Simulating Brownian Motion on the Surface of a

Sphere

Oum S. Lahade1,2, Nicholas A. Vickers2, Sean B. Andersson2

1Research Triangle High School, 3106 East, NC-54, Durham, NC 277092Boston University, Boston, MA 02215

BACKGROUND

METHODOLOGY

Single Particle Tracking (SPT) is a class of tools used by scientists to study the motion of biological

macromolecules moving in their native environments, including within living cells. A typical SPT

experiment has three main steps. The first step is to acquire a time series of images. The second step is

to localize the particle in the images. Finally, the last step is to link the localized positions into

trajectories.

SPT has allowed scientists to understand many biological phenomena. For example, SPT was used

to understand how HIV-1 enters macrophages2 as seen in Figure 1B. STP is also used as a method to

investigate molecular dynamics, structure, and functions in a cellular context. Finally, SPT has allowed

scientists to build models of complex structures and develop and test hypotheses of how these

complexes perform their biological roles in health as well as in disease states.

Significant development in both optical microscope design and the algorithms used to estimate their

trajectories has allowed scientists measure biological processes with greater precision and accuracy.

This continual development has pushed SPT microscopes into three-dimensional tracking; however,

there is currently no way to test three-dimensional single particle tracking with known ground truths in

an experimental setting.

Brownian Motion is the random movement of microscopic particles in fluid resulting from their

collisions with surrounding particles as seen in Figure 2. Brownian motion is exhibited in many

biological systems including cell membranes. This process can approximately be described by equation

1. X is the position of the particle over time, D is the diffusion coefficient, and ∆𝑡 is the discrete time-

step.

𝑋[𝑘+1] = 𝑋𝑘 +𝑊, 𝑊~𝑁 0,2𝐷∆𝑡 (1)

OBJECTIVES

The goal of this project is to create an algorithm to generate Brownian motion constrained to the

surface of a sphere.

A)B)

Figure 1. A) Acquisition of single particle trajectories from image stacks. B) Single particle tracking of HIV-1 entering macrophages2.

Figure 2. Brownian Motion of a particle in two dimensions.

റ𝑎 ∙ 𝑏 = റ𝑎 |𝑏| ∗ cos 𝜃𝜃 = 90°

റ𝑎 ∙ 𝑏 = 0

റ𝑎 × 𝑏 = റ𝑐Where A, B, and C are all

orthogonal to one another.

റ𝑞 = 𝑣𝑒𝑐𝑡𝑜𝑟 𝑡𝑜 𝑛𝑒𝑤 𝑝𝑜𝑖𝑛𝑡

റ𝑝 = 𝑟 ∗ (റ𝑞

റ𝑞)

Figure 5. Brownian Motion on the surface of a sphere is synthesized through a projection method. A) Identifying the most recent position on the sphere. B)Creating a plane tangent to point on the sphere. C) Obtaining a line lying on the tangent plane that will act as one axis for movement in the plane. D) Usingthe cross product to obtain a line orthogonal to both others that will serve as the other axis for movement within the plane. With both axis it is possible tomove to any point on the tangent plane. E) Using the given equations independently for each axis to determine a new point on the tangent plane. F)Projecting this point onto the surface of the sphere.

A) B) C)

D)E) F)

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑁 0,2𝐷Δ𝑡𝑟𝜃 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝜃 =𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑟

sin 𝜃 =𝑡𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡

𝑟𝑡𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡 = 𝑟 ∗ sin 𝜃

𝑡𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡 = 𝑟 ∗ sin(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑟)

I would like to thank my mentors Sean Anderson and Nicholas Vickers for choosing this project for me and providing insightful advice. I would also like to

thank Boston University, particularly the RISE program, for providing the opportunity for me to do research.

EXPERIMENTAL SETUP

Figure 4. Optical microscope setup. Left: Thesynthetic trajectories are generated on thecomputer and sent to the controller. Thecontroller in turn sends this information tothe amplifier which sends current to thepiezo actuators in the stage (top right) whichmove the stage in three dimensions toreplicate the synthetic motion. Particlemovement on the stage is monitored by thecamera mounted to the microscope (bottomright). Both the images from the camera andpositions that the piezo actuators are in aresent back to the computer which analyzesthe difference between the known groundtruth and the particles realized trajectory.

RESULTS

Future directions are:

• Creating a general model applicable to many different cell types

• Extrapolating this process to encompass different shapes that common cells can take one, such as

ellipsoids

• Use this algorithm to test three-dimensional single particle tracking microscopes and algorithms with

known ground truths

SUMMARY AND FUTURE DIRECTIONS

ACKNOWLEDGMENTS

REFERENCES

• Single Particle Tracking is a powerful tool that has allowed scientists to understand many biological

phenomena.

• Brownian motion is a model for the random motion of particles within fluid. It results from the

bombardment of particles.

• Analyzing SPT techniques and algorithms involves comparing particle movement to known ground

truths.

• This project is the first step in taking this process to three dimensions as it involves creating known

trajectories on the surface of spheres, our model of a cell.

MATLAB was used to create the script that generates the synthetic Brownian motion trajectory.

Variables to be changed include the diffusion coefficient (D), the time series time-step (DT), the number

of points in each trajectory (N), the number of trajectories (num_trajectories), the initial point

(initial_point) , and the frequency.

[1] Vickers, Nicholas & B. Andersson, Sean. (2019). Brownian Motion using a Piezo Actuated Microscope Stage. Biophysical Journal. 116. 135a-136a. 10.1016/j.bpj.2018.11.751. [2] Li, Q.; Li, W.; Yin, W.; Guo, J.; Zhang, Z.P.; Zeng, D.; Zhang, X.; Wu, Y.; Zhang, X.E.; Cui, Z. Single-Particle Tracking of Human Immunodeficiency Virus Type 1 Productive Entry into Human Primary Macrophages. ACS Nano 2017, 11, 3890–3903.