Download Supplementary Materials Metabolic Flux Determination in Perfused

Supplementary Materials
Metabolic Flux Determination in Perfused Livers by Mass Balance Analysis: Effect of
Fasting
Mehmet A. Orman1
Kosuke Arai3
Martin L. Yarmush2,3
Ioannis P. Androulakis1,2
Francois Berthiaume2,*
Marianthi G. Ierapetritou1,*
1: Department of Chemical and Biochemical Engineering, Rutgers, The State University of New
Jersey, Piscataway, NJ 08854, USA
2: Department of Biomedical Engineering, Rutgers, The State University of New Jersey,
Piscataway, NJ 08854, USA
3: Center for Engineering in Medicine/Surgical Services, Massachusetts General
Hospital, Harvard Medical School, and the Shriners Hospitals for Children, Boston, MA 02114,
USA
*: Corresponding authors. Emails: [email protected], [email protected]
1
A. Stoichiometry of Hepatic Metabolic Network
The liver metabolic network involving all possible major liver-specific pathways such as
gluconeogenesis, glycolysis, urea cycle, fatty acid metabolism, pentose phosphate pathway, TCA
cycle, glycogen metabolism and amino acid metabolism is given in Table SI. For more detailed
explanations about the enzymes and the number of steps in each reaction, see references (Banta
et al. 2007; Chan et al. 2003; Lee et al. 2000; Lee et al. 2003).
B. Mathematical Analysis
Solution Space of Steady State Fluxes
The flux distribution is calculated by using the mass balances of internal metabolites (Varma and
Palsson 1994). It is generally assumed that the internal metabolites are at pseudo steady state,
since metabolic transients are fast compared to environmental changes (Varma and Palsson
1994). Therefore, the mass balance is rewritten as follows:
S .v  0
vi  0
i  irreversible reactions
(1)
where v is the flux distribution vector, and S is the stoichiometric matrix. One of the common
problems in solving system (1) to obtain unknown fluxes is that the rank of S is generally smaller
than the number of unknown fluxes which leads to an underdetermined system. To overcome
this limitation, a range of possible values for each flux is determined using linear programming
where each flux is maximized or minimized while allowing all other flux values to vary (Burgard
et al. 2001; Llaneras and Picó 2007; Mahadevan and Schilling 2003). To account for
experimental errors, the equation is modified such that each of the measured fluxes allow to very
between (vmin and vmax) as follows:
2
Maximize / Minimize v j ,
j  unknown fluxes
Subject to S .v  0
vmmin  vm  vmmax ,
vi  0,
m  measured fluxes
(2)
i  irreversible fluxes
Problem (2) is solved 2*nu times where nu is total number of unknown fluxes and each time, a
flux distribution vector is obtained.
Monte Carlo Sampling
Calculating the range of each flux using problem (2), the solution space of steady state fluxes can
be described as follows:
S .v  0
v min  v  v max , v  all fluxes
(3)
To determine a representative sample of flux distributions of the solution space (Equation 3),
Monte Carlo Sampling is used. We adopted an algorithm provided by Wiback and her coworkers
(Wiback et al. 2004) and modified to add constraints given in equation (3). This method has
been widely used to analyze metabolic networks and its applications were recently reviewed by
Schellenberger and Palsson (Schellenberger and Palsson 2009). The algorithm is straightforward.
After assigning uniformly random values for independent variables within the known ranges, the
dependent variables are linearly calculated using the stoichiometric matrix. The results are
random flux distributions. Back-calculating the dependent flux values can result in invalid flux
distributions (Wiback et al. 2004) since the solution space obtained from sampling analysis can
be larger than the actual one defined by equation (3) (see Figure S1) due to the fact that only the
independent fluxes are bounded in sampling analysis. The valid flux distribution is one that
meets the “range criteria” given in equation (3). In this study, the number of independent
3
variables is 31 since we use a system with 83 fluxes (or variables) and a stoichiometric matrix
having a rank of 52. The steady state flux space obtained from the sampling analysis is larger
than the actual one which is shown in Figure S2 and the valid flux distributions are the ones in
the actual solution space.
