Analyzing Spatial Point Patterns in Biology

Dr. Maria Byrne

Mathbiology and Statistics Seminar

September 25th, 2009

Analyzing Spatial Point Patterns in

Membrane Biology

Dr. Maria Byrne

Mathbiology and Statistics Seminar

September 25th, 2009

OutlineA. Biomembranes

B. lipid organization?

C. Statistical Analysis of Point Patterns- Ripley’s K

D. Conclusions and Questions

Overview

The micro-organization of lipids and proteins within the cell membrane is an important but open question.

We investigate ways in which statistical methods could be used to determine existence and properties of lipid organization in cell membranes.

A. Biomembranes

Biological Membranes

Structurally composed of phospholipids. Phospholipid: hydrophobic part and hydrophilic part

Naturally form bilayers.

Biological Membranes

Structurally composed of phospholipids. Phospholipid: hydrophobic part and hydrophilic part

www.bioteach.ubc.ca

Biological Membranes

Structurally composed of phospholipids. Phospholipid: hydrophobic part and hydrophilic part

http://academic.brooklyn.cuny.edu/biology

Sea of phospholipids, with a diverse variety of proteins and lipids.

B. Lipid Organization

Are lipids and proteins arranged randomly

throughout the biomembrane, or is there micro-organization?

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Lipid-Lipid Interactions

Gel Domains: Phospholipids with long, ordered chains

Fluid Domains: Phospholipids with short, disordered chains

Cholesterol : Gel domains form a liquid ordered phase

Domain Formation In Model Membranes

The Lipid Raft Hypothesis The cell membrane phase separates into liquid-ordered domains and liquid-disordered domains.

Liquid-Ordered Domains

- “lipid rafts”

- enriched in glycosphingolipids and cholesterol

- act to compartmentalize membrane proteins: involved in signal transduction, protein sorting and membrane transport.

Applications/Relevance

Immune system: Lipid domains are putatively required for antigen recognition (and antibody production).

Vascular system: lipid domains are putatively required for platelet aggregation.

HIV: lipid domains are putatively required to produce virulogical synapses between T-lymphocytes that ennable replication

Cancer: Ras proteins, implicated in 30% of cancers, are thought to signal by compartmentalizing within different domains

Hypotheses for lipid organization:

• Random / homogeneous distributions• Complexes/Oligomers• Exotic organizations

Investigate how to quantitatively distinguish (a) from (b) below:

Future Directions

Investigate how to quantitatively distinguish (a) from (b) below:

Future Directions

Investigate how to quantitatively distinguish (a) from (b) below:

Investigate other models for domain formation:

Oligomerization (e.g., mass action)Cell-controlled organization Protein “corals”

Lipid Raft Controversy

• Lipid rafts: Elusive or Illusive? (S. Munro, Cell, 2003)• Recent controversy surrounding lipid rafts. (M. Skwarek, Arch Immunol Ther Exp ,

2004)– “Although there have been many articles concerning LRs, there is still controversy about

their existence in the natural state, their size, definition, and function.” • Lipid Rafts: Real or Artifact? (M. Ediden, Science Signaling Opening Statement,

2001)• Lipid rafts: contentious only from simplistic standpoints (J. Hancock, Opinion in

Nature Reviews Mol Cell Bio, 2004)• Lipid Rafts Exist as Stable Cholesterol-independent Microdomains in the Brush

Border Membrane of Enterocytes (Hansen et al, Journal of Biol Chem, 2001)• Special Issue: Lipids Rafts (BBA, 2005)

– The controversy arises from the fact that rafts have proven frustratingly difficult to precisely define in cells. We do not yet have an unambiguous picture of raft size, stability, or protein and lipid composition. It is also not clear whether rafts exist in cell membranes constitutively or form only in a regulated manner.

Adding to complexity/lack of consensus:

• Many cell types

• Different types of cell membranesmembranes of organellesplasma membrane

• Inner verses outer membrane layers

C. Statistical Analysis of Point Patterns

•The positions of N molecules is precisely described by 2N numbers in continuous space.

Prior, Muncke,Parton and Hancock

Yet…

The organization of lipids are determined by physical and biological parameters that may greatly constrain the set of possible distributions.

Example: if the distribution of lipids is genuinely random, the entire distribution can be described with just the lipid density.

Ripley’s K

Aggregation and Segregation

Aggregation and Segregation

Domain Radius

• A positive value of H(r) indicates clustering over that spatial scale.

• A negative value of H(r) indicated dispersion over that spatial scale.

• The maximum value of H(r) occurs at r = the radius of maximal aggregation.

• To what extent does this maximum report on the domain radius?

Domain Radius

• To what extent does this maximum report on the domain radius?

Method used in • R.G. Parton et al., J. Cell Biol., 2004

• Hancock and Prior, Trends Cell Biol, 2004

• Zhang et al., Micron, 2006

Domain Radius To what extent does this maximum report on

the domain radius?

Prediction correct within a factor of 2, but the radius is usually over-estimated.

Domain Radius Prediction correct within a factor of 2, but the

radius is usually over-estimated.

Why: the radius of maximal aggregation occurs when the effects of aggregation within a domain begin to be offset by the dispersion outside the domain, which occurs at a radius somewhere beyond the domain boundary.

Domain Radius Solution: remove accumulative effects by taking the derivative of

H(r).

Instead of identifying the domain radius, we identify the domain diameter by finding the value of r when the derivative is minimized.

Domain Radius

Application: K-ras Nanoclusters

Experimentally derived point pattern with an immunogold density 625 m-2.

Application: K-ras Nanoclusters

Nanoclusters of constant size (~16nm).

Each contains ~3.2 gold particles.

Noise: 56% protein monomeric

Application: K-ras Nanoclusters

Method:

(1) Use Ripley’s K to estimate domain size for the experimental image. (14 nm)

(2) Use Monte Carlo generated images to estimate error due to noise.

Effect of “Non-Random” Noise

•Points non-randomly dispersed within domains.(significant effect!)

•Domains not approximately disk-shaped.(significant effect!)

•Interconnected or finger-like domains.(method: useless)

Immediate challenge: how to ‘parametrize’ non-random noise?

D. ConclusionsAnd Questions

Conclusions• Ripley’s K can be used to determine

domain radius if points are arranged randomly within domain.

• The measure is robust to random noise.

• The measure is sensitive to systematic / patterned noise.

Question• It’s relatively straight-forward to measure the extent

to which a pattern holds.

• Determining which patterns hold is an ad hoc process.– Are the particles aggregated?– Are the particles separated?– Are the particles arranged in predetermined ways?

• More difficult to know if any pattern holds.

• To what extent can the problem of “finding pattern” be framed so that it may be (more) well-defined? In general, how can one identify pattern?

A Random Donor is Excited

FRET Efficiency is Related to the Distance of the Nearest Acceptor

Compare with the Hausdorff Measure

The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set.

Compare with the Hausdorff Measure

The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set.

Compare with the Hausdorff Measure

),( YXd H

),(infsup),,(infsupmax yxdyxd

XxYyYyXx

Compare with the Hausdorff Measure

Compare with the Hausdorff Measure

The Hausdorff metric will measure one half the domain separation as long as the domain separation is greater than the domain radius.

Let X={inter-domain space} Y={domain space}