University of Ljubljana
Faculty of Mathematics and Physics
Biljana Stojković
Mentor: Prof. Dr Igor Poberaj
Ljubljana, December 4th, 2012
Outline
Introduction
Microrheology
Optical tweezers
Passive Microrheology
Active Microrheology
Rheology of bacterial network
Future work
 Microrheology  Rheology
Rheology is the study of the deformation and flow of a
material in response to applied force.
solid
polymers
bacteria
DNA
materials
properties
gels
foams
fluid
V
I
S
K
O
E
L
A
S
T
I
C
Applying oscillatory shear strain:
𝜸 𝒕 = 𝜸𝟎 𝒔𝒊𝒏(𝝎𝒕)
Resultant shear stress: 𝜎 𝑡 = 𝜎0 𝑠𝑖𝑛(𝜔𝑡 + 𝛿)
𝜎 𝑡 = 𝛾0 𝐺 ′ 𝜔 sin 𝜔𝑡 + 𝐺 ′′ 𝜔 cos(𝜔𝑡)
𝐺 ∗ 𝜔 = 𝐺 ′ 𝜔 + 𝑖𝐺 ′′ 𝜔
𝐺′′
tan 𝛿 =
𝐺′
𝐺′′
𝜂=
𝜔
Microrheology is
“rheology on the micrometer length scale”
» Microscopic probe particles
» Locally measure viscoelastic parameters
» Study of heterogeneous environments
» Requires less than 10 microliters of sample
» Biological samples – limited amount of material
» Important for fundamental reaserch and in industrial applycations
Current techniques can be divided into two main categories:
•
active methods that involve probe manipulation
•
passive methods that rely on thermal fluctuations of the probe
Technique in microrheology
Optical tweezers technique
 Tightly focused laser beam
 Dielectric particles with
surrounding medium
higher
refraction
index
that
 Wavelength of the laser  size of the object being trapped
 Maximum force strenght is in the range of 0.1-100 pN
 Powerful laser beam (power on sample 10 − 100 mW)
 Microscope objective with high numerical aperture
(𝑁𝐴 = 𝑛 sin 𝜃 ≥ 1)
of
How we could describe the trapping of dielectric bead?
 R<<λ, point dipol
λ R
𝑭𝒕𝒓𝒂𝒑 = −𝒌𝒙
 R>>λ, ray optics
Optical tweezers set-up
Force calibration
• Bead is held in stationary trap
• Equation of motion:
𝑑2 𝑥
𝑑𝑥
𝑚 2 = −𝑘𝑥 − 𝛾
+ 𝐹(𝑡)
𝑑𝑡
𝑑𝑡
• Power Spectral Density (PSD):
𝑋(𝑓)
2
𝑘𝐵 𝑇
= 2 2
𝛾𝜋 (𝑓𝑐 + 𝑓 2 )
𝒌
𝒇𝒄 =
𝟐𝝅𝜸
𝑿(𝟎)
𝟐
𝟒𝜸𝒌𝑩 𝑻
=
𝒌𝟐
Force calibration
• Boltzman statistic
• In the equilibrium, the probability density of
the 1D particle position:
𝜌 𝑥, 𝑦 = 𝐶
𝑈(𝑥)
−𝑘 𝑇
𝑒 𝐵
• Trap potential can be obtained from
normalization histogram of trapped particle
postition as:
𝜌(𝑥)
𝑈 𝑥 = −𝑘𝐵 𝑇 ln
𝐶
• Fit parabola with:
𝑦 = 𝑎𝑥 2 + 𝑏
𝒌𝒙 = 𝒂/𝒌𝑩 𝑻
 Passive microrheology
• Brownian motion
• Two ways for determination shear modulus:
𝑟 2 (τ) = [𝑟 𝑡 + τ − 𝑟 𝑡 ]2
1.
