Tuesday, June 11, 2013
Geometric interpretation of additive variance
My intuitive understanding of additive variance:
Total variance = Var(X) + Var(Y)
where Var(X) = sum of (x_i - mean)^2 .
So, Var(X) and Var(Y) are made of the square areas around their mean. Total variance is the total area. Additive indicate X and Y are independent.
Co-variance will be the rectangles from this view.
Var(x+y) = Var(x) + Var(y) + 2Cor(X,Y)
I need to add a diagram to illustate this.
Total variance = Var(X) + Var(Y)
where Var(X) = sum of (x_i - mean)^2 .
So, Var(X) and Var(Y) are made of the square areas around their mean. Total variance is the total area. Additive indicate X and Y are independent.
Co-variance will be the rectangles from this view.
Var(x+y) = Var(x) + Var(y) + 2Cor(X,Y)
I need to add a diagram to illustate this.
Sample R code for bootstrapping residues for regression
y = rnorm(100) + rep(c(0, 1), each = 50)
x = rep(c(1,2), each =
50)
my = mean(y)
res = lm(y~x)$resid
# Log LR test
lr.obs = sum((y-my)^2) - sum(res^2) # Can replace with "F" like statistic
lr.star = c()
for(boot in 1:1000){
y.star = my + sample(res, replace = T)
res.star = lm(y.star~x)$resid
lr.star[boot] = sum((y.star-my)^2) - sum(res.star^2)
}
pval = mean(lr.star>= lr.obs)
This sample code was provided by Dr. Dipen Sangurdekar. Boostrapping residue of the empirical data is considered to be more reasonable estimation of p-values.
Monday, June 10, 2013
Notes, Costanzo SGA 2010
Costanzo et al 2010, SGA genetic interaction dataset
Costanzo10 model double mutant colony size (growth fitness) as $C_ij = f_ij * t * s_ij * e$.
The double mutant fitness $f_ij = f_i * f_j + epsilon$. Hence,
So, for perfect synthetical lethals $f_ij$=0, and $f_i= f_j= 1$, we should have $epsilon = -1$.
Costanzo10 defines negative interaction as $epsilon<0, and positive interaction as $epsilon>0$. Because the measure is based on loss of function assay, the actual function of the wildtype genes are just opposite.
Extreme examples of negative interactions are synthetic lethality. Positive interactions may indicate antagonizing activities (Thanks to my friends for pointing this out).
It is interesting that Costanzo10 recovered only 30% of the published synthetic lethality. It is possible growth defect of most single deletion mutant in synthetic lethal pairs is often de-emphasized.
The SGA genetic interaction dataset is available for download in various text formats
(see Data Files S1-S5) or through a web-based database, DRYGIN (http://drygin.ccbr.utoronto.ca, (S2)). SGA data can also be explored through the SGAExpress browser (http://www.ailab.si/sgaexpress).
For stringent cutoff, epsilon < -0.12 with p<0.05 and epsilon > 0.16 with p < 0.05.
#genetic network in pairs
GIN.pairs.tb = read.csv("sgadata_costanzo2009_stringentCutoff_101120.csv", header=F,
colClass=c("character","character","character","character",NA,NA,NA))
names(GIN.pairs.tb) = c("ORF1", "name1", "ORF2", "name2", "epsilon", "stddev", "p")
#epsilon = fij - fi * fj
PositiveGIN.pairs.tb = GIN.pairs.tb[GIN.pairs.tb$epsilon>0, ]
NegativeGIN.pairs.tb = GIN.pairs.tb[GIN.pairs.tb$epsilon<0, ]
str(PositiveGIN.pairs.tb)
str(NegativeGIN.pairs.tb)
Costanzo10 model double mutant colony size (growth fitness) as $C_ij = f_ij * t * s_ij * e$.
The double mutant fitness $f_ij = f_i * f_j + epsilon$. Hence,
Costanzo10 defines negative interaction as $epsilon<0, and positive interaction as $epsilon>0$. Because the measure is based on loss of function assay, the actual function of the wildtype genes are just opposite.
Extreme examples of negative interactions are synthetic lethality. Positive interactions may indicate antagonizing activities (Thanks to my friends for pointing this out).
