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Modelling red blood cell optical trapping by machine learning
improved geometrical optics calculations
R. Tognato,1,3, D. Bronte Ciriza,2,3, O. M. Marag`o,2, and P. H. Jones1,*,
1 Department of Physics and Astronomy, University College London, Gower
Street, London, WC1E 6BT, UK
2 CNR-IPCF, Istituto per i Processi Chimico-Fisici, Messina, I-98158, Italy
3 The authors contributed equally to this work
* philip.jones@ucl.ac.uk
Abstract
Optically trapping red blood cells allows to explore their biophysical properties, which
are affected in many diseases. However, because of their nonspherical shape, the
numerical calculation of the optical forces is slow, limiting the range of situations that
can be explored. Here we train a neural network that improves both the accuracy and
the speed of the calculation and we employ it to simulate the motion of a red blood cell
under different beam configurations. We found that by fixing two beams and controlling
the position of a third, it is possible to control the tilting of the cell. We anticipate this
work to be a promising approach to study the trapping of complex shaped and
inhomogeneous biological materials, where the possible photodamage imposes
restrictions in the beam power.
Introduction
In 1970 Arthur Ashkin first demonstrated how to manipulate and confine microscopic
particles suspended in water through radiation pressure [3]. Following the first
demonstration of optical trapping, Ashkin and collaborators developed the single-beam
gradient trap, today known as optical tweezers (OT) [5,40]. The basic principles of OT
utilise the fact that light carries momentum which can be harvested to manipulate
microscopic particles in solution. In its conventional and simplest set-up, OT focus a
collimated Gaussian laser beam to a diffraction-limited spot where it can trap
microparticles. Soon after the first demonstration of OT, Ashkin et al. employed them
to manipulate biological particles like bacteria and erythrocytes without causing
damage [4, 6].
In humans, erythrocytes, or red blood cells (RBCs), are anucleated cells responsible
for the oxygen delivery to tissues and organs. Mature and healthy RBCs have a
biconcave disk shape that minimizes the membrane bending energy. Typically, RBCs
have diameter of 6-8
µ
m, a peripheral thickest portion of 2-3
µ
m, and a central dimple
0.8-2 µm thick [18]. The excess surface area and membrane elasticity render the cell
elastic and permits the RBC to pass through the microvasculature by deforming [30].
Alterations in the RBCs’ membrane elasticity are implicated in severe disfunctions of
the microcirculation (e.g., capillaries can be entirely clogged, triggering tissue necrosis
or organ damage and failure) [1]. The RBCs’ alteration can be genetically inherited [16],
a consequence of a pathogen infection [41], a metabolic disorder [2], or due to radiation
treatment [25]. Very recently, it has also been correlated to SARS-Cov2 infection [31].
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In the last decades, OT have been widely applied in RBC research to investigate
biochemical and biophysical properties of both healthy and unhealthy RBC via single-
or multi-beam OT [7]. In these studies, researchers have adopted two main approaches
to trapping: the indirect trapping, where handles as silica or polystyrene microspheres
are used to manipulate the RBC [36], and the direct trapping, where the light beam
directly traps the RBC [6]. Regardless of the mechanism used for trapping, the nature
of biological samples makes them particularly susceptible to photodamage. To minimise
this, infrared light in the second biological window (wavelength around 1064 nm) is
generally preferred for the experiments [8].
As the cell is significantly larger than the incident wavelength, the geometrical optics
approximation (GO) models properly the beam cell interaction [21, 33, 38]. GO assumes
that the beam can be discretized in a series of rays that carry a fraction of the total
momentum and by calculating and summing up the scattering of all rays, it is possible
to compute the total force applied. However, even though GO simplifies considerably
the theoretical treatment compared to a full wave optical approach [27], an accurate
calculation requires consideration of a large number of rays, with an associated high
cost in computational time. Simulating the Brownian dynamics requires repetition of
the force calculation at each time step sequentially, which becomes prohibitively slow if
the force calculation is not optimized [10]. The fact that the calculation is sequential
and that the shape is complex prevents the use of conventional approaches to speed up
the calculation (e.g., parallelization and interpolation based approaches).
Machine learning, and in particular neural networks (NN), are emerging across a
variety of research fields as a powerful technique to solve challenging problems. Backed
by their ability to learn from previous examples in order to make new predictions, NN
are contributing to biology [29], food sensing control [17], and even to containment of
epidemics [35]. In fact, NN have recently been demonstrated as an useful technique to
increase both the speed [32] and the accuracy [9] of optical forces calculations when
compared to GO, allowing the study of more complex systems through Brownian
dynamics simulations. While these previous works consider spheres [32] and
ellipsoids [9], there is no evident reason to remain constrained to these relatively simple
shapes. Indeed, the computation time saving that could be achieved by using NN for
force calculation of particles with more complex shapes makes this a particularly
attractive application.
