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Time-sharing optical tweezers is a versatile technique to realize multiple traps for manipulating biological cells and macromolecules. It has been based on an intuitive hypothesis that the trapped viscoelastic object does not “sense” blinking of the optical beam. We present a quantitative analysis using mechanical modeling and numerical simulation, showing that the local stress and strain are jumping all the time and at all locations with the jumping amplitude independent of the recovery time of the viscoelastic material and the jumping frequency. Effects of the stress and strain jumping on the object deformation and the internal energy dissipation are analyzed.
© 2014 Optical Society of America
1. Introduction
Optical tweezers is a useful tool for studying micromechanical properties of biological cells and macromolecules in the micro-rheology, biomedicine, cell mechanics and single-molecule biophysics [1–5] with the advantage that the force exerted to the trapped particle by the laser beam is in the piconewton range and in the same order of magnitude as that of the interaction forces in the interior of the biological cells and between the cells. A variety of approaches have been proposed for manipulating the trapped particle with the trapping beam moving, oscillating or jumping in space or blinking in time. In many cases two or more trapping beams are employed to stretch a cell or to fold and unfold a biomolecule. As the trapping beam can be positioned with high accuracy and can scan multiple spots at high frequencies by an acousto-optic deflector, the multiple traps can be realized in the optical tweezers in the time-sharing regime [1,6]. The laser beam can scan multiple micro-beads attached to the biological object for stretching or buckling with the accurately controlled beam positions and gradient forces. The time-sharing optical tweezers proposed by Block et al is simple, efficient and versatile and has been widely used in the experiments, where a single laser beam scans a set of positions in space at a high frequency (~100 Hz or higher) [1,6]. The basic assumption of the time-sharing tweezers is that the trapped biological viscoelastic particle may be insensitive to “blinking” of the laser beam. They “sense” essentially the time-average of the optical intensity distribution on the multiple focal spots, e.g. a steady multiple-trap [1,6,7]. These intuitive concepts are not sufficient however for modeling the optical tweezers [8–10]. To our best knowledge there does not exist a rigorous analysis on the mechanics in the time-sharing optical tweezers. This analysis becomes necessary as the dynamic viscoelasticity of bio-material object in optical tweezers is explored in many recent experiments [11,12].
In this paper we use the viscoelastic mechanics and the continuum medium mechanics to analyze the responses of the biomaterial object in the time-sharing optical tweezers. We provide the quantitative description on the stress, strain and deformation in a one dimensional (1D) model, and the simulation for 3D objects with the finite element method (FEM). We are limited in this paper to discuss the case that the trapping beam jumps among multiple spots, but does not sweep nor traverse continuously the trapped object. Such a jumping can be implemented by an acousto-optic deflector with the square wave control signal, as reported recently in a dual-trap optical jumping tweezers for stretching a biological cell. Our analysis is valid for both the cases where the biological object is attached to polystyrene micro-beads, and is directly shined by the jumping trapping beam without attaching beads [7–10].
2. Temporal response of a viscoelastic 1D rod
Multiple traps can be realized by scanning the trapping beam of the optical tweezers on multiple spots in the time-sharing regime for manipulating the viscoelastic biological object. We analyze in this paper the dual-trap with the trapping beam jumping between two spots. The result of this analysis can be extended readily to the multiple traps.
We first consider a 1D micro-rod, which could be a cytoskeletal filament or a macro-molecule. The rod lay along the x-axis, and is tugged by a constant stress σ0 in the + x-direction at its end with the stress σ(L) = σ0. The other end of the rod is free of load, σ(0) = 0. When the rod material is homogeneous and linear elastic, the local stress satisfying the boundary condition is linearly distributed in the rod as σa(x) = σ0x/L, as shown in Fig. 1. An elementary section of the rod dx receives stretching stresses σ(x) and σ(x + dx) on its two sides respectively in opposite directions, as shown in Fig. 1, and is deformed with the local strain , where E is the Young’s modulus. The total elongation of the rod is a sum of the local strains over the rod length computed as , which is the half of the elongation by the same stress σ0 but applied to both ends of the rod in opposite directions, or by the same stress σ0 applied at one end of the rod with the other end fixed. In addition to the elongation, the object is moved as a rigid body under the total stress σ0 according to the Newton's law of motion.