It is interesting that Costanzo10 recovered only 30% of the published synthetic lethality. It is possible growth defect of most single deletion mutant in synthetic lethal pairs is often de-emphasized.
The SGA genetic interaction dataset is available for download in various text formats
(see Data Files S1-S5) or through a web-based database, DRYGIN (http://drygin.ccbr.utoronto.ca, (S2)). SGA data can also be explored through the SGAExpress browser (http://www.ailab.si/sgaexpress).
For stringent cutoff, epsilon < -0.12 with p<0.05 and epsilon > 0.16 with p < 0.05.
#genetic network in pairs
GIN.pairs.tb = read.csv("sgadata_costanzo2009_stringentCutoff_101120.csv", header=F,
colClass=c("character","character","character","character",NA,NA,NA))
names(GIN.pairs.tb) = c("ORF1", "name1", "ORF2", "name2", "epsilon", "stddev", "p")
#epsilon = fij - fi * fj
PositiveGIN.pairs.tb = GIN.pairs.tb[GIN.pairs.tb$epsilon>0, ]
NegativeGIN.pairs.tb = GIN.pairs.tb[GIN.pairs.tb$epsilon<0, ]
str(PositiveGIN.pairs.tb)
str(NegativeGIN.pairs.tb)
Friday, June 7, 2013
Renewal/repair effect on network robustness
Aging increases heterogeneity. Renewal and repair should increase network homogeneity. So, renewal/repair should increase robustness (high G).
However, Barlow&Proschan96 argues that renewal/repair also leads approximate exponential function for the pdf of mortality (page 15 and section 3). Exponential pdf give CDF in form of $1 - exp(- \lambda x)$, so its surviorship or viability is also exponential. From this perspective, renewal/repair should lead to constant mortality rate and G->0.
Why are there two opposite conclusions?
Point-repair (age-based component repair)
Heat-shock proteins
Proteosome
Ubiquitin?
Block-repair
Lysosome-like vacuole, mitochondria
Random repair versus targeted repair
Limiting modules in power-law network
For fixed decaying rate, small degree essential node -> faster earlier rates, large-degree essential node -> fat tail.
However, Barlow&Proschan96 argues that renewal/repair also leads approximate exponential function for the pdf of mortality (page 15 and section 3). Exponential pdf give CDF in form of $1 - exp(- \lambda x)$, so its surviorship or viability is also exponential. From this perspective, renewal/repair should lead to constant mortality rate and G->0.
Why are there two opposite conclusions?
Point-repair (age-based component repair)
Heat-shock proteins
Proteosome
Ubiquitin?
Lysosome-like vacuole, mitochondria
Random repair versus targeted repair
Limiting modules in power-law network
For fixed decaying rate, small degree essential node -> faster earlier rates, large-degree essential node -> fat tail.
Thursday, June 6, 2013
Xue 07, MSB, a modular network model of aging
My notes on Xue 2007, a modular network model of aging.
Xue07 examined PPI in fruit fly and human brain aging. Xue07 used simulation and stated that "aging might preferentially attack key regulatory nodes that are important for the network stability, implicating a potential molecular basis for the stochastic nature of aging".
Xue07 identified negative (N) and positive (P) correlations during aging based on expression data, and then identified the PPI subnetwork for these NP correlations. Xue07 then used clustering method to identify 'network modules'.
Its method paper is in
Xue07 examined PPI in fruit fly and human brain aging. Xue07 used simulation and stated that "aging might preferentially attack key regulatory nodes that are important for the network stability, implicating a potential molecular basis for the stochastic nature of aging".
Xue07 identified negative (N) and positive (P) correlations during aging based on expression data, and then identified the PPI subnetwork for these NP correlations. Xue07 then used clustering method to identify 'network modules'.
Its method paper is in
Xia K, Dong D, Xue H, Zhu S,Wang J, Zhang Q, Hou L, Chen H, Tao R, Huang Z, Fu Z, Chen YG, Han JD (2006) Identification of the proliferation/differentiation switch in the cellular network of
multicellular organisms. PLoS Comput Biol 2: e145
Wednesday, June 5, 2013
Basic concepts and references on aging and reliability, Barlow & Proschan 1996
IFR: increasing failure rate
DFR: decreasing failure rate
The failure rate function $r(t)$ is defined as: $r(t) = f(t) / S(t) = f(t) / (1- F(t))$ in Barlow & Proschan96 on page 10. Function $f(t)$ should be the pdf of death, and $F(t)$ is the failure distribution (page 2), and $S(t)$ should be viability.