2. Linear response theory:
𝑥 𝜔 = 𝛼(𝜔)𝐹(𝜔)
𝛼 𝜔 = 𝛼 ′ 𝜔 + 𝑖𝛼 ′′ 𝜔
𝛼
′′
𝜔 𝑆(𝜔)
𝜔 =
4𝑘𝐵 𝑇
2
′
𝛼 𝜔 =
𝜋
𝐺∗
∞
∞
cos (𝜔𝑡)
0
1
𝑓 =
6𝜋𝛼𝑎
0
𝛼 ′′ 𝜔′ sin(𝜔′ 𝑡)𝑑𝜔′ 𝑑𝑡
 Active microrheology
 One-particle active
Oscillations of trap:
𝑥𝑡 𝑡 = 𝐴 sin 𝜔𝑡
The response of the bead is:
The equation of motion:
𝑥 𝑡 = 𝐷 𝜔 sin 𝜔𝑡 − 𝛿 𝜔
𝑘 + 𝑘𝑚 𝑥 𝑡 + 6𝜋𝜂𝑎𝑥 𝑡 = 𝑘𝐴𝑠𝑖𝑛(𝜔𝑡)
The viscoelastic moduli are calculated as:
𝐺′
𝑘𝑚
𝑘 cos 𝛿(𝜔)
𝜔 =
=
−1
6𝜋𝑎 6𝜋𝑎 𝑑(𝜔)
𝑘 sin 𝛿(𝜔)
𝐺′′ 𝜔 = 𝜔𝜂(𝜔) =
6𝜋𝑎 𝑑(𝜔)
 Active microrheology
 Two-particle active
 The displacements od the probe particle:
𝑥
𝑦
2
2
1
𝜔 = 𝐴𝐼𝐼 (𝜔)𝐹𝑥 (𝜔)
1
𝜔 = 𝐴⊥ (𝜔)𝐹𝑦 (𝜔)
 The same displacements can be also expressed directly as:
𝑥
2
𝜔 =𝛼
𝜔 𝐹𝑥
2
𝜔 −𝑘
2
𝑥
2
+ 𝛼𝐼𝐼 𝐹𝑥
1
𝜔 −𝑘
1
𝑥
1
𝑦
2
𝜔 = 𝛼 (2) (𝜔) 𝐹𝑦
2
𝜔 −𝑘
2
𝑦
2
+ 𝛼⊥ 𝐹𝑦
1
𝜔 −𝑘
1
𝑦
1
2
 Active microrheology
Mutual response functions:
𝟏
𝜶𝑰𝑰 =
𝟒𝝅𝑹𝑮(𝝎)
𝟏
𝜶⊥ =
𝟖𝝅𝑹𝑮(𝝎)
Single particle response functions:
𝜶
𝟏
=𝜶
𝟐
𝟏
=
𝟔𝝅𝒂𝑮(𝝎)
Complex viscoelastic modulus:
𝐺 𝜔 =
𝐺′
𝜔
+ 𝑖𝐺 ′′
𝑘𝑚
𝜔 =
+ 𝑖𝜔𝜂
6𝜋𝑎
Rheology of bacteria network
Bacteria – single cell organisms
Different modes:
• Free floating mode
• Formation of biofilms
Biofilms
Free-floating organisms attach to a surface
Colonies of bacteria embedded in an extracellular matrix (EPS)
EPS consist of:
• Polymers and proteins
• accompanied with nucleic acids and lipids
EPS:
 Protect microorganisms from hostile enviroment
 Support cells with nutrients
 Allow comunication between cells
Biofilm development
Stationary phase
Death phase
Log phase
Lag phase
Complexity of biofilm arises:
• Spatial heterogeneities in extracellular chemical concentration;
• Regulation of water content of the biofilm by controling the
composition of EPS matrix;
• Spatial
heterogeneities
on
gene expression
heterogeneities in polymer and surfactant production
creates
The production and assembly of cells, polymer, cross-links
and surfactants result in a structure that is heterogeneous and
dynamic.
Why is this study important
• Biofilm mechanics is important for survival in some enviroments
• Well-known viscoelasticity of bioflims can provide insight into the
mechanics of biofilms
• Quantitative measure of the “strength” of a biofilm could be useful for:
• Development of drugs for inhibition of biofilm growth
• In identifying drug targets
• Characterizing the effect of specific molecular changes of biofilms.
Future work
We will use optical tweezers to study viscoelastic
properties of different biological samples;
We want to understand fundamentally how
the
viscoelasticity changes on different lenght scales on different
frequencies;
The methods will be first tested on water;
The final testground will be viscoelastic characterization
of bacterial biofilms at different stages of biofilm evolution.
References
• Annu. Rev. Biophys. Biomol. Struct. 1994. 23.’247-85
• Annu. Rev. Condens. Matter Phys. 2010.1:301-322.
• Natan Osterman, Study of viscoelastic properties, interparticle potentials and self
ordering in soft matter with magneto-optical tweezers, Doctoral thesis, University
Ljubljana, 2009.
• Natan Osterman, TweezPal – Optical tweezers analysis and calibration software,
Computer Physics Communications 181 (2010) 1911–1916
• Oscar Björnham, A study of bacterial adhesion on a single – cell level by means of
force measuring optical tweezers and simulations, Department of Applied Physics
and Electronics, Umeå University, Sweden 2009
• Mark C. Williams, Optical Tweezers: Measuring Piconewton Forces, Northeastern
University