly to zero as the interface propagates. During propagation, the torque at the interface remains zero. The cases for the semi-coiled form nucleation and propagation are similar to the above mentioned. When the semicoiled form nucleates at the stuck end at the time intervals 4, 12, 18 shown in Fig. 7, the torque reaches a positive maximum at the stuck end. Then its propagation makes the normal form below this transition point rotate left handedly, because the torque on the normal filament is negative. Here the radius and pitch of the normal left handed forms are m and m, respectively, and when it rotates at a constant frequency 1.3 below the transition point, the accelerating time is about s. Again this time is too short to be observed in the experiment. When the semi-coiled form nucleates at the stuck end, the torque reaches a positive maximum, when the interface propagates, the torque drops to zero in about s. In Hotani s experiment, for the normal semi-coiled transition the time interval of observation is about 5/9 s, and for the normal curly transition it is about 1/36 s. These time intervals are too long to observe the torque dropping process. 3 Summary and conclusion Based on the above detailed analysis of the forces acting on the filament during the phase transition in the experiment of Hotani [16, we can draw the following conclusions: (1) The motion of bacteria is propelled by the propulsive force F which is generated by the bacterial flagellar filament rotation and the viscosity of the fluid. Flagellar filament parameters such as wavelength, radius and rotation frequency of the filament can influence motion velocity. (2) Phase transitions of the bacterial flagellar filament in both vivo and vitro environments can be induced by mechanical forces. It is motivated by the bacterial flagellar motor in the former case, and it is motivated by the flow of viscous fluid in the latter case. (3) During the phase transition of the isolated bacterial flagellar filament, the torque T and bending moment M 1 of the filament in each transition cycle reach their peak values at the stuck end to nucleate the new phase. Therefore the formation of a polymorphic region at the fixed end is a result of the torque and bending moment. (4) During the growth of the new phase, T and M 1 at the interfaces are zero, which means the Maxwell forces are zero and the two coexisting phases have the same free energy. Other force components contribute to the elastic deformation of the filament only. References 1. Ehrenberg, C.G.: Die Infusionsthierchen als Vollkommene Organismen Leopold Voss. Leipzig, Germany (1838) 2. Wang, X.L., Sun, Q.P. : Phase transition in biological systems. Adv. Mech. 40(1), (in Chinese) (2010) 3. Wang, X.L.: Mechanical analysis and free energy construction of phase transition in bacterial flagellar filaments. Ph.D. Thesis, The Hong Kong University of Science and Technology (2006)

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