Fig. 1 Linear distribution of stress in a 1D viscoelastic rod at time t = 0.5 ms after a constant stress σ0 = 1 Pa in + x-direction is applied at the right end of the rod, computed by finite element method.
Consider now the time sharing tweezers with the radiation stress jumping from one end to the other end of the rod. In the first half of the jumping period, the stress σ0 is applied at x = L in + x-direction, resulting in a linear distribution of local stress σa = σ0x/L. In the second half of the jumping period the stress σ0 is jumped to x = 0 and is in the –x-direction. The local stress distribution is inversed to σb = σ0(L-x)/L. Thus, at any point x in the rod the local stresses are always stretching, and are jumping over the time between σa and σb, except at the midpoint where σa = σb. The jumping amplitude Δσ = |σa-σb| increases linearly with the distance to the midpoint, and is a maximum of σ0 at the two ends. In the rod segment of the location x > L/2 we have σa > σb, and in the rod segment of the location x < L/2 we have σa < σb. Therefore, the loading state in the jumping tweezers is quite different from that in the equivalent steady dual traps, where the local stress is constant in time and uniform over the rod length. This difference may be meaningful depending on the viscoelastic properties of the trapped object.
The cumulated local strain in the viscoelastic rod is also jumping with the local stress jumping over the time. We use the standard linear solid (SLS) model [13] for the viscoelastic material, as shown in Fig. 2, where E1 and E2 are the Young’s modules of spring 1 and 2, η3 is the viscosity of dashpot 3, the two arms experience the same strain ε. The total stress is the sum of that in each arm, σ = σ1 + σ2. In the Maxwell arm, spring 1 and dashpot 3 receive the same stress σ1 = σ3. Their strains are added ε = ε1 + ε3. By considering the stress jumping from σb to σa as a sum of a jump up from 0 to σa and a jump down from σb to 0, we express the local strain at a point x as the sum of the corresponding recovery and creep functions as (see Appendix)
withwhere ε0 is the cumulative strain at time t0 and at the point x, τσ is the recovery time defined asIn the first half period of the jumping, the stress at the point x jumps up from σb to σa and is then kept constant. At time t = t0 + (T/2) the stress jumps down from σa to σb and is then kept constant for another half period T/2. In the second half period of the jumping, the strain ε(t, x) may be still computed by Eq. (1) only with σa and σb alternated and a new cumulative strain ε0. In the beginning of each half of the stress jumping period, the strain jumps instantly by Δε = Δσ / (E1 + E2) as dashpot 3 does not have time to react. After the jump, the strain increases or decreases exponentially depending on the coefficient K in Eq. (1). Note that the strain jumping amplitude Δε is independent of the viscosity η3 and the jumping frequency f = 1/T.We performed the finite element method (FEM) simulation with Comsol multiphysicsTM software for an optical jumping tweezers with a constant external stress of σ0 = 1 Pa jumping between the two ends of a rod at frequency f = 1 KHz. The rod was 8 μm long with rectangular section of 4 × 2 μm2. The SLS viscoelastic material was with the spring elastic module E1 = 156 Pa, Poisson’s ratio ν = 0.49 and dashpot viscosity η3 = 30 mPa·s in the Maxwell arm and the spring elasticity E2 = 15.6 Pa in the parallel arm [14,15]. The recovery time was τσ = 2.1 ms, which is much shorter than τσ = 100–300 ms of the Red Blood Cell (RBC) estimated in the micropipette aspiration experiments [16]. This choice of parameters was for showing the explicit variation of the local strain with the time. The computation of the stress and strain along the rod was performed at a time step of 1/100 ms with the FEM. The results were in good agreement with the analytical expression in Eq. (1), except the presence of high frequency (> f) oscillating noise due to the 3D nature of the rod in the FEM model, so that the data shown in this paper were interpolated by averaging per five adjacent points. From the analysis and the numerical simulations we made the following observations:
1. The local stress and strain follow exactly Eq. (1). With the local stress jumping as a square function, the local strain first follows the stress jumping, and then increases or decreases exponentially. From Eq. (1) the local strain ε tends to σa/E2 with t → ∞ in the first half period of jumping, and to σb/E2 in the second half period of jumping. The strain ε increases when K < 0 and decreases when K > 0. As E1 >> E2 in this model, in all the first half jumping periods K ≈ε0 - (σa/E2) < 0, the strain increases. In all the second half jumping periods K ≈ε0-(σb/E2) the strain increases, as the cumulated strain ε0 is small and K < 0 in first period of jumping. When ε0 becomes large over time, K becomes positive, the strain then decreases with time, as can be seen in Fig. 3(a).