Barlow&Proschan96 quote $r(t)$ as "force of mortality" by Steffernson 1930. The reference is missing in the book's reference list), "hazard rate", "intensity function". I found a JSTOR examination of actuarial science in 1910 with a question of comparing 'force of mortality' and 'rate of mortality'. So, this concept should goes back much earlier.
Parzen, E, 1962, stochastic process, holden-Day, San Francisco
Gumbel 1958, Statistics of extremes. Columbia Univ Press.
Discussed exponential distribution and limiting distribution.
Lawrence Leemis, Reliability-Probabilistic models and statistical methods, second edition, 2009.
Its explanation on series and parallel structure functions are really clear.
Gavrilov 1991
DFR: decreasing failure rate
The failure rate function $r(t)$ is defined as: $r(t) = f(t) / S(t) = f(t) / (1- F(t))$ in Barlow & Proschan96 on page 10. Function $f(t)$ should be the pdf of death, and $F(t)$ is the failure distribution (page 2), and $S(t)$ should be viability.
Barlow&Proschan96 quote $r(t)$ as "force of mortality" by Steffernson 1930. The reference is missing in the book's reference list), "hazard rate", "intensity function". I found a JSTOR examination of actuarial science in 1910 with a question of comparing 'force of mortality' and 'rate of mortality'. So, this concept should goes back much earlier.
Parzen, E, 1962, stochastic process, holden-Day, San Francisco
Gumbel 1958, Statistics of extremes. Columbia Univ Press.
Discussed exponential distribution and limiting distribution.
Lawrence Leemis, Reliability-Probabilistic models and statistical methods, second edition, 2009.
Its explanation on series and parallel structure functions are really clear.
Gavrilov 1991
Tuesday, June 4, 2013
Teaching microsatellite instability, replication slippage using beads
Microsatellites are short tandem DNA repeats tend to have high mutation rates due to DNA replication slippage.
An analogy to help students understand this phenomena is to ask them to count beads on beaded bracelets. If all the beads are the same (tandem repeats), we can easily slipped one or two beads. If beads are colored, our counting can be more accurately tracked. This analogy worked in Bio125, Spring of 2013.
An analogy to help students understand this phenomena is to ask them to count beads on beaded bracelets. If all the beads are the same (tandem repeats), we can easily slipped one or two beads. If beads are colored, our counting can be more accurately tracked. This analogy worked in Bio125, Spring of 2013.
Notes, Vaupel, Manton, Stallard 1979, failty model
Reading notes on Vaupel, Manton, Stallard 1979, Demography, The impact of heterogeneity in individual frailty on the dynamics of mortality.
This is really a classic paper to study the effect of heterogeneity on aging.
.JPG)
VMS79 defined frailty $z$ as $\mu(z) = z \mu$, where $mu$ is the mortality rate (termed force of mortality in the paper).
VMS79 used a gamma distribution to model frailty. They found that frailty among the survivors at age $x$ is gamma distributed with the same shape parameter $k$ at birth. The second parameter is
$\lamda(x) = \lambda + H(x)$.
This is really a classic paper to study the effect of heterogeneity on aging.
.JPG)
VMS79 defined frailty $z$ as $\mu(z) = z \mu$, where $mu$ is the mortality rate (termed force of mortality in the paper).
VMS79 used a gamma distribution to model frailty. They found that frailty among the survivors at age $x$ is gamma distributed with the same shape parameter $k$ at birth. The second parameter is
$\lamda(x) = \lambda + H(x)$.
Notes, Vaupel & Yashin,1999, From homogeniety to heterogeneity,
VY99 used a general frailty model to account for heterogeneity in the population.
The $s(x,z) can be specified in "relative-risk model", $\mu(x,z)= z \mu(x)$.
Reference:
Vaupel and Yashin, 1999, From homogeneity to heterogeneity:Unobserved population heterogeneity.
