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Expectation Maximization Methods for Metabolic Pathway Analysis

Dissertation Proposal

Fil Rondel

Committee: Alex Zelikovsky

Pavel Skums

Murray Patterson

Artem Rogovskyy

July 1^st 2022

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Part I - Introduction

Introduction

Metabolism
Metabolic pathways
Enzymes
Metabolic pathway activity
Problem formulation
Challenges
Enzyme participation
Contributions
Publications�

Estimating enzyme participation and metabolic pathway activity for microbial communities
Estimating enzyme participation and metabolic pathway activity for wild and lab mice
Conclusions

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Metabolism

Living organisms need energy to grow, move, and think
Metabolism is

series of complex biochemical (metabolic) processes that regulate energy in organisms
organised into distinct metabolic pathways to either maximise the capture of energy or minimise its use�

Understanding metabolism is key to understanding life and has been a subject of fascination with scientists for more than 150 years

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Metabolic Pathways

Metabolic pathways are structured parts of metabolism

maximise the capture of energy
prevent its uncontrolled combustion
E.g.

Glycolysis – breaks down glucose to generate energy
Photosynthesis – used by organisms to convert light energy into chemical energy�

Metabolic processes are catalyzed by specific proteins called enzymes
Metabolic pathways are regulated by enzymes and substrates

Lipases – help digest fats in the gut
Amylase – helps process starches into sugars

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Enzymes

Catalyze biochemical reactions in organisms
Enzyme activity is directly related to substrates available
Enzyme Commission (EC) numbers
EC classes describe specific enzyme functions

EC 3 enzymes are hydrolases (use water to break up molecules)
EC 3.4 peptide bond acting hydrolases
EC 3.4.11 hydrolases that cleave off the amino-terminal end from a polypeptide
EC 3.4.11.4 cleave off the amino-terminal end from a tripeptide

More enzymes and substrates → faster metabolism

Enzymes are catalysts of biochemical reactions and their activity is directly related to substrates available. More enzymes and substrates means faster metabolism. Enzymes have very complex and descriptive names, but for simplicity sake I will refer to them by their Enzyme Commission numbers otherwise known as EC numbers. For example enzymes with top level EC group 3 are hydrolases, which means they use water to break up molecules. If that three is followed by the group 4 it means that the enzymes are hydrolases that act on peptide bonds. Followed by 11 means it's hydrolases that cleave off the amino-terminal end from a polypeptide. Finally, if it ends in a 4, it means it cleaves off the amino terminal end from a tripeptide, specifically.

http://www.dynamicscience.com.au/tester/solutions1/biology/enzrates.html

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Metabolic Pathway Activity

Pathway activity levels can be estimated using

RNA-seq reads
Gene expression data
Enzyme activity

Different metabolic pathways aren’t always unique

Overlap – they have genes in common
May use same enzymes

Common enzymes may not perform same functions

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Problem formulation

Given RNA-seq reads from a biological sample
Estimate

Gene expression
Enzyme expression
Metabolic pathway activity level

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Challenges

Estimating pathway activity poses challenges

Genes in common → issues distinguishing between enzymes
Enzymes are orthologs

Groups of enzymes based on the sequence homology, not on the reactions they catalyze�

Sample-specific challenge:

Contigs can participate in two different enzymes
Enzymes are distinguished by any other contigs in the sample�

Results in groups of indistinguishable enzymes
Pathway activity estimation algorithm converges inconsistently

Here are a few challenges that I've encountered. First of all as I said previously, some metabolic pathways overlap which means they have genes in common which leads to issues distinguishing between enzymes. Additionally, some enzymes are orthologs. Orthologs in case of enzymes means groups that are based on sequence homology, and not on the reactions they catalyze. Which means if the same contiguous sequence or contig for short, participates in two different enzymes, they have to be distinguished by any other contigs available. Ortholog enzymes become indistinguishable, which leads to the pathway activity estimation algorithm to converge inconsistently.