Fig. 3 Local stress and strain in a 1D rod with an external load jumping at 1 KHz. (a) Recovery time τσ = 2.1 ms at x = 6 μm where σa = 0.75 Pa and σb = 0.25 Pa; (b) τσ = 1.1 s at point x = 8 μm where σa = 1.0 Pa and σb = 0 Pa.
2. The jumping of the local stress and strain is omnipresent. Even when the rod elongation is “saturated”, the local strain still jumps with the same jumping amplitude, as can be seen in Fig. 3(a). Increasing the recovery time reduces the after-jump exponential variation of the local strain, but the jumping amplitudes of the strain Δε = Δσ / (E1 + E2) remain unchanged. In Fig. 3(b) the recovery time was set as τσ = 1.1 s. Thus, the mean value of the strain increases slower with the time, compared with that shown in Fig. 3(a), but the strain jumping amplitude is still large.
3. Due to the special 1D geometry, the rod is elongated as a smooth exponential function of time without jumping, as shown in Fig. 4 computed with the FEM. As the linear stress distribution along the rod is symmetrical with respect to the midpoint x = L/2, when the stressat a point x jumps from σb to σa, the stress at the point L-x jumps from σa to σb. Therefore, the strain jumps at these two points are canceled by the summation ε(x) + ε(L-x) according to Eq. (1). Moreover, both ε(x) and ε(L-x) can be computed by Eq. (1) with alternated σa and σb, so that as σa + σb = σ0, when t → ∞ the sum ε(x) + ε(L-x) = (σ0/E2)[1 - exp(-t/τσ)] increases exponentially and tends to σ0/E2. As a sum of the strains at all the pairs of symmetrical points from x = 0 to x = L/2 the rod elongation tends to σ0L/2E2. This is similar to the elongation of a rod tugged at one end by a constant stress σ0 and to that obtained in the steady dual-trap with a half of the external stress σ0/2 applied to the two ends of the rod.
Fig. 4 Elongation of the entire rod with recovery time τσ = 2.1 ms when the stress load was jumping at 1 KHz and stopped at t = 15 ms.
4. There is an energy dissipation associated to the presence of the local stress and strain jumping all the time in the trapped biological particle. Figure 5 shows a local stress-strain hysteresis loop at one end point of the rod when the rod elongation is “saturated”. At point A the local stress is σb and the local strain is minimal. Then, the stress jumps up to σa and the strain is jumped to point B. In the following half jumping period σa is constant, the strain exponentially increases to reach point C. At the end of the first half jumping period, the stress jumps down from σa to σb, the strain is jumped to point D and then recovered back exponentially to point A. The creep response follows the trajectory [ABC] and the recovery response is drawn by trajectory [CDA]. The stored energy is zero in each cycle as the local strain returns to its starting point A. The energy dissipated in one jumping cycle of the material internal friction corresponds to the area within the parallelogram ABCD. The lengths [BC] and [DA] depend on the viscosity of the material from Eq. (1). In Fig. 5 the recovery time was set to 2.1 ms. For the material of a higher viscosity with a longer recovery time, the after-jump variation of the strain in each half jumping period T/2 is smaller, the lengths [BC] and [DA] are shorter and the energy dissipation is smaller.