In this work, we train a NN to enhance the speed and accuracy of the optical force
calculation for RBC. This permits a numerical exploration of the Brownian dynamics of
a RBC, potentially allowing to study in a more complete manner different trapping
configurations. More efficient trapping configurations employ less laser power and
therefore reduce the risk of photo damaging the trapped cells.
1 Methods
1.1 Model and geometrical optics calculations
In our model we consider a Gaussian beam propagating along the opposite direction of
the force of gravity (+
z
direction). The wavelength (1
.
064
µ
m) is selected to match the
vast majority of experiments and in agreement with previous works on trapping
RBCs [2, 42]. The OT parameters are the ones of a typical OT experiment (beam power
5 mW, numerical aperture 1.3).
The RBC is assumed to be in its healthy biconcave disk conformation, and the
parameters describing the shape of the RBC are those reported by Evans et al. [18]. A
radius (
r
) of 3
.
91
µ
m, a central dimple with a thickness (
tmin
) of 0
.
81
µ
m, and a thickest
portion, located at 2.76 µm from the axis of symmetry, with a thickness (tmax) of
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2
.
52
µ
m. According to the Evans-Fung model, the thickness (
Z
) of a section of the RBC
reads:
Z(ρ) = r1−ρ
r2
·C0+C2ρ
r2
+C4ρ
r4(1)
where
ρ
is the radial distance from the axis of symmetry,
r
is the cell radius and
C0
,
C2
and C4(0.81, 7.83, -4.39, respectively) are numerical values related to the observable
parameters that describe the cell morphology [39].
As RBCs are significantly larger than the wavelength of the incident light, the
optical forces acting on them can be calculated with GO. We perform this calculation
with the specialized software OTGO [11]. For biological samples, such as RBCs, that
have a low refractive index contrast with the typical suspending medium, the fraction of
power that is reflected after a scattering event is very low (
<
0
.
001) [22], therefore in our
ray tracing calculations, only the first two scattering (refraction) events are considered.
1.2 Diffusion tensor
The erratic motion of a particle trapped in liquid in an OT set up is influenced by the
fluid’s resistance, by the thermal noise, and by the external deterministic forces exerted
by the OT [19,27]. For non-spherical objects, a single scalar diffusion coefficient is not
enough to describe the statistics of the random motion. It is necessary to use a 6 x 6
diffusion tensor (D), which depends on the particle shape and orientation [27]:
D=Dtt Dtr
Drt Drr(2)
where
Dtt
,
Drr
and
Drt
=
Dtr
T
are 3 x 3 blocks and the subscripts ‘r’ and ‘t’ refer to
the particle’s rotational and translational degrees of freedom, respectively.
Although an analytical expression for (D) exists for simple shapes like spheres,
ellipsoids or cylinders [24], the RBC morphology is more complex and requires numerical
methods for its determination. Here, we used the bead model technique developed by
De La Torre et al., exploiting the widely used software winHYDRO++ [15, 20]. In the
bead model, a series of spheres are used to approximate the size and the total volume of
the RBC. From the bead model, winHYDRO++ calculates the 6x6 tensor (
Ξ
) encoding
the hydrodynamic resistance of the non-spherical particle. We then obtain the diffusion
tensor Dvia the generalised Einstein relationship [14]:
D=kBTΞ−1(3)
where kBis the Boltzmann constant, and Tis the temperature of the system.
In the present study, the bead model is constructed in a strict sense, filling the
volume of the RBC considering only spheres of equal sizes. In this case, the centre of
diffusion of the particle coincides with the centre of mass of the particle, and the
numerical output for the diffusion tensor reads:
Dtt =
7.43 ×10−14 −4.83 ×10−20 6.23 ×10−21
5.93 ×10−21 7.43 ×10−14 5.99 ×10−21
−8.74 ×10−20 −5.42 ×10−19 6.28 ×10−14
(4)
Drt =Dtr
T=
−6.18 ×10−15 −2.52 ×10−15 −1.75 ×10−15
−2.52 ×10−15 8.85 ×10−16 −2.72 ×10−15
−1.75 ×10−15 −2.72 ×10−15 −2.20 ×10−16
(5)
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Drr =
4.04 ×10−33.63 ×10−11 1.06 ×10−10
1.04 ×10−94.04 ×10−3−7.85 ×10−10
1.02 ×10−10 3.17 ×10−10 3.36 ×10−3
(6)
where the units are
m2
s
,
rad·m
s
, and
rad2
s
respectively. Notice that the diagonal terms of
Dtt
and
Drr
that indicate the diffusion coefficient along a specific direction (i.e., x,y,z)
and a specific axis (i.e., x,y,z) are several orders of magnitude larger than the off
diagonal terms highlighting the shape-induced directional dynamics typical of
non-spherical particles [23].