Singular Value Decomposition
A matrix including all valid steady state flux distributions obtained from Monte Carlo sampling
is constructed (matrix A). To identify the basic trends in the matrix A, SVD analysis is
performed. SVD analysis decomposes the flux distribution matrix into three matrices (Henry and
Hofrichter 1992; Klema and Laub 1980):
A  U V T
(4)
where U and V are orthonormal matrices with columns corresponding to left and right singular
vectors of A, respectively; and Σ is a diagonal matrix containing the singular values (σi) of
matrix A, arranged in descending order. The magnitude of the singular values in Σ indicates the
relative contribution of the singular vectors in reconstructing the matrix A. U is a combination of
the orthonormal bases of the column space and left null space of A. Therefore, in this analysis,
the matrix U is further investigated, since each column of U represents a dominant pattern in the
flux vector solution space.
To determine the relative contribution of each column of U in reconstructing the matrix A, the
following weight value is used:
wi 
i
 i
i  1,..., ne
(5)
i
where wi is the weight of eigenvector i, σi is the corresponding singular value, and ne is the
number of nonzero singular values or number of eigenvectors.
4
The first eigenvector corresponds to a valid flux distribution since it satisfies both mass and
reversibility constraints given in Equation (1). In a solution cone described by a set of vectors,
the first eigenvector comes up through the geometric center of the cone, and is close to the part
of the cone where the vectors are densely accumulated (Price et al. 2003). Moreover, the first
eigenvector contributing most in reconstructing the matrix of flux vectors also gives the
dominant flux-pattern. Subsequent eigenvectors define the direction of most variance in the
subspace. Although all subsequent eigenvectors obey the mass balance constraints, they do not
necessarily satisfy reaction directionality constraints. Each subsequent vector can identify key
network branch points or tradeoffs in the network and represent how tradeoff points influence the
flux distribution. A more detailed analysis about the physiological meaning of eigenvectors can
be found in reference (Price et al. 2003).
To simplify the interpretation of initial eigenvectors that capture most of the variation in the
matrix of flux samples, basis rotation method can be applied to these eigenvectors (Barrett et al.
2009). They are rotated using the orthogonal varimax rotation procedure (Kaiser 1958) where
the variance or sum of the squared vector loadings or coefficients across the column is
maximized.
The first 5 eigenvectors and their fractional contributions obtained from different number of
samples (104 and 105) are given in Table SII.
Statistics
Comparisons were performed using a 2-tailed Student’s t-test, and p<0.05 was the criterion used
for statistical significance.
5
Note:
In this work, all optimization problems are solved using MATLAB (Mathworks Inc,
Massachusetts) and GAMS with the interface program MAT-GAMS in a Pentium(R) processor
at 2.80 GHz with 1.00 GB of memory.
Tables
Table SI. Hepatic Network Model.
Table SII. The weight values of first five eigenvectors.
Figure Legends
Figure S1. Monte Carlo sampling used to identify the flux vectors within the steady state
flux space. Larger square represents the flux space obtained from sampling method since only
independent fluxes are bounded. Smaller square (grey color) is actual flux space where all fluxes
are bounded. Each dot represents a random solution.
Figure S2. The steady state flux space of hepatic metabolism in fasted state (A) and fed sate
(B) obtained from the flux spectrum approach (grey space) and sampling analysis (space
surrounded by black lines).
6
References
Banta S, Vemula M, Yokoyama T, Jayaraman A, Berthiaume F, Yarmush ML. 2007.
Contribution of gene expression to metabolic fluxes in hypermetabolic livers induced
through burn injury and cecal ligation and puncture in rats. Biotechnology and
Bioengineering 97(1):118-137.
Barrett CL, Herrgard MJ, Palsson B. 2009. Decomposing complex reaction networks using
random sampling, principal component analysis and basis rotation. Bmc Systems Biology
3.
Burgard AP, Vaidyaraman S, Maranas CD. 2001. Minimal reaction sets for Escherichia coli
metabolism
under
different
growth
requirements
and
uptake
environments.
Biotechnology Progress 17(5):791-797.
Chan C, Berthiaume F, Lee K, Yarmush ML. 2003. Metabolic flux analysis of cultured
hepatocytes exposed to plasma. Biotechnology and Bioengineering 81(1):33-49.
Henry ER, Hofrichter J. 1992. Singular Value Decomposition - Application to Analysis of
Experimental-Data. Methods in Enzymology 210:129-192.