Fig. 5 Hysteresis loop of stress-strain at the rod end x = 8 μm in a time range from 12 to 15 ms when the rod elongation is saturated. Recovery time τσ = 2.1 ms. Stress jumping frequency 1 KHz
In addition to the elongation, the trapped object is accelerated, moved and vibrated as a rigid body following the jumping stress. The vibration is observable in the experiment [7,8] at a low jumping frequency from 1 Hz to a few tens of Hz. At jumping frequency of 100 Hz or higher, the object vibration is not observable by naked eyes, but still persists with a reduced amplitude because of the shortening of the jumping period.
3. Spatial response of 3D object
In most experiments the trapped and manipulated object is 3D. When only one parameter, such as cell’s elongation under one external load, is measured in the experiment, the 1D spring-dashpot models may be appropriate for describing the viscoelasticity of the material, regardless of the 3D shape and size of the object [7,11]. In this sense, the analysis and the basic concepts for a 1D rod introduced in Section 2 are still valid for a 3D object. A complete analysis should consider the 3D nature of the object and define the materials in the object 3D structure with viscoelastic material models. Usually, the FEM should be used to compute deformation of the trapped object due to the 3D geometry complexity. The 3D deformation of a biconcave shape RBC in the steady dual-trap optical tweezers has been analyzed using the FEM [2,17].
For simulating the time sharing optical tweezers we set the viscoelastic material parameters and the jumping external load to a biconcave RBC trapped in a dual-trap jumping optical tweezers. The cell was filled with the viscoelastic material of the SLS model with the spring elastic module E1 = 0.9 Pa, Poisson’s ratio ν = 0.49 and dashpot viscosity η3 = 30 mPa·s in the Maxwell arm and the spring elasticity E2 = 0.45 Pa in the parallel arm [14,15]. The recovery time was τσ = 0.1 s. The biconcave shape was defined by the classical expression as that in [17], with the diameter of the RBC platelet disc of 7.8 μm. The biconcave RBC is trapped with its platelet parallel to the x-z plane defined by the jumping trapping beam [17]. A uniform stress of 1 Pa was applied to the cell at all the points on the surface with x ≥ 3.6 μm in the directions normal to the surface, and then jumped to all the points on the surface with x ≤ −3.6 μm at the jumping frequency of 1 KHz.
We used the continuum mechanics to compute the behavior of the 3D trapped viscoelastic cell and performed a full FEM simulation in one jumping cycle of the external load. As shown in Section 2 the local stress and local strain are jumping all the time and at all the locations following the jumping of the external load in the time-sharing tweezers in the 1D rod. The 3D object will have the similar temporal response as that of 1D object. The presentations for the stress and strain tensors with 3 × 3 components respectively in the 3D cell by 2D figures were a challenge. As the system, including the RBC and the laser beam, is symmetric with respect to the x-z plane, Fig. 6(a) (Media 1) shows a time sequence of the distributions of the first principal stress σ1 in a cross section of a biconcave RBC in the x-z plane, as a function of time at a time step of 1/100 ms in one jumping cycle of 1 ms of the external load. The color encodes the values of the first principal stress. At each point in the 3D space there exist mutually perpendicular directions, along which the shear stresses are zero. The normal stresses along the three principal stress directions are the principal stresses which were computed as the eigenvalues of the stress tensor arranged in the order as σ1 ≥ σ2 ≥ σ3. We choose to present the first principal stress σ1 rather than the pressure, because the latter is mostly related to the volume change of the cell material and is not of interest for the RBC [17]. In Fig. 6(a) the arrows show the acceleration vectors, which correspond to the total forces acting on the volume elements of the cell at different locations. Figure 6(b) (Media 2) shows a time sequence of the distributions of the first principal strain ε1 in a cross section of a biconcave RBC in the x-z plane in one jumping cycle of 1 ms of the external load. The principal strains ε1 ≥ ε2 ≥ ε3 were computed as the eigenvalues of the strain tensor at each point. The computation of the strain was under the small deformation condition. From Fig. 6 we see that both the instantaneous local stress and strain were ‘propagated’ from the loading side to the internal of the cell in each half of the jumping period of 0.5 ms. The local stress and strain were strong mostly in a horizontal zone defined by the two extremities of the surface which received the normal external load. The ‘propagation’ of the stress and strain was slow down and ‘diffused’ in the central circular region where the RBC platelet disc is concaved toward the x-z plane. The stress propagation can be ‘reflected’ from the cell interface where the mechanical load was free.