https://genomebiology.biomedcentral.com/articles/10.1186/gb-2002-3-3-preprint0002

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4908344/

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Enzyme participation

All enzymes are important in order for metabolic pathway to continue the biochemical reactions
Some enzymes are more active than others

Different substrates available
Different temperatures

More activity → higher participation in pathway activity
Quantification of enzyme expression as well as enzyme participation in metabolic pathways can improve metabolic pathway activity estimate

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Contributions

Exploring static enzyme-in-metabolic-pathway participation coefficients
Developing Expectation-Maximization (EM) based method to infer non-static enzyme-in-metabolic-pathway participation coefficients
Integrating enzyme participation coefficients into the metabolic pathway activity estimation pipeline to improve estimation accuracy
Differential analysis of metabolic pathway activity from microbial communities
Downstream analysis of pathway activity for infected and uninfected wild and lab mice

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Publications

Pipeline for analyzing activity of metabolic pathways in planktonic communities using metatranscriptomic data�
F. Rondel, R. Hosseini, B. Sahoo, S. Knyazev, I. Mandric, F. Stewart, I. I. M ̆andoiu, B. Pasaniuc, A. Zelikovsky, ”Estimating Enzyme Participation in Metabolic Pathways for Microbial Communities from RNA-Seq Data,” Proc. of International Symposium on Bioinformatics Research Applications (ISBRA), 2020, Lecture Notes in Bioinformatics 12304, 335-343�
Analyzing differential metabolic pathway activity levels using enzyme participation in metabolic pathways from RNA-seq data of Mus musculus and Peromyscus leucopus (In works)

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Part II - Microbial community analysis

Introduction
Estimating enzyme participation and metabolic pathway activity for microbial communities

Previous work
EMPathways
EMs
Data
Results

Estimating enzyme participation and metabolic pathway activity for wild and lab mice
Conclusions

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Previous work

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EMPathways

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First and Second EM

Expectation maximization algorithm is an iterative method to find maximum likelihood estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation or E step which creates a function for the expectation of the log likelihood evaluated using the current estimate for the parameters and the maximization step or M step which computes parameters maximizing the expected log likelihood found on the E step. These estimates are in turn used to determine the distribution of the latent variables in the next E step, and so on until the difference, epsilon, between the previous step and the current is sufficiently small, which is when it converges. During the first EM the contig to enzyme correspondence is computed where contigs and enzymes are nodes and their correspondences are directed weighted edges. Once the expressions of enzymes are computed, second EM does the same thing, except this time the enzymes and metabolic pathways are nodes and are linked by directed weighted edges.

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EM for enzyme participation (3rd EM)

Calculating and normalizing weights on edges

Enzymes -to-pathway participation ← 1/pathway size

Weights are calculated according to E and M step:

E-step��
M-step

However, we discovered two EMs just weren't enough. Using a static uniform enzyme participation across all enzymes showed some promise, but not much. So we decided to use another expectation maximization based step to evaluate each enzymes participation coefficients for every metabolic pathway. The weights on enzyme to pathway edges are initialized as 1 over the pathway size, which is a number of enzymes the pathway is known to include. Then the weights are calculated continuously according to E and M steps until convergence. First part of E step is finding the expected enzyme coefficients by initializing them as 1 over the size of pathway, then calculating the product of the enzyme participation coefficient and the quotient of current enzyme expressions and expected enzyme expressions, latter being the sum of products of metabolic pathway expressions and 1 over the pathway size.

The M step then calculates new estimations for each enzyme in pathway participation coefficient by normalizations for each enzyme and pathway.