Kaiser H. 1958. The varimax criterion for analytic rotation in factor analysis. Psychometrika
23(3):187-200.
Klema V, Laub A. 1980. The singular value decomposition: Its computation and some
applications. Automatic Control, IEEE Transactions on 25(2):164-176.
Lee K, Berthiaume F, Stephanopoulos GN, Yarmush DM, Yarmush ML. 2000. Metabolic Flux
Analysis of Postburn Hepatic Hypermetabolism. Metabolic Engineering 2(4):312-327.
Lee K, Berthiaume F, Stephanopoulos GN, Yarmush ML. 2003. Profiling of dynamic changes in
hypermetabolic livers. Biotechnology and Bioengineering 83(4):400-415.
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Llaneras F, Picó J. 2007. An interval approach for dealing with flux distributions and elementary
modes activity patterns. Journal of Theoretical Biology 246(2):290-308.
Mahadevan R, Schilling CH. 2003. The effects of alternate optimal solutions in constraint-based
genome-scale metabolic models. Metabolic Engineering 5(4):264-276.
Price ND, Reed JL, Papin JA, Famili I, Palsson BO. 2003. Analysis of metabolic capabilities
using singular value decomposition of extreme pathway matrices. Biophysical Journal
84(2):794-804.
Schellenberger J, Palsson BO. 2009. Use of Randomized Sampling for Analysis of Metabolic
Networks. Journal of Biological Chemistry 284(9):5457-5461.
Varma A, Palsson BO. 1994. Metabolic Flux Balancing - Basic Concepts, Scientific and
Practical Use. Bio-Technology 12(10):994-998.
Wiback SJ, Famili I, Greenberg HJ, Palsson BO. 2004. Monte Carlo sampling can be used to
determine the size and shape of the steady-state flux space. Journal of Theoretical
Biology 228(4):437-447.
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Tables
Table SI. Hepatic Network Model.
Enzymes and explanations
Reaction No
Glycolytic and Gluconeogenic Pathways
Reaction 1a,b,*
Hexokinasea , Glucose-6-Paseb
Glucose + Pi <==> Glucose-6-P
Reaction 2a,b,*
Phosphoglucose isomerase a,b
Glucose-6-P <==> Fructose-6-P
a
b
Reaction 3 a,b,* PFK-1 , Fructose-1,6-Pase-1
Fructose-6-P + Pi <==> Fructose-1,6-P2
Triose P-isomerase, fructose biphosphate
Reaction 4 a,b,* aldolase
Fructose-1,6-P2 <==> 2 Glyceraldehyde-3-P
Glyceraldehyde-P dehydrogenase, 3phosphoglycerate kinase,
Reaction 5 a,b,* phosphoglyceromutase, enolase
Glyceraldehyde-3-P + Pi + NAD+ + ADP <==> ATP + NADH + PEP
Reaction 6 a
Pyruvate kinase
ADP + PEP ==> ATP + Pyruvate
Reaction 7 a
PDH
NAD+ + Pyruvate ==> CO2 + NADH + Acetyl-CoA
Reaction 8 b
PEPCK
Oxaloacetate + GTP ==> CO2 + PEP + GDP
Reaction 9 b
Pyruvate carboxylase
CO2 + ATP + Pyruvate ==> Pi + ADP + Oxaloacetate
Pentose Phosphate Pathway
Glucose-6-P dehydrogenase and 3 additional
Reaction 10
steps
Glucose-6-P + 12 NADP+ + 7 H2O ==> 6 CO2 + 12 NADPH + Pi + 12 H+
9
Lactic acid production
Reaction 11
Lactate dehydrogenase
NAD+ + Lactate <==> NADH + Pyruvate
TCA cycle
Reaction 12
Citrate synthase
Oxaloacetate + Acetyl-CoA ==> Citrate + CoA-SH
Reaction 13
Aconitase, isocitrate dehydrogenase
NAD+ + Citrate <==> CO2 + NADH + a-ketoglutarate
Reaction 14
a-ketoglutarate dehydrogenase
NAD+ + CoA-SH + a-ketoglutarate ==> CO2 + NADH + Succinyl-CoA
Succinyl-CoA synthase and succinate
Reaction 15
dehydrogenase
Pi + GDP + Succinyl-CoA + FAD <==> GTP + Fumarate + FADH2
Reaction 16
Fumarase
Fumarate <==> Malate
Reaction 17
Malate dehydrogenase
NAD+ + Malate <==> NADH + Oxaloacetate
Urea cycle
Reaction 18
Arginase
Arginine ==> Urea + Ornithine
Carbonate dehydratase, carbamoyl-P synthase,
Reaction 19
ornithine transcarbamylase
CO2 + 2 ATP + Ornithine + NH4+ <==> 2 Pi + 2 ADP + Citrulline
Argininosuccinate synthetase,
Reaction 20
argininosuccinase.