1.3 Particle dynamics simulation
The simulation of the dynamics of the RBC is based on the works of M. X. Fernandes et
al. [19], and described in reference [27]. Two reference frames are defined: a particle
reference frame Σ
p
, which has an origin that coincides with the particle’s centre of mass
(CM) and the centre of diffusion (CD), and a laboratory reference frame Σlthat is
centred at (0,0,0) and which axes are oriented along
ˆx
,
ˆy
and
ˆz
, Fig. 1-a. At time
t
, the
RBC’s CD is located at rCD(t)=[xCD(t), yCD (t), zCD(t)]. The cell orientation can be
described by the angles α1(t), β1(t) and γ1(t) defined with respect to the particle unit
vector ˆxp(t) = [ˆxp,x(t),ˆxp,y(t),ˆxp,z(t)], ˆyp(t) = [ˆyp,x(t),ˆyp,y(t),ˆyp,z(t)],
ˆzp(t) = [ˆzp,x(t),ˆzp,y(t),ˆzp,z(t)]. Dis obtained in the particle reference frame, that is
centred at rCD(t) and the axes are oriented along ˆxp,ˆypand ˆzp
To simulate the free diffusion of an arbitrarily shaped particle from time tto the
time step
t
+ ∆
t
, initially one has to calculate the increment of the particle position and
orientation in Σp(t):
∆xp
∆yp
∆zp
∆αp
∆βp
∆γp
=√2∆t
wx
wy
wz
wα
wβ
wγ
(7)
where [
wx, wy, wz, wα, wβ, wγ
]
T
are white noise terms, random numbers obtained from a
multivariate normal distribution with zero mean and covariance D. Successively, the
increments of the particle position calculated in Σ
p
has to be transformed to Σ
l
. This is
given by the transformation matrix:
MΣp→Σl(t) =
ˆxp,xˆyp,xˆzp,x
ˆxp,yˆyp,yˆzp,y
ˆxp,zˆyp,zˆzp,z
(8)
Therefore, the finite difference equation to update the particle position in Σlis:
xCM(t+ ∆t)
yCM(t+ ∆t)
zCM(t+ ∆t)
=
xCM(t)
yCM(t)
zCM(t)
+MΣp→Σl(t)
∆xp
∆yp
∆zp
(9)
Once the new particle position is calculated, one has to update the particle
orientation from Σp(t) to Σl(t), which is effectively a rotation of the particle unit
vectors. This rotation, for small angles, is expressed in Σpby the rotation matrix:
Rp(∆αp,∆βp,∆γp) = Rp,x(∆αp)Rp,y(∆βp)Rp,z(∆γp) (10)
where
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Rp,x(∆αp) =
1 0 0
0 cos(∆αp)−sin(∆αp)
0 sin(∆αp) cos(∆αp)
(11)
Rp,y(∆βp) =
cos(∆βp) 0 sin(∆βp)
0 1 0
−sin(∆βp) 0 cos(∆βp)
(12)
Rp,z(∆γp) =
cos(∆γp)−sin(∆γp) 0
sin(∆γp) cos(∆γp) 0
0 0 1
(13)
Transforming this rotation matrix to Σ
l
, we obtain the unit vectors representing the
orientation of the particle at the end of the time step:
[ˆxp(t+ ∆t),ˆyp(t+ ∆t),ˆzp(t+ ∆t)] = [ˆxp(t),ˆyp(t),ˆzp(t)]Rp(∆αp,∆βp,∆γp) (14)
As the last step, the rotation matrix has to be updated:
MΣp→Σl(t+ ∆t) = MΣp→Σl(t)Rp(∆αp,∆βp,∆γp) (15)
However, in the current situation, we must also account for the optical forces (F)
and torques (T) exerted by the optical trap on the centre of mass of the RBC.