Fig. 6 One frame of excerpt at t = 0.4 ms in the videos sequence of one cycle of the external load jumping at frequency: 1 KHz. Viscoelastic recovery time τσ = 0.1 s. (a) Color: the first principal stress; Red arrows: amplitudes and directions the acceleration vectors (Media 1); (b) Color: normalized first principal strains (Media 2)
Obviously, in the 3D biconcave shape RBC the local stress tensors and the local strain tensors cannot be canceled by any summation at any geometrically symmetrical points as that in the 1D rod. As a result, the 3D displacement field in the cell would jump and oscillate along different directions over time at the jumping frequency of the external load. In addition, the cell would vibrate as a rigid body with the external jumping stress as observed in Ref [7]. This is different from that in the case of the steady dual trap. The cell’s elongation measured in Refs [7,8]. may be in fact a time average value.
The RBC’s deformation tended slowly to a saturated state. Instead of the temporal variation of the displacement variables, we show in Fig. 7 the deformation of the RBC trapped by a steady dual-trap. Two trapping beams were applied to the cell directly with the separation distance of 7.3 μm of the two beam focuses. The light scattered by the cell was computed by the RF module in Comsol multiphysicsTM. The radiation stress was computed from the total electromagnetic field via the Maxwell stress tensor. The trapping beams was of NA = 1.25 propagating in –z direction. Beam power was 40 mW and wavelength was 1.06 μm. Refractive indices of the cell n1 = 1.378 and of the buffer solution n2 = 1.335. In the simulation the RBC was of layered structure with the membrane modeled as the hyper-elastic Mooney-Revlin material and the cytosol modeled as linear elastic material [17]. As in the static state the acceleration vectors did not exist we projected the three principal stresses at every point into the x-z plane and computed the sum of their component vectors, which was represented by the red arrows in Fig. 7. Note that the stresses are concentrated in the membrane shell and are mostly tangent to the membrane. In the cytosol the stresses were weak as shown by the colors and arrows in Fig. 7.
Fig. 7 Normalized principal stress distribution and deformation in the cross section of the RBC in the steady dual-trap. Red arrows: principal stresses, Color and shape: deformation.
4. Conclusion
We have performed the analytical and numerical analysis showing that in the time-sharing optical tweezers with a jumping laser beam the local stresses and local strains in the trapped object are jumping all the time and at all locations. In the viscoelastic standard linear solid model the jumping amplitude of the strain is independent of the material viscosity. A longer recovery time of the material can reduce the variation of the strains in the period between the stress jumps and reduce the energy dissipation. In the dual-trap jumping optical tweezers trapping a 1D rod, the rod is deformed smoothly and exponentially with time. The final elongation of the object is the half of that in the static dual-trap tweezers. The 3D model with the FEM is described with the temporal variations of the stress and strain tensor distributions and the final deformation of a biconcave RBC. When only the 1D deformation of a 3D object is measured in the experiment, the object can be approximately modeled by the 1D spring-dashpot model. In this case, the basic concepts in the 1D model can be applied.
Appendix
From the relations σ1 = E1ε1, σ2 = E2ε2, and σ3 = η3 we obtain a differential equation as:
where the recovery time for constant stress τσ and the relaxation time for constant strain τε areWhen the stress is jumped up from 0 to σa at time t = t0, the local strain jumps immediately from 0 to σa/(E1 + E2) as springs 1 and 2 are deformed immediately, while dashpot 3 does not move. Then, under the constant stress σa the strain increases exponentially by the deflection velocity of dashpot 3 as
This is the creep function.When the stress is jumped down from σb to 0 at t = t0, the strain is jumped down from ε0 to ε0 - σb/(E1 + E2) with the instant reactions of springs 1 and 2. Then, the strain is decreased exponentially with the relaxation of dashpot 3. This is the recovery function as
Acknowledgments
The research is supported by Discovery grant from Natural Science and Engineering council of Canada.
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