ATP + Citrulline + Aspartate ==> 2 Pi + Fumarate + Arginine + AMP
Amino acid metabolism
Reaction 21
Alanine aminotranferase
NAD+ + Alanine <==> NADH + Pyruvate + NH4+
10
Reaction 22
Serine dehydratase
Serine ==> Pyruvate + NH4+
Transaminase,
Reaction 23
3-mercaptopyruvate sulfurtransferase
NAD+ + Cysteine + H2SO3 <==> NADH + Pyruvate + NH4+ + H2S2O3
Reaction 24
Threonine 3-dehydrogenase, acetyl-CoA ligase NAD+ + CoA-SH + Threonine ==> NADH + Acetyl-CoA + Glycine
Glycine hydroxymethyltranferase, glycine
Reaction 25
cleavage system
NAD+ + 2 Glycine <==> CO2 + NADH + NH4+ + Serine
Methylmalonyl-CoA epimerase,
Reaction 26
Methylmalonyl-CoA mutase
S-Methylmalonyl-CoA <==> Succinyl-CoA
Reaction 27
Lysine metabolism (8 steps)
5 NAD+ + CoA-SH + FAD + Lysine ==> 2 CO2 + 5 NADH + FADH2 + 2 NH4+ + Acetoacetyl-CoA
Reaction 28
Phenylalanine hydroxylase
Phenylalanine + Tetrahydrobiopterin + O2 ==> Dihydrobiopterin + Tyrosine
Reaction 29
Tyrosine metabolism (5 steps)
NAD+ + 2 O2 + Tyrosine ==> CO2 + NADH + Fumarate + NH4+ + Acetoacetate
Reaction 30
Glutamate dehydrogenase
NAD+ + Glutamate <==> NADH + a-ketoglutarate + NH4+
Reaction 31
Glutaminase
Glutamine ==> NH4+ + Glutamate
Reaction 32
Ornithine metabolism (2 steps)
NADP+ + NAD+ + Ornithine ==> NADPH + NADH + NH4+ + Glutamate + H+
Proline oxidase, 1-pyrroline-5-carboxylate
Reaction 33
dehydrogenase
0.5 NADP+ + 0.5 NAD+ + 0.5 O2 + Proline ==> 0.5 NADPH + 0.5 NADH + Glutamate
Reaction 34
Histidine metabolism (4 steps)
Histidine + THF ==> NH4+ + Glutamate + 2-formimino-THF
Reaction 35
Methionine metabolism (5 steps)
2 ATP + NAD+ + CoA-SH + Serine + Methionine ==> CO2 + Pi + NADH + NH4+ + Cysteine + PPi + Adenosine + Propinoyl-CoA
Reaction 36
Propinoyl-CoA carboxylase
CO2 + ATP + Propinoyl-CoA ==> Pi + ADP + S-Methylmalonyl-CoA
11
Reaction 37
Aspartate aminotransferase
NAD+ + Aspartate <==> NADH + Oxaloacetate + NH4+
Reaction 38
Asparaginase
Asparagine ==> NH4+ + Aspartate
Valine metabolism (7 steps)
0.5 NADP+ + 3.5 NAD+ + FAD + 2 H2O + valine ==> 2 CO2 + 0.5 NADPH + 3.5 NADH + FADH2 + NH4+ + Propinoyl-CoA + 3
H+
Reaction 39
Isoleucine Metabolism (6 steps)
0.5 NADP+ + 2.5 NAD+ + FAD + 2 H2O + isoleucine ==> CO2 + 0.5 NADPH + 2.5 NADH + Acetyl-CoA + FADH2 + NH4+ +
Propinoyl-CoA + 3 H+
Reaction 40
Leucine Metabolism (6 steps)
0.5 NADP+ + ATP + 1.5 NAD+ + FAD + H2O + leucine ==> 0.5 NADPH + Pi + ADP + 1.5 NADH + Acetyl-CoA + FADH2 + NH4+
+ 2 H+ + Acetoacetate
Reaction 41
Fatty acid metabolism
Reaction 42
Hepatic Lipase
Palmitoylglycerol + ATP + NAD+ <==> Pi + 3 Palmitate + ADP + NADH + glycerol
