Analyzing Spatial Point Patterns in Biology Dr. Maria Byrne Mathbiology and Statistics Seminar September 25 th, 2009

Analyzing Spatial Point Patterns in Membrane Biology Dr. Maria Byrne Mathbiology and Statistics Seminar September 25 th, 2009

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

Overview The micro-organization of lipids and proteins within the cell membrane is an open question. We investigate ways in which statistical methods could be used to determine existence and properties of lipid organization in cell membranes. We: Anne Kenworthy & lab John F. Hancock

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

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

Sea of phospholipids, with a diverse variety of proteins and lipids. Singer-Nicolson Fluid Mosaic Model

B. Lipid Organization

Are lipids and proteins arranged randomly throughout the biomembrane, or is there micro- organization? Model Membranes Subcellular 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.

Heetderks and Weiss 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

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

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.

Hypotheses for lipid organization: Random / homogeneous distributions Complexes/Oligomers Exotic organizations Other models for domain formation: Oligomerization (e.g., mass action) Cell-controlled organization Protein “corals”

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.

Hypotheses for lipid organization: Random / homogeneous distributions Complexes/Oligomers Exotic organizations

Can we quantitatively distinguish (a) from (b) below?

… even with noise? Can we quantitatively distinguish (a) from (b) below?

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

Ripley’s K Ripley’s K is the second moment property of a spatial point pattern (the expected # of points within a distance r of another point) normalized by the # of points per area. K(r) =  N p i (r)/ ·1/n where p i is the i th point and the sum is taken over n points.

Ripley’s K K(r) = 1/n  N pi (r)/ The function measures the number of points within a radius of each point, normalized by the average point density.

Ripley’s K K(r) = 1/n  N pi (r)/ The expected value of K(r) for a Poisson distribution is  r 2. The function measures the number of points within a radius of each point, normalized by the average point density.

Ripley’s K K(r) = 1/n  N pi (r)/ The expected value of K(r) for a Poisson distribution is  r 2. Deviations from this expectation indicate scales of clustering and dispersion. The function measures the number of points within a radius of each point, normalized by the average point density.

Normalized Ripley’s K K(r) can be normalized so that its expected value is r : L(r)=  [K(r)/  ]. K(r) = 1/n  N pi (r)/ And further normalized so that its expected value is 0 : H(r)=L(r)-r

Ripley’s K Ripley’s K reports on the extent of aggregation and dispersion that occurs.

Typical H(r) For Aggregated Points

Domain Radius A positive value of H(r) indicates clustering over that spatial scale. A negative value of H(r) indicates dispersion over that spatial scale.

Domain Radius A positive value of H(r) indicates clustering over that spatial scale. A negative value of H(r) indicates dispersion over that spatial scale. To what extent can we use Ripley’s K to determine the domain radius?

Domain Radius The maximum value of H(r) (the radius of “maximal aggregation”) has been used to estimate the domain radius. For example: R.G. Parton et al., J. Cell Biol., 2004 Hancock and Prior, Trends Cell Biol, 2004 Zhang et al., Micron, 2006

Radius of Maximal Aggregation To what extent does the max of H(r) report on the domain radius?

Radius of Maximal Aggregation To what extent does the max of H(r) report on the domain radius? Prediction correct within a factor of 2, but the radius is often over-estimated.

Why: the radius of maximal aggregation occurs when the effects of aggregation within a domain are just offset by the dispersion outside the domain, which occurs at a radius somewhere beyond the domain boundary. Radius of Maximal Aggregation

Domain Radius Needed: remove accumulative effects by taking the derivative of H(r). H'(r) Also, instead of identifying the domain radius, we identify the domain diameter by finding the value of r when the derivative is minimized.

Domain Radius H MAX H' MIN

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 Random Noise Domain Interaction Monomer Fraction

Effect of Random Noise Domain Interaction Monomer Fraction

Effect of “Non-Random” Noise

Immediate challenge: how to ‘parametrize’ non-random 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. Conclusions And 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.

Questions Determining which patterns hold is an ad hoc process. –Are the particles aggregated? –Are the particles separated? –Are the particles arranged in any specified, highly specific way? More difficult to know if any non-specified pattern holds. To what extent can the problem of “finding pattern” be framed so that it may be better defined? In general, how can one identify “pattern”?