Applications in Computational Biology. Stephen J Eglen University of Cambridge

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2 Applications in Computational Biology Stephen J Eglen University of Cambridge

3 Mathematics Is Biology s next microscope, only better; Biology Is Mathematics next Physics, only better. (J Cohen, 24 PLOS Biol)

4 Cambridge Computational Biology Institute (CCBI) Launched in 23, with support across University. Wellcome Trust PhD programme in Mathematical Genomics and Medicine MPhil programme in Computational Biology Key faculty: Prof. Simon Tavaré (Cancer Research Institute), Dr Gos Micklem (Genetics), Dr Julia Gog (DAMTP)

5 Classification of breast cancers Curtis C et al. (212) The genomic and transcriptomic architecture of 2, breast tumours reveals novel subgroups. Nature 486: How to move beyond morphological markers (and two key markers) used currently in clinics?

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7 Disease dynamics: flu spread Gog JR, Ballesteros S, Viboud C, Simonsen L, Bjornstad ON, Shaman J, Chao DL, Khan F, Grenfell BT (214) Spatial Transmission of 29 Pandemic Influenza in the US. PLoS Comput Biol 1:e13635.

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10 For each of i= cities, prob of outbreak given by: T{1 P i (T)~(1{e {l i (T) ) P t~1 e{l i (t) l i (t)~h i w e vx i Bb Ni a zb si i z(b d zb ds I i )N m 1 P Nj nd{c i, j j[l " # e C P A j=i N n j d{c i, j

11 Most parsimonious model Parameter Description Units Value 95% Confidence interval Background transmission β (Δt) rate β d β ds µ γ ε Spatial transmission coefficient (Δt) 1 (km) 1 ε Boost to β d when schools are open Exponent of dependence on recipient population size Exponent of distance in gravity model kernel Strength of density normalisation (Δt) 1 (km) 1 ε none none none λ i (t) = β + (β d + β ds I i )N i µ *, +, j Λ γ d i, j γ d i, j j i - /./ ε

12 Detecting correlations Catherine Cutts (J Neurosci 214)

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15 (Demas et al. 23, P11 mouse)

16 Correlation index A B 25 WT (n=13) B2(TG) (n=17) 2 A Correlation index 15 1 B Δt c = N A,B[ t,+ t]t N A N B 2 t inter unit distance (µm) (Wong et al. 1993; Xu et al. 211)

17 Measuring correlation Distance measures and cost functions 1 Victor and Purpura (1997) 2 ISI-distance (Kreuz et al., 27a) 3 Hunter-Milton similarity (Hunter and Milton, 23) 4 Van Rossum (21) 5 SPIKE (Kreuz et al., 213) Cross-correlation based 6 Coincidence index (Pasquale et al., 28) 7 Altered Coincidence index* 8 Cross correlation coefficient (Pasquale et al., 28) 9 Schreiber et al. (23) similarity coefficient 1 Altered Schreiber et al. similarity coefficient* 11 Kerschensteiner and Wong (28) cross-correlation 12 Jimbo and Robinson index (Jimbo et al., 1999) Synchrony not from cross-correlation 13 Correlation index (Wong et al., 1993) 14 Activity pair (Eytan et al., 24) 15 Unitary events analysis (Grün et al., 22) 16 Event synchronization (Kreuz et al., 27b)* 17 Joris et al. (26) correlation index Information theory 18 Mutual information (Li, 199) 19 Mutual information with smoothing* Measures from shot-noise process 2 Coherence (at zero) (Eggermont, 21) 21 Spike count correlation (Eggermont, 21) 22 Smoothed spike count correlation (Kruskal et al., 27)* 23 Spike count covariance (Eggermont, 21) Measures assuming a marked point process 24 Stoyan s K mm function (Stoyan and Stoyan, 1994) 25 Isham s mark correlation function (Isham, 1985) 26 Ripley s K mm function (Ripley, 1976) 27 Simpson (1949) index 28 Simpson (1949) index no correction 29 Stoyan s mark covariance function (Stoyan, 1984) 3 Mark variogram (Cressie, 1993) 31 Mark covariance function (Cressie, 1993) 32 Mark conditional expectation (E; Schlather et al., 24) 33 Mark conditional variance (V; Schlather et al., 24) 34 Mark conditional standard deviation (Schlather et al., 24)

18 Which method is best? 34 measures in literature + 1 from us => 35. Phase 1: six necessary properties: 1. Symmetric 2. Robust to variations in firing rate 3. Robust to amount of data 4. Bounded [-1, +1] 5. Robust to variations in bin width (Δt)

19 1.6 1 Measure Measure Measure 8 Twenty-one measures rejected as they depend on firing rate. (autocorrelation of Poisson trains) Measure 12 v1 Measure 16 Measure Measure 14 Measure 17 Measure Measure 15 Measure 18 Measure Measure 25 4 Measure Measure Measure Measure Measure 3.55 Measure 31 1 Measure 33 1 Measure measure Firing rate (Hz) 1.1 Measure Measure Measure 35 measure Firing rate (Hz) measure Firing rate (Hz) measure Firing rate (Hz)

20 Short-list from Phase 1 r(a, B) N i 1 N i 1 (A i A)(B i B) N (A i A) 2 i 1 (B i B) 2, 1.Spike count correlation coefficient 2. Kerschensteiner and Wong (28) 3. Kruskal et al. (27) 4. Spike time tiling coefficient

21 TA: the proportion of total recording time which lies within ±Δt of any spike from A. TB calculated similarly. A 2Δt Time (s) T TA is given by the fraction of the total recording time (black) which is covered (tiled) by blue bars. Here TA is 1/3. PA: the proportion of spikes from A which lie within ±Δt of any spike from B. PB calculated similarly. B A Time (s) T PA is the number of green spikes in A (3) divided by the total number of spikes in A (5). Here PA is 3/5. TC = 1 2 PA - TB 1-PA TB + PB - TA 1-PBTA

22 Phase 2: Desirable properties Desirable properties: D1: Ignore periods when both neurons are inactive. D2: minimal assumptions on structure. D3: aside from Δt, minimise number of parameters Four methods: Kerschensteiner and Wong correlation (D1, D2) Tiling coefficient (D1, D2, D3) Spike count correlation (D2, D3) Kruskal et al. binless correlation measure (D2, D3)

23 Wild Type A 1 min Cx36/Cx45 dko Correlation Index Cx45 KO B Tiling coefficient C Wild Type (.31 Hz, n=5) Cx45 ko (1.6 Hz, n=4) Cx36/45 dko (1.99 Hz, n=6) Intercell distance (µm) Intercell distance (µm) (Blankenship et al. 211)

24 Correlation Index A P (.74 Hz) P15 (.29 Hz) P21 (.2 Hz) P3 (.15 Hz) Ferret Tiling measure B Intercell distance (µm) Intercell distance (µm) Mouse C D Correlation Index P9 P15 (.3 Hz) (.96 Hz) P21 (.7 Hz) 6 week (2.4 Hz) Tiling measure Intercell distance (µm) Intercell distance (µm)

25 Summary Correlation measure already applied in recent papers. Code available, easy to use. Too specific to retinal waves (Eisenman et al, J Neurophys 215)? Bayesian hierarchical modelling framework for comparing correlation curves on Biorxiv

26 Acknowledgments Funding: EPSRC, Wellcome Trust, USIAS, BBSRC Data/code: See also posters by: Stephen Kissler (disease dynamics) Ellese Cotterill (burst analysis) Linda Bowns (motion detection).