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Third EM

Initialize with uniform participation
EM 3: Enzyme expression + Pathway activity level → enzyme-in-pathway participation
Update enzyme-in-pathway participation
EM2: Enzymes → Pathways
Repeat EM3-EM2 until convergence �

RESULT: more accurate estimation of

Enzyme-in-pathway participation
Metabolic pathway activity

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Datasets

26 samples taken over course of 2 days in Gulf of Mexico

2 Depths, 2 meters and 18 meters
13 samples per depth
Every 4 hours

Samples RNA was extracted and sequenced
RNA-seq data processed through EMPathways pipeline
Output of EMPathways is a list of values:

Expression of each metabolic pathway in time series
Output is processed further to receive a list of upregulated metabolic pathways

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Datasets

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Sample data

RNA-seq data

Aligned and assembled
181 metabolic pathways in time series, 13 time points, 2 depths
Normalized values that represent metabolic pathway expression

For each time-point there are 6 conditions:

PAR - photosynthetically active radiation
pTemp - potential temperature (C °)
Density - water density
Oxygen - amount of oxygen in water
PSU - practical salinity unit, in other words - salinity of water
Chl - Chlorophyll, levels of algae in water in ug/L

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Challenge - different EM converging points

Run EM over the same sample 5 times:

Enzymes with stable expression
Enzyme with non stable expression

Same reads → hard to distinguish two enzymes
Find the group indistinguishable enzymes for editing the dictionary:

Collapse the enzymes into a group
Update the dictionary with updated enzymes

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Finding pairs

Given: multiple enzyme expression estimates
Find: enzyme groups/pairs
Algorithm:

Remove stable enzymes
Unstable enzymes:

Find difference between the frequencies of two samples
Same difference over multiple samples → enzyme pair
Belong to the same orthology

There are some exceptions

A

B

B - A

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Exceptions in finding pairs

Some enzymes group up with multiple different enzymes groups
For example: EC:4.2.1.51

is paired with EC:4.2.1.91
is paired with EC:5.4.99.5

EC:4.2.1.51 shares individual orthologies with one and another
All three do not share a common orthology group

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Exceptions in finding pairs

There are triplets and quadruplets

The sum adds up to the same value over the 13 samples

Triplets and quadruplets → large orthology groups

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Results

Measured the performance of the model with fault discovery rate.

The FDR conceptualizes the rate of type I errors

Generated 200 random permutations of conditions
Multiple regression with each of the 200 permutations
Marginal regression with 200 condition permutations
95% Confidence interval

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Correlation of pathways activity level with environmental parameters (old pipeline)

	Salinity	Temp	Oxygen	Chl	PAR	Density	MLR
1. # of significantly correlated pathways	8	14	5	8	1	4	10
2. 95% randomized CI	1-10	1-11	1-8	0-7	0-6	1-8	0-8
3. The most correlated pathway	ec00364	ec00310	ec00281	ec00281	ec00740	ec00623	ec00623

In this table is a correlation of pathways activity level with environmental parameters that was produced using the old pipeline without enzyme participation coefficients. In line one is a number of significantly correlated pathways with every environmental parameter. In line 2 is the 95% randomized confidence interval for the significantly correlated pathways with each randomly permuted environmental parameter and then line 3 are the most correlated pathways with each environmental parameter. Salinity, oxygenation, photosynthetically active radiation, as well as density are well within the interval, which just essentially means that randomizing the environmental parameters 200 times in some cases may result in higher correlation. The number of metabolic pathway activity levels significantly correlated with chlorophyll levels is eight which is only one more than the upper bound of the 95% randomized confidence interval, which isn't very confidence inspiring. The multiple regression column is last, which is a regression of all environmental parameters onto each metabolic pathway activities. And as you can probably tell is also not great. Finally the number of pathway activities significantly correlated with temperature is 14 which is three more the randomized confidence interval, which is the best we got, and yet still not very exciting.