The $s(x,z) can be specified in "relative-risk model", $\mu(x,z)= z \mu(x)$.
Reference:
Vaupel and Yashin, 1999, From homogeneity to heterogeneity:Unobserved population heterogeneity.
*** Yeast genomic data resources for network and robustness (in progress)
Seattle Yeast Resource Center
Provided genomes, mass spec, RNA seq data. Description of the experimental conditions are not clear.
http://www.yeastrc.org/g2p/about.do
GenAge
Expression data from S. cerevisiae under serial stresses, including H2O2 time course
http://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE15936
Yeast
fitness Giaever lab, 2002 Nature
http://www.nature.com/nature/journal/v418/n6896/extref/nature00935-s1.htm
http://genomics.lbl.gov/YeastFitnessData/websitefiles/cel_index.html
(The site provide .cel files, but I did not locate the fitness files?! How did He14Ibuprofen obtain their fitness values?).
yeast resources pages
http://yeast.uni-graz.at/index.php?item=research
http://www.yeastgenome.org/download-data/sequence
Yeast essential and non-essential genes
/data_download/literature_curation/phenotypes.tab #contain this information in an irregular way.
Winzeler99Science and yeast deletion database also provide these information (cited by Hwang, Lin, Chang, Mori, Juan, Huang, 2009 Molecular Biosystems, Predicting essential genes based on network and sequence analysis)
Yeast RLS and CLS data
Kaeberlein and Kennedy labs.
bipixe
http://downloads.yeastgenome.org/published_datasets/Myers_2004_PMID_16420673/README
http://www.yeastgenome.org/download-data/published-datasets
Yeast CV GFP from Alon's paper
http://www.ncbi.nlm.nih.gov/pubmed/?term=zakrzewska+robustness
http://www.yeastgenome.org/cache/genome-wide-analysis.html
CV of GEO expression data
Genetic network (Costanzo2009).
PPI
LOH screen by Gottschling lab
Phenotypic capacitance, The phenotypic capacitance was used directly from the Levy & Siegal study (S28) and captures variability across a range of morphological phenotypes upon deletion of each of
the non-essential genes
Yeast morphology Database
Princeton genomics
http://genomics-pubs.princeton.edu/
BYxRM
http://genomics-pubs.princeton.edu/YeastCross_BYxRM/data/BYxRM_PhenoData.txt
Predicted prions
http://www.utsouthwestern.edu/labs/halfmann/publications/
ChEMBL, chemical database
https://www.ebi.ac.uk/chembl/
- Hellenmyer 2008 science chemical screen
- Rugly 2006 JB, literacture curated PIN dataset (I did not find it in Biogrid publication download)
- Lipson 2009 NBT, single molecular sequencing on trancriptome, supplment table 2 provide stddev and mean that can be used for CV=stddev/mean.
- CV of microarray based gene expression
- Gasch 2000
- Genetic network, Costanzo 2010
- PPI
- Power06 CLS life span
- LevySegel08 phenotypic potential
- Yeast GFP CV data, Newman et al
- Supplementary information From the following article Quantification of the yeast transcriptome by single-molecule sequencing Doron Lipson, Tal Raz, Alix Kieu, Daniel R Jones, Eldar Giladi, Edward Thayer, John F Thompson, Stan Letovsky, Patrice Milos & M
Yeast human gene homologs
http://yeastmine.yeastgenome.org/yeastmine/bagDetails.do?scope=all&bagName=Human+genes+with+yeast+homologs
Monday, June 3, 2013
Aging of a power-law network
.JPG)
2013 Dec 9. Power-law network can lead to the Gompertz-Makeham model of aging. The Makeham term is basically the non-aging term.
Chebyshev's inequality
I am amazed at the generality of Chebysheve's inequality that provides an outbound to any distribution. This means that I may use it to find the bound for some network-related distributions.
Zakrzewska 2011 MCB, tradeoff between growth rate and cellular robustness
Zakrzewska11 found a inverse correlation between growth rate and stress tolerance using yeast deletion collection. Cellular functions required for stress tolerance, independent of the growth rate, include vesicular transport, Rpd3 histone deacetylase complex, mitoic cell cycle.
Zakrzewska11 studied heat, acid, oxidative stresses as stress factors.
