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ideal viscous material, there are two forces acting on the bead. The first is the small, stochastic driving force generated the thermal energy kT. The second force is a coun- teracting frictional force that provides resistant to each step the bead takes and is dependent on the radius of the bead a and the viscosity of the liquid η.

In each instant, the thermal energy drives the bead to step in a random direction with a small amplitude. The next step, occuring in the next instant, has an uncoorelated direction and has a different, random step size. Einstein found that for a freely diffusing bead, the mean squared displacement (MSD) hr2_{(τ)i is related to the the diffusion}

coefficient D by (Einstein, 1905)

∆r2(τ)= 2nDτ (2.36)

where n defines the dimensionality of the observed process. In this work, we use video microscopy to observe diffusion in two dimensions. Thus, h∆r2(τ)i = 4Dτ. The diffusion coefficient is given by the Einstein relation (Einstein, 1905)

D =M kT (2.37)

where M is the hydrodynamic mobility of the particle. For a spherical particle in a Newtonian fluid, the mobility is given by Msphere = (6πaη)−1, and results in the

well-known Stokes-Einstein relation (Rubinstein and Colby, 2003)

D= kT

6πaη (2.38)

The Stokes-Einstein relation is powerful to passive microrheology in that it connects the calculated MSD to the viscosity of the liquidη through Eq. 2.36

η= 2kT

3πah∆r2_(τ)iτ (2.39)

Single Particle Diffusion in Elastic Solids. In an elastic material, the bead is driven by collisions with the background material, which is generated by thermal energy

kT. However, an ideal elastic material stores all of this energy and instantaneously exerts a restoring force with equal amplitude in the opposite direction. Thus, the MSD of a particle embedded in such a material is constant and independent of time:

∆r2(τ)_solid =K (2.40)

In Section2.5.2, we wrote the viscoelastic stress response to an oscillatory strainγ(t) =

γ0sin(ωt) as σ(t) = γ0(G0sinωt+G00cosωt). Evaluating at the limiting cases for an ideal liquid and solid, we have (Wirtz, 2009)

Ideal Liquid→ G0 = 0, G00 =ωη → η= σ(t) γ0cosωt 1 ω ↔ η = 2kT 3πah∆r2_(τ)iτ (2.41) Ideal Solid→ G0 =G, G00 = 0 → G= σ(t) γ0sinωt ↔ G= 2kT 3πah∆r2_(τ)i (2.42)

where in Eq. 2.42we have set h∆r2_(τ)i

solid =K =2kT/3πaG. Hence, we have arrived at an expression for ideal solids, analogous to Eq. 2.39 for ideal liquids, that can be used to determine the elastic shear modulus by measuring the MSD of embedded beads.

Thus, by measuring the MSD for beads of size a that are stuck to the coverslip (e.g. a noise floor sample), these expressions for G and η can be used to estimate the maximum shear modulus and viscosity that our instrumentation is sensitive to. We perform these calculations in Fig. 6.6 for our high throughput microscopes.

2.7.2 Single Particle Diffusion in Viscoelastic Materials

As described above in Eq. 2.10, the general form for the response functions of viscoelastic materials can be written in terms of the complex shear modulus as

G∗(ω) = G0(ω) +iG00(ω) (2.43)

Recently, a generalization to the Stokes-Einstein relation (Eq. 2.38) was developed to describe these response functions (Mason et al., 1997). The generalized Stokes-Einstein relation (GSER) relates the frequency-dependent complex modulus to a frequency- dependent complex diffusion coefficient, D∗(ω) through

G∗(ω) = kT

6πaD∗_(ω) (2.44)

It is conventional in this formulation to useω as the frequency in units of Hz, such that

τ = 1_{/ω is the timescale in units of sec. In terms of the time-scale dependent MSD,} h∆r2_{(τ)i, the complex modulus can be written as (Mason, 2000)}

G∗(ω) = 2kT

3πah∆r2_{(1/ω)iΓ[1 +α(ω)]} (2.45) where Γ is the gamma function and α(ω) is the logarithmic slope of h∆r2(τ)i eval- uated at τ = 1/ω: α(ω) ≡ d lnh∆r 2_(τ)i d lnτ τ=1/ω

. The shear modulus, G0(ω), and the loss modulus, G00(ω), are computed through G0(ω) = G∗(ω) cos

πα(ω) 2 and G00(ω) = G∗(ω) sin πα(ω) 2

elastic solid whenα = 0 and for a Newtonian fluid when α= 1 (recall η=G00(ω)/ω).

A note about data for passive microbead rheology of cells in this work. In raw form, passive microbead data are position and time measurements produced by the particle tracking software Video Spot Tracker (VST; cismm.org). The mean-squared displacement (MSD) of the particle trajectory is calculated using

∆r2(τ) =

[x(t+τ)−x(t)]2+ [y(t+τ)−y(t)]2

(2.46)

for a predetermined vector of timescales [τ1, τ2, τ3, ..., τT].

The above formulations for passive microbead rheology assume a fully embedded bead in either a Newtonian liquid, an elastic solid, or a viscoelastic material. In Ch.

6, we assess the accuracy of our Array High Throughput microscope to measure the viscosity of standard solutions; in this case, beads are fully embedded in the solutions, and application of Eqn. 2.39 is appropriate. In Ch. 4 and 5 we use Eqn. 2.45 and Eqn. 2.42 to estimate the “effective” shear modulus G0 for a bead that is externally attached to the cell surface (Fig. 4.8 and Fig. 5.8, respectively). Because the use of these equations violates the assumption of a fully embedded bead, we limit our implementation in this work to these two instances.

In Ch. 7 we analyze passive bead rheology (PBR) data for cells conditions by comparing the distributions of the MSD at the τ = 1 timescale.