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Correlation of pathways activity level with environmental parameters (new pipeline)

	Salinity	Temp	Oxygen	Chl	PAR	Density	MLR
1. # of significantly correlated pathways	31	22	19	18	14	30	22
2. 95% randomized CI	1-8	0-8	0-6	0-6	0-6	1-8	0-7
3. The most correlated pathway	ec00071	ec00195	ec00622	ec00460	ec00360	ec00071	ec00626

Now let's look at the exact same table generated by the updated pipeline that utilized enzyme participation coefficients to improve metabolic pathway activity levels estimation. The improved accuracy of metabolic pathway estimation has resulted in much higher number of metabolic pathway activity levels that correlate significantly with all environmental parameters. Not a single number of significantly correlated pathways is even close to the upper bound of the 95% randomized confidence interval. Salinity levels significantly correlated with 31 metabolic pathway activity levels in time series across 13 time points. Temperature changes significantly correlated with 22 metabolic pathways. Oxygen with 19. Chlorophyll concentrations with 18. Water density correlated with a whole 30, and last but not least, a multiple linear regression analysis was able to correlate 22 metabolic pathway variations to changes in 6 environmental parameters together.

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Enzyme-in-Pathway Participation

ec00020	D1: 12	D1: 16	D1: 20	D2: 00	D2: 04	D2: 08	D2: 12	D2: 16	D3: 00	D3: 04	D3: 12	AVE	STD
EC:1.2.4.1	12.82	21.68	20.64	33.71	35.76	30.38	21.78	23.71	32.4	28.07	21.98	25.72	6.6
EC:1.2.7.1	0.51	6.18	15.43	6.69	4.97	9.32	13.14	9.61	7.87	12.95	2.54	8.11	4.37
EC:1.2.7.3	13.99	21.46	20.32	26.74	28.96	24.87	21.26	22.22	27.08	24.44	26.7	23.46	4.02
EC:1.8.1.4	7.61	12.92	11.24	16.94	16.65	14.39	12.93	16.92	19.16	14.03	22.16	15	3.78
EC:2.3.1.12	12.82	21.68	20.64	33.71	35.76	30.38	21.78	23.71	32.4	28.07	21.98	25.72	6.6
EC:4.1.1.32	12.82	21.68	20.64	33.71	35.76	30.38	21.78	23.71	32.4	28.07	21.98	25.72	6.6
EC:4.1.1.49	14.78	23.66	23.38	32.19	36.13	37.34	26.62	28.41	35.9	33.66	25.61	28.88	6.6
EC:1.1.1.37	18.14	19.76	26.62	17.9	18.93	30.78	20.27	20.43	22.97	22.13	44.21	23.83	7.43
EC:1.1.1.41	72.88	72.85	70.78	71.2	68.42	38.66	45.68	60.11	62.77	61.29	27.09	59.25	14.74
EC:1.1.1.42	19.96	24.06	22.58	21.52	23.68	19.95	22.48	22.32	22.95	21.92	42.38	23.98	5.95
EC:1.1.5.4	0	0	0	29.35	0	0	0	20.53	0	0	0	24.94	4.41
EC:1.2.4.2	10.1	13.02	10.76	11.91	10.91	11.72	12.75	14.08	14.74	10.13	25.75	13.26	4.21
EC:1.3.5.1	21.35	27.74	28.74	34.65	39.51	30.74	29.4	29.56	36.38	33.32	46.73	32.56	6.43
EC:2.3.1.61	10.1	13.02	10.76	11.91	10.91	11.72	12.75	14.08	14.74	10.13	25.75	13.26	4.21
EC:2.3.3.1	86.31	41.26	66.16	28.14	39.2	260.4	209	93.27	70.39	107.9	96.4	99.85	68.92
EC:2.3.3.8	19.96	24.06	22.58	21.52	23.68	19.95	22.48	22.32	22.95	21.92	42.38	23.98	5.95
EC:4.2.1.2	14.54	18.81	19.68	23.77	28	20.3	19.67	20.16	24.74	22.7	32.79	22.29	4.72
EC:4.2.1.3	33.31	29.83	34.13	23.43	28.96	41.1	44.43	37.46	35.39	38.11	69.02	37.74	11.35
EC:6.2.1.4	19.96	24.06	22.58	21.52	23.68	19.95	22.48	22.32	22.95	21.92	42.38	23.98	5.95
EC:6.4.1.1	14.54	18.81	19.68	23.77	28	20.3	19.67	20.16	24.74	22.7	32.79	22.29	4.72