Therefore, taking into account Fand T, the increments of the particle orientation and
position in Σpare:
∆xp
∆yp
∆zp
∆αp
∆βp
∆γp
=D
kBT∆t
Fx,p
Fy,p
Fz,p
Tx,p
Fy,p
Tz,p
+√2∆t
wx
wy
wz
wα
wβ
wγ
(16)
which then need to be transformed back into Σl. However, Fand Tare calculated in
Σl, and therefore they must be transformed to Σpvia the matrix MT
Σp→Σl.
The finite difference scheme is combined with the NN or with the GO code to
calculate the optical forces and torques acting on the RBC and to simulate the
Brownian dynamics of the optically trapped particle. We estimate the time step, ∆
t
, for
the Brownian motion simulation from the trap stiffness reported by Tognato et al. [38],
and from the diffusion properties of a healthy RBC obtained here. The typical time
scale on which the restoring force acts is given by τOT =γ
k, while the momentum
relaxation time is given by τm=m
γ. To assure numerical stability, ∆tmust fall in
between these two characteristic time scales (
τOT ≫
∆
t≫τm
) [27]. From the diffusion
tensor D, one can extract the diffusion properties of the RBC along a specific direction
(Di), then through the fluctuation-dissipation theorem one can obtain γi. For example,
for the x-direction one obtains γx=kBT
Dx∼5×10−8kg
s. Therefore, considering a trap
stiffness in this direction
kx∼
1
.
6
pN
µm
, one obtains
τOT ∼
3
×
10
−2
s. On the other hand,
given a mass of ∼1×10−11kg for a typical healthy RBC, one obtains τm∼4×10−4s.
A similar estimation can be made for the other directions, and contemplating the
magnitude of the other terms in D, a time step ∆t= 0.001s is adequate to assure the
numerical stability [27].
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1.4 Neural network architecture and training
The neural network (NN) architecture is composed of one input layer with 6 neurons
representing position and orientation of the cell (x,y,z, cos(θ), sin(θ), ϕ), one output
layer with 6 neurons representing the force and torque components (
Fx
,
Fy
,
Fz
,
Tx
,
Ty
,
Tz
),
and 7 hidden layers in between with 256 neurons each, Fig. 1-b. While Σ
p
encodes the
orientation of the particle by using 3 angles, because of symmetry, the orientation of the
RBC can be completely defined by the polar angle ϕand the azimuthal angle θ, as
shown in Fig. 1-a.
The training data consists of 4
×
10
6
different points in the 5D space of parameters
(
x
,
y
,
z
,
θ
,
ϕ
). 90% of these data points are used as a training data set while the remaining
10% is kept as a testing data set to evaluate the accuracy of the NN. The training data
are generated via GO calculations made in OTGO [11]. The cell is placed in uniformly
distributed positions in a cube of side 8µm centred at the origin of the Cartesian
coordinates system (i.e. −4µm≤x≤4µm, −4µm≤y≤4µm and −4µm≤z≤4µm).
Simultaneously, to account for the possible different orientation of the RBC within the
trap, the cell is uniformly and randomly oriented in an interval for −π≤θ≤πand
0≤ϕ≤π
2. The training data are generated for the simplest case of a single-beam OT
with the geometrical focus centered at (0,0,0) and a beam power of 5mW.
The NN is trained in Python using Keras (version 2.2.4-tf) [13]. The training of the
NN is divided into 5 different steps. The data pre-processing and the model definition,
which are done only once, and the loading of the data, the training step, and the
evaluation of the performance, that are carried out iteratively. The training data,
generated as previously described, contains data in different units and scales. While the
position scale is in the order of ∼1×10−6m, the forces are on the range of
∼1×10−12N, and the torques are around ∼1×10−18 N·m. To achieve an efficient
training of the NN, we need to apply a pre-processing step where the variables must be
rescaled around unity and θ, that ranges from −πto π, is expressed in terms of sines
and cosines to avoid inconsistencies around 2π. Shuffling the data and dividing them
into a validating and training set is the final step of the pre-processing. In our case, the
training data set contains 5.4×106points while 6 ×105points are reserved for the
testing data set. In this work, we employ fully connected NNs where each neuron is
activated by a sigmoidal function. Defining the model implies choosing the number of
layers and the number of neurons per layer. Among the explored architectures, the one
consisting of 7 hidden layers provides the best results (in terms of accuracy, training
time, and speed).