The study used BY4741, two hours exposure to 39C, 10C, 0.15mM H2O2, 1.2mM sorbic acid at pH5.0. The control culture is at 30C. The second stress was 327mM acetic acid at pH3.0, 10mM H2O2, or 48C for 10 minutes.
Zakrzewska11 used barcoded yeast deletion collection and Affymetrix chip to conduct the genome-wide measures. Each knock-out strain is usually represented by 4 values. Both growth rate and survival rates were measured by barcodes. Synthetic complete medium was used. The first time point is t=-6 hours, with cell density of 2x10^5 cells/ml. The second timepoint is t=-3 hours. The 38C heat shock started at t=-3 hours and lasted till t=0 hours. Linear regressions were applied with these 3 time points, and its slope is the calculated growth rate (1/hr).
Zakrzewska11 provided Excel data of growth rate and survival data for 4066 null mutants in file 'mc-E10-08-0721-s06.xls'. The 'growth rates' sheet,
contain 'mu30' and 'mu38', which are probably the growth rates (slopes of -6, -3, and 0 hr hybridization signals). Mostly 'mu30' > 'mu38' as expected, but sometimes 'mu30' < 'mu38' that may suggest heat-tolerance or heat-adaptation. The 'survival % 95% CI' sheet contains data that were measured after a 10 minute treatment with 10 mM H2O2 (oxi), 327 mM acetic acid at pH 3.0 (acid) or 48oC (heat), compared with CFU counts of an untreated control.
This was done after growth at 30C, as well as after a 3 hour 38C pretreatment. Each temperature has 3 replicates. The columns of "<95%" and ">95%" are not clear to me and I wonder what they mean.
I downloaded 'log transformed viability for website.xls' and 'raw data for website.xls'. The first seems to be the calculations, and the second seems to contain the original raw data.
Note: Deep sequencing may provide better information than bar-code and hybridization.
References:
http://www.molbiolcell.org/content/22/22/4435.full
http://www.uva.nl/over-de-uva/organisatie/medewerkers/content/s/m/g.j.smits/g.j.smits.html
Zakrzewska11 studied heat, acid, oxidative stresses as stress factors.
The study used BY4741, two hours exposure to 39C, 10C, 0.15mM H2O2, 1.2mM sorbic acid at pH5.0. The control culture is at 30C. The second stress was 327mM acetic acid at pH3.0, 10mM H2O2, or 48C for 10 minutes.
Zakrzewska11 used barcoded yeast deletion collection and Affymetrix chip to conduct the genome-wide measures. Each knock-out strain is usually represented by 4 values. Both growth rate and survival rates were measured by barcodes. Synthetic complete medium was used. The first time point is t=-6 hours, with cell density of 2x10^5 cells/ml. The second timepoint is t=-3 hours. The 38C heat shock started at t=-3 hours and lasted till t=0 hours. Linear regressions were applied with these 3 time points, and its slope is the calculated growth rate (1/hr).
Zakrzewska11 provided Excel data of growth rate and survival data for 4066 null mutants in file 'mc-E10-08-0721-s06.xls'. The 'growth rates' sheet,
contain 'mu30' and 'mu38', which are probably the growth rates (slopes of -6, -3, and 0 hr hybridization signals). Mostly 'mu30' > 'mu38' as expected, but sometimes 'mu30' < 'mu38' that may suggest heat-tolerance or heat-adaptation. The 'survival % 95% CI' sheet contains data that were measured after a 10 minute treatment with 10 mM H2O2 (oxi), 327 mM acetic acid at pH 3.0 (acid) or 48oC (heat), compared with CFU counts of an untreated control.
Raw data are available at
http://www.uva.nl/over-de-uva/organisatie/medewerkers/content/s/m/g.j.smits/g.j.smits.htmlI downloaded 'log transformed viability for website.xls' and 'raw data for website.xls'. The first seems to be the calculations, and the second seems to contain the original raw data.
Note: Deep sequencing may provide better information than bar-code and hybridization.
References:
http://www.molbiolcell.org/content/22/22/4435.full
http://www.uva.nl/over-de-uva/organisatie/medewerkers/content/s/m/g.j.smits/g.j.smits.html
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