Now for enzyme participation coefficients. In this table are enzyme in pathway participation coefficients for metabolic pathway ec20. For every data point in every pathway the participation level is estimated separately for every enzyme. In this table are participation levels of all expressed enzymes in pathway EC 20. The participation level of any enzyme does not significantly change from one datapoint to another meaning standard deviation is significantly smaller than the mean for all enzymes. If an enzyme is not expressed in a sample then the participation is not defined and the participation level is reported is zero. That means that the only data points that need to be taken into the account are the ones with non-zero participation levels when computing mean and standard deviation over all data points.

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Part III - Rodent community analysis

Introduction
Estimating enzyme participation and metabolic pathway activity for microbial communities
Estimating enzyme participation and metabolic pathway activity for wild and lab mice

EM models
Data sets
Results

Enzyme coefficients
Comparing group metabolic pathway expression
Pathway activity table

Conclusions

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Initial pipeline

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EMPathways pipeline

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Data Sets

RNA extracted from 4 groups of rodents
RNA-seq reads
IsoEM2

Transcript expressions

Transcript gene mapping
Mapped to gene, enzyme, pathways for further analysis
4 groups of mice

3 infected and 3 uninfected

Mus musculus
Peromyscus leucopus

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Enzyme coefficients

ec00062	WS_1	WS_2	WS_3	Average	StDev	WH_1	WH_2	WH_3	Average	StDev
EC:1.1.1.211	0.08483	0.08255	0.08154	0.08297	0.00169	0.07992	0.07949	0.07321	0.07754	0.00376
EC:4.2.1.134	0.09528	0.09273	0.09386	0.09396	0.00128	0.09587	0.08763	0.08563	0.08971	0.00543
EC:1.1.1.35	0.03545	0.03325	0.03408	0.03426	0.00111	0.03396	0.03003	0.02923	0.03107	0.00253
EC:1.3.1.93	0.09528	0.09273	0.09386	0.09396	0.00128	0.09587	0.08763	0.08563	0.08971	0.00543
EC:2.3.1.16	0.07277	0.06689	0.06721	0.06896	0.00331	0.06804	0.06149	0.06175	0.06376	0.00371
EC:2.3.1.199	0.09528	0.09273	0.09386	0.09396	0.00128	0.09587	0.08763	0.08563	0.08971	0.00543
EC:3.1.2.22	0.11754	0.12052	0.11921	0.11909	0.00149	0.11801	0.12478	0.12822	0.12367	0.00519
EC:1.3.1.38	0.11754	0.12052	0.11921	0.11909	0.00149	0.11801	0.12478	0.12822	0.12367	0.00519
ec00062	LS_1	LS_2	LS_3	Average	StDev	LH_1	LH_2	LH_3	Average	StDev
EC:1.1.1.211	0.07611	0.07564	0.07962	0.07712	0.00217	0.07214	0.07137	0.06937	0.07096	0.00143
EC:4.2.1.134	0.08968	0.09392	0.08879	0.09080	0.00274	0.08621	0.08510	0.08544	0.08558	0.00057
EC:1.1.1.35	0.02749	0.02697	0.02646	0.02697	0.00051	0.02663	0.02661	0.02736	0.02686	0.00043
EC:1.3.1.93	0.08968	0.09392	0.08879	0.09080	0.00274	0.08621	0.08510	0.08544	0.08558	0.00057
EC:2.3.1.16	0.06534	0.06643	0.06407	0.06528	0.00118	0.06752	0.06742	0.06776	0.06757	0.00018
EC:2.3.1.199	0.08968	0.09392	0.08879	0.09080	0.00274	0.08621	0.08510	0.08544	0.08558	0.00057
EC:3.1.2.22	0.12275	0.12035	0.12346	0.12219	0.00163	0.12638	0.12688	0.12736	0.12687	0.00049
EC:1.3.1.38	0.12275	0.12035	0.12346	0.12219	0.00163	0.12638	0.12688	0.12736	0.12687	0.00049