The iterative part of the training starts by loading the training data and applying
the training step where the NN weights are optimised to minimise the loss function. We
use the mean squared error as the loss function and the Keras implementation of the
Adam optimiser [13]. Once the training dataset is fully explored, the difference between
the NN calculation and the validating dataset (defined as the mean square difference) is
computed. The iterative step is repeated until this difference reaches its minimum value
and we consider that the model is fully trained. The training of the NN is done in a
GPU type NVIDIA GeForce RTX 2060 with 16 GB of memory. The processor of the
computer is an Intel Core i7-10700, and it has 16 GB of RAM.
2 Results
2.1 Single beam Optical Tweezers
To evaluate the effectiveness of our approach, we start by testing the ability of the NN
to predict the forces and torques acting on an RBC in a single beam OT (SBOT). We
compare the NN predictions (trained with data generated using 4 ×102rays) and the
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GO calculations considering 4 times more rays (1.6×103rays) at 1 ×105random
positions and orientations. The 2D density plots shown in Fig. 1-c and -d illustrate the
agreement between the NN and GO in predicting the optical forces (regression
coefficient 0.998, R2= 0.996) and torques (regression coefficient 0.999, R2= 0.996),
respectively. We further demonstrate the accuracy of the NN by comparing our NN
(trained with data generated with 4
×
10
2
rays) with the GO calculation (considering a
greater number of rays). Fig. 1-f shows the normalised root mean squared error
(NRMSE) between the predictions of the NN trained with 4 ×102rays and the GO
calculations with different numbers of rays (up to 5
×
10
3
rays). The NRMSE decreases
as the number of rays increases. The forces and torques calculated with 5 ×103rays
result more similar to the NN output than to the forces obtained with a total of 4
×
10
2
rays, meaning that the NN is able to increase the accuracy of the force and torque
prediction, even for an object with such a complex shape.
Figure 1. a) Definition of particle (yellow) and laboratory (white) reference frames and
rotation angles (
α
,
β
,
γ
) of the cell around the laboratory reference frame; b) schematic
depiction of the neural network. The input layer contains six neurons describing the cell
position and orientation, and the output layer has six neurons describing the components
of force and torque acting on the cell. In between are seven hidden layers (i= 7), each
of them with 256 neurons (
j
= 256). c-d) Density plots comparing the magnitude of the
total force (
FNN
tot
) and torque (
TNN
tot
) predicted with NN with those calculated with the
GO method (
FGO
tot
) and (
TGO
tot
). Regression lines are shown in red. e) Log-Log plot of
the normalised root mean squared error (NRMSE) between
FNN
tot
and
FGO
tot
, and
TNN
tot
and
TGO
tot
as a function of the number of rays used in the GO calculation. For each data
point, the NN employed remains the same (trained with 4 ×102rays).
2.2 Double beam Optical Tweezers
Since the NN is trained for a SBOT, one may think it can only predict the optical forces
and torques for a SBOT. However, the NN can be used multiple times to simulate
multi-beam optical tweezers. In fact, the NN can predict the forces generated by a
single beam on different locations on the cell, and the total force acting on the centre of
mass of the cell may then be calculated as the vector sum of each contribution. Here we
consider a double-beam optical tweezers (DBOT) where the two beams’ geometric foci
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are positioned 5.06µm apart along the x-axis, similar to the experiments conducted by
Agrawal et al. [2], Fig. 2-a.
Figure 2. a) Schematic depiction of an RBC trapped by a double-beam OT. b-c)
Comparison between the GO calculation and the NN prediction for the b) torque-
rotation curve for rotation around the x-axis and c) force-displacement curve along
the x-direction. (d) Comparison of the probability distribution obtained with the GO
calculation and with the NN prediction for a RBC in a DBOT. (e) Cell orientations in
the numerical simulation for both GO and NN.
Fig. 2-(b-c) shows
Tx
(
α
) and
Fx
(
x
) calculated with GO and predicted with the NN
for a cell in its folded configuration (i.e., cell major axis parallel to the optical axis)
trapped in a DBOT. In both cases, the NN predictions (solid line) agree well with the
GO method (dots), demonstrating the possibility to use the NN for multi-beam optical
traps. We therefore conclude that this approach can be extended to predict forces and
torques generated by a three- and four-beam OT, situations in which the GO
calculation is considerably slower given the very large number of light rays required.
We now investigate the cell’s dynamics within a DBOT using both NN and GO to
compute the optical forces. The simulation of the Brownian dynamics follows the
strategy explained in the Methods section (Particle dynamics simulation) where now,
the force and torque considered is the sum of the contributions of each of the beams.