WS / WH - wild sick / wild healthy

LS / LH - lab sick / lab healthy

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Results

Clear differences between group metabolic pathway expressions

Averages of infected and uninfected groups show a significant difference

ec00053, ec00280, ec00500

Ascorbate and aldarate metabolism
Valine, leucine and isoleucine degradation
Starch and sucrose metabolism�

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Differentially expressed pathway ec00053

Ascorbate and aldarate metabolism

	Healthy	Healthy	Healthy	Average	STDev	Sick	Sick	Sick	Average	STDev	Average	STDev
Wild ec00053	11.322	11.946	12.896	12.055	0.793	33.636	12.688	11.848	19.391	12.344	15.723	8.168
Lab ec00053	45.066	44.674	34.24	41.327	6.140	30.887	34.548	22.335	29.257	6.268	35.292	0.090

For ec00053:

Decreased expression for sick lab mice while (almost) no change for wild mice
Increased expression in lab mice comparatively with wild mice

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Differentially expressed pathway ec00280�Valine, leucine and isoleucine degradation

	Healthy	Healthy	Healthy	Average	STDev	Sick	Sick	Sick	Average	STDev	Average	STDev
Wild ec00280	176.159	180.124	171.589	175.957	4.271	176.788	177.81	179.145	177.914	1.182	176.936	2.184
Lab ec00280	136.395	135.017	135.33	135.581	0.722	163.634	163.142	167.456	164.744	2.362	150.162	1.159

For ec00280:

Elevated expression for sick lab mice while no change for wild mice
Reduction of expression in lab mice comparatively with wild mice

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Differentially expressed pathway ec00500

Starch and sucrose metabolism

	Healthy	Healthy	Healthy	Average	STDev	Sick	Sick	Sick	Average	STDev	Average	STDev
Wild ec00500	157.897	159.69	160.486	159.358	1.326	114.175	153.072	155.825	141.024	23.293	150.191	15.533
Lab ec00500	145.837	129.389	138.745	137.990	8.250	89.675	133.198	87.802	103.558	25.686	120.774	12.329

For ec00500:

Reduction of expression for sick lab mice while (mostly) no change for wild mice
Reduction of expression in lab mice comparatively with wild mice

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Part IV - Conclusions

Introduction
Estimating enzyme participation and metabolic pathway activity for microbial communities
Estimating enzyme participation and metabolic pathway activity for wild and lab mice
Conclusions

Future work
Acknowledgements

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Conclusions

EM approach works:

More conditions can be correlated with the improved metabolic pathway expression of microbial community

3rd EM estimating enzyme-in-pathway participation improves the metabolic pathway expression evaluation of microbial and mice communities
Pathway activity correlates significantly with environmental parameters for microbial communities
Enzyme participation in pathways can be measured, but does not vary significantly across metabolic pathways

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Future work

Analysis of mice metabolic pathways and enzyme activity and participation
Differential analysis of enzyme and pathway activity between M. musculus and P. Leucopus

Are there any pathways that change in wild mice from healthy to sick while being stable in lab mice?

Comparing stability of enzyme participation coefficients across microbial communities and mice groups

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Acknowledgements

CS Department

Alex Zelikovsky
Murray Patterson
Roya Hosseini
Sergey Knyazev
Babatunde Bello

School of Medicine

Bogdan Pasaniuc

CS Department

Ion Mandoiu

Department of Microbiology

Frank Stewart

Igor Mandric

College of Veterinary Medicine and Biomedical Sciences

Artem Rogovskyy