Fig. 2-(d) shows the probability distribution of the centre of mass of the cell for a total
simulation time of 5s, while Fig. 2-(e) shows the orientation of the cell with respect to
the fixed reference frame as a function of the simulation time. It is important
mentioning that in the current configuration a rotation around the y-axis (
β
) would be
a rotation around the cell axis of symmetry and therefore completely irrelevant. By
extracting the average values for each degree of freedom, it is possible to compare the
final equilibrium configuration obtained with the NN and with GO. Indeed, the average
values obtained with the predictions of the NN and GO methods agree well and are also
in good agreement with previously reported values [38], Table 1.
Moreover, the biggest advantage of using the NN for numerical simulations is a
consistent decrease in the simulation time required to achieve the same precision (the
NN is two orders of magnitude faster). Since the NN shows a higher computational
efficiency, hereafter, we make use of the NN prediction to simulate the Brownian
dynamics of an optically trapped RBC.
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GO NN
x2,eq(µm) 0.01 ±0.05 0.01 ±0.05
y2,eq(µm) 0.00 ±0.05 0.00 ±0.04
z2,eq(µm) −0.20 ±0.08 −0.18 ±0.08
ϕ2,eq(◦) 90.65 ±2.11 90.27 ±1.44
θ2,eq(◦)−90.45 ±1.06 −90.09 ±0.92
Table 1. Equilibrium position and orientation for a RBC in a double-beam OT as
found with geometrical optics (GO) and with neural networks (NN). For each parameter
we report the average and the standard deviation.
Figure 3. a) Translational autocorrelation function. The solid lines are exponential fits.
Cxx (t)
,
Czz (t)
, decay as single exponential while
Cyy (t)
as double exponential. b)
y−α
correlation shown as density plot. c) Normalised cross-correlation function between the
rotation around the x-axis (
α
) and the y-displacement (red line exponential fit). Both
Fy(α)
d) and
Tx(y)
e) reveal unstable equilibrium when the cell is tilted of 90°around
the x-axis (i.e. RBC in its folded position)
We therefore move to extract quantitative information on the trap constants.
Initially we analyse the hydrodynamics of the RBC, since non-spherical particle could
have an intrinsic roto-translation coupling due to their peculiar shape [23]. In our case,
the diffusion tensor Ddoes not show any strong roto-translation coupling; therefore, we
do not expect to find any strong correlation in the cell’s motion intrinsically due to the
RBC’s hydrodynamics. Still, optically trapped non-spherical particles could show
roto-translation coupling in their motion as previously observed by others. In this
framework, the normalised auto-correlation function (ACF) has been successfully used
to extract quantitative information about the trapping constants [28,34].
We first evaluate the spatial ACFs (Cxx (τ), Cyy (τ), Czz (τ)) of the particle centre
of mass trajectories. Cxx (τ) and Czz (τ) decay as a single exponential with
characteristic decay frequencies
ωx
= 28
s−1
and
ωz
= 6
.
4 s
−1
. Contrariwise,
Cyy (τ)
is
well fitted with a double exponential with characteristic frequencies ωy,1= 42 s−1and
ωy,2= 2.7 s−1, Fig. 3-a. We associate the fast decay rate to the translation, while the
slower decay can be related to rotation around the
x
-axis (
α
) induced by a motion along
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the y-direction. The normalised cross-correlation function between αand yFig. 3-b
further confirm a roto-translation coupling (amplitude -0.368) [26]. Fig. 3-c shows a
density plot of the rotation around the x-axis (α) as function of the motion along the
y
-direction. Here it can be seen a moderate negative correlation which suggests that the
RBC rotates as it moves away from yeq,2, and undergoes to an “oscillating” motion
about the equilibrium configuration where it is stably confined. To better comprehend
this correlation we simulate Fy(α) (Fig. 3-d) and τx(y) (Fig. 3-e) which undoubtedly
shows the coupling between the motion along
y
and
α
. Actually, the cell in its “folded”
position (i.e. α= 90◦) is constantly subjected to a force along the y-direction that
moves the particle away from yeq,2which in turns induces a rotation around the
x-direction. On the other hand, as extensively described by Tognato et al., the
transverse forces and torques components confine the cell in its “folded”
configuration [38]. The overall consequence of these stable and unstable equilibria is a
“circulating motion” of the cell within the optical trap about the equilibrium
configuration. This would suggest that the coupling is intrinsically due to particle shape
and to the optical trap rather than to the hydrodynamic of the particle.
Lastly, we extract average values and the standard deviations for the force constants
(k2,x =ωxkbT
Dxx = 0.166 ±0.024 pN
µm·mW ,k2,y =ωykbT
Dyy = 0.218 ±0.025 pN
µm·mW ,
k2,z =ωzkbT
Dzz = 0.005 ±0.001 pN
µm·mW ). These values are in excellent agreement with a
previously reported work [38]. Similarly to the translational motion, we calculate
Cαα (τ) and Cγγ (τ), Fig. S1. Cαα (τ) and Cγγ (τ) decay as a single exponential and
the respective trap constant are: kα=ωαkbT
Dαα = 0.352 ±0.096 pN·µm
rad·mW and
kγ=ωγkbT
Dγγ = 1.587 ±0.382 pN·µm
rad·mW . We do not analyse the dynamics around βsince
the cell is not confined about this axis.
2.3 Triple beam Optical Tweezers
As previously suggested, one of the greatest advantage of using a NN instead of GO is
the significant lowered computation time, especially when a very high number of light
rays is needed (e.g. a triple- or four beams optical tweezer). Now, we exploit this
feature to investigate the equilibrium orientation and position of a RBC with a
reconfigurable triple-beam OT.
If directly trapped, a healthy biconcave RBC can assume two different and
alternative orientations within the optical trap depending on the number of beams used
for trapping [7, 38]. In a double-beam OT, the major axis of a RBC is parallel to the
optical axis and the beam foci are contained in the cell, known as “folded”
configuration [2]. On the contrary, if three or four beams arranged in symmetric
configurations are used (i.e. beams foci on the vertex of equilateral triangle or a square),
the major axis of the cell is confined to be orthogonal to the optical axis (i.e. α= 0◦),
configuration referred to as ‘flat’ configuration [37]. Here, we sought for alternative (and
intermediate) RBC equilibrium configurations in respect to the well-known “folded” and
“flat” ones.
We consider a trap configuration that is intermediate to those traps able to trap the
cell in its “folded” or “flat” configuration. We consider a triple-beam optical tweezers
(TBOT) composed by three identical and tightly focused Gaussian laser beams. Two
beams are always arranged along the
x
-axis in a diametrically opposite location on the
thickest portion of the cell (white crosses in Fig. 4-a). A third beam (yellow cross in Fig.
4-a) can be translated over the thickest portion of the cell and is used to counteract
Tx
generated by the two fixed beams. For simplicity, henceforth, the position of the moving
beam is described by a polar co-ordinates system in the x−yplane. Its location is
defined by a single angle (
ξ
), and the distance from the origin is fixed and equal to the
radius of the thickest portion of the cell (2.76µm), Fig. 4-a.
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Next, we proceed with the identification of the positional and translational equilibria.
As a first step in our investigation, we simulate a force-field acting on the cell for
ξ= 45◦to appreciate the effect of the potential landscape on the RBC. In this
simulation, the cell is in its “flat” configuration and located at z= 0. It can be seen
that the light pattern creates a very complex force-field (Fig. 4-b). Non-negligible
optical forces act simultaneously along the
x−
and
y
-direction for every location of the
cell. The complexity of the force-field makes extremely difficult the identification of the
equilibrium positions (i.e., point in space where a specific force component vanishes
with negative slope). This process would requires several reiterations for every degree of
freedom rendering the process labour intensive. However, noting that if a particle is
subjected to an optical potential it falls into the equilibrium position/orientation, it
would be possible to identify the equilibrium configuration studying its dynamics as
suggested by Cao et al [12]. From symmetry arguments, the effect of different locations
of beam 1 can be understood restricting ξin the interval [0◦,90◦] as schematically
depicted in Fig. 4-a. Moreover, since we are looking for alternative equilibrium
configurations (or to a transition from a “flat-like” to “folded-like” configuration), it is
also rationale to disregard every position where two beams are too close to each other
(i.e.
ξ <
15
◦
), which should induce a “folded” configuration. Thus, the position of beam
1 can be restricted to 15
◦≤ξ≤
90
◦
. To evaluate the effect of the reconfigurable optical
trap
ξ
is sampled every 15
◦
, and for each
ξ
, the Brownian dynamic is simulated for a 10
s trajectory starting from a RBC positioned in its ‘flat’ configuration (θ= 0◦and
ϕ= 0◦) centred at (0,0,0). The simulation finishes once the cell equilibrates around a
stable position and orientation. The final position and orientation are then given as the
average position and orientation with the standard deviation of the last second of the
simulation. Fig. 4-c shows the 3D trajectories of the RBC’s CM obtained from the
simulations carried out for different
ξ
. Here, while
x
and
y
equilibrium positions remain
close to the origin for different angles, the equilibrium in zdoes depend on ξ. In
particular, for ξ < 30◦,zeq <0µm and for ξ > 45◦,zeq >0µm, Fig. 4-c. We anticipate
that for
ξ≤
30
◦
, the cell is in its ‘folded’ configuration, Fig 4-d and Fig. 4-f. This is due
to a combination of the light intensity distribution and the cell configuration within the
trap. In fact, when the cell is in its “folded” position, the cell’s major axes are parallel
to the direction of propagation of the light beam. In this condition, more highly
converging “light rays” strike the biggest faces of the RBC. This increases significantly
the gradient force (Fg). Simultaneously, while in folded position, the scattering force
(Fs) decreases appreciably because of the smaller geometrical cross-section of the cell.
However, if ξincreases, this effect is less pronounced since the light rays strike the cell
less symmetrically, and for ξ= 30◦,zeq ∼ −0.2µm. Conversely, for ξ≥45◦a net shift
in the axial position is evidenced (
zeq ∼
0
.
8
µ
m), and this is due to a sequential shifting
from the “folded-like” configuration to a “flat-like” configuration, Fig. 4-c (ii) and Fig.
4-d (i). Much more interesting is the analysis of the rotational equilibrium. In Fig. 4-d
are shown the polar orientation (ϕ) of the cell as a function of the simulation time for
different locations of the moving beam (i.e. various ξ). It is evident that ξstrongly
influences the final polar orientation of the cell, Fig. 4-d. In particular, as beam 1
approaches beam 2, the cell tilts more until it reaches the “folded” configuration (i.e.
ϕ= 90◦) for ξ= 30◦. Analysing the final orientation of the cell in more detail, it is
possible to discriminate between three different regions. When the two beams are close
to each other, ξ≤30◦, the cell is in the “folded” configuration. If 30◦≤ξ≤75◦, the
RBC’s tilting seems to vary linearly with ξ, from a “folded-like” configuration to
“flat-like” configuration. The last region is for ξ≥75◦, where the cell tilting cannot be
decreased further, Fig. 4-d. It is also interesting to note the minor effect that
ξ
has on
θ. In particular, for ξ= 45◦is possible to obtain the highest cell’s tilting around the
z-axis, Fig. 4-e. For every other ξ, the tilting of the RBC around the z-direction
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decreases towards θ= 90◦.
Figure 4. a) Schematic depiction of the triple-beam optical trap and the polar co-
ordinates system used to identify the position of the moving beam. (b) Force-field for an
RBC on the x-y plane for
ξ
= 45
◦
. The colour code indicates the total force acting on the
x-y plane, while the grey arrows indicate the direction of the force. (c) Three-dimensional
trajectories of the cell centre of mass over a simulation time of 10 s for different
ξ
, and
the average values for the last second of simulation. d) Polar (
ϕ
) and e) azimuthal (
θ
)
orientation of the RBC as a function of the simulation time. Average orientations are
measured over the last second of the simulation. The error bar represents the standard
deviation. f,g) Final equilibrium configuration for a RBC in the reconfigurable triple
beam optical trap for
ξ
= 15
◦
and 90
◦
respectively. The blue dot indicates the center of
mass of the RBC while the red stars indicates the beams’ foci.
3 Conclusions
The biomechanical properties of RBC are affected in different diseases and OT have
been successfully used to trap RBC and to extract information about the membrane
properties. While studying the optical forces applied on a trapped RBC can be done,
the computation is extremely time consuming. In this work we have demonstrated that
by using NN, one can significantly increase the speed of calculation without
compromising the accuracy. This enhancement of the force calculations has allowed us
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to explore systems that were almost impossible to tackle with the conventional method
for optical force calculation. In particular, we have focused on the analysis of the
dynamics of trapped RBC with multiple beams. This can potentially allow
determination of the best trapping configuration and to minimize the incident laser
power and therefore reduce the risk of photodamaging the trapped cells.
4 Funding
D.B.C., O.M.M. and P.H.J. acknowledge financial support from the European
Commission through the MSCA ITN (ETN) Project “ActiveMatter”, Project Number
812780. D. B. C and O. M. M acknowledge funding from the European Union
(NextGeneration EU), through the MUR-PNRR project SAMOTHRACE
(ECS00000022) and PNRR MUR project PE0000023-NQSTI.
5 Acknowledgments
The authors would like to thank Prof. Giovanni Volpe and Dr. Agnese Callegari
(University of Gothenburg) for fruitful discussions on machine learning enhanced
geometrical optics calculations.
6 Disclosures
The authors declare no conflicts of interest.
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