Reaction 43
Glycerol-3-P dehydrogenase
NAD+ + glycerol <==> Glyceraldehyde-3-P + NADH
Reaction 44
Fatty acid oxidation (7x4 steps)
ATP + 7 NAD+ + Palmitate + 8 CoA-SH + 7 FAD ==> 2 Pi + 7 NADH + 8 Acetyl-CoA + 7 FADH2 + AMP
Reaction 45
Fatty acid synthesis (7x4 steps)
14 NADPH + 7 ATP + 8 Acetyl-CoA + 14 H+ ==> 14 NADP+ + 7 Pi + Palmitate + 7 ADP + 6 H2O
Reaction 46
Thiolase (Ketogenesis)
2 Acetyl-CoA <==> 2 CoA-SH + Acetoacetyl-CoA
Reaction 47
HMG-CoA synthase and lyase (Ketogenesis) Acetoacetyl-CoA ==> CoA-SH + Acetoacetate
Reaction 48
B-OH-butyrate dehydrogenase (Ketogenesis)
NADH + Acetoacetate <==> NAD+ + B-OH-butyrate
Electron-chain reactions
Reaction 49
Electron transport system
3 Pi + 3 ADP + NADH + 0.5 O2 + H+ ==> 3 ATP + NAD+ + H2O
Reaction 50
Electron transport system
2 Pi + 2 ADP + FADH2 + 0.5 O2 ==> 2 ATP + FAD + H2O
12
Protein synthesis or degradation
Reaction 51**
Albumin synthesis or degradation
Glycogenesis/Glycogenolysis
Reaction 52
Glycogen metabolism (3 steps)
glycogen <==> Glucose-6-P
External Reactions
Reaction 53
glycogen <==>
Reaction 54
Palmitoylglycerol <==>
Reaction 55
O2 <==>
Reaction 56
CO2 <==>
Reaction 57
albumin <==>
Reaction 58
Glucose <==>
Reaction 59
Lactate <==>
Reaction 60
Urea <==>
Reaction 61
NH4+ <==>
Reaction 62
Aspartate <==>
Reaction 63
Glutamate <==>
Reaction 64
Serine <==>
13
Reaction 65
Asparagine <==>
Reaction 66
Glycine <==>
Reaction 67
Glutamine <==>
Reaction 68
Histidine <==>
Reaction 69
Threonine <==>
Reaction 70
Arginine <==>
Reaction 71
Alanine <==>
Reaction 72
Proline <==>
Reaction 73
Tyrosine <==>
Reaction 74
Cysteine <==>
Reaction 75
valine <==>
Reaction 76
Methionine <==>
Reaction 77
isoleucine <==>
Reaction 78
Ornithine <==>
Reaction 79
leucine <==>
Reaction 80
Lysine <==>
Reaction 81
Phenylalanine <==>
Reaction 82
Acetoacetate <==>
14
Reaction 83
B-OH-butyrate <==>
Notes:
a or b stands for a reaction in glycolysis or gluconeogenesis, respectively.
* These reactions indicate futile cycle (reversible reaction) composed of two different reactions, one of which is from glycolytic pathway and the other from
gluconeogenic pathway.
** Both protein synthesis and degradation are given in one reaction, which results in reversible reaction.
15
Table SII. The weight values of first five eigenvectors.
N=104
N=105
Weights
Fasted
Fed
Fasted
Fed
w1
0.724645
0.723939
0.653732
0.654452
w2
0.085830
0.085985
0.118614
0.118727
w3
0.069688
0.069972
0.074555
0.073390
w4
0.049681
0.049363
0.048616
0.048944
w5
0.037997
0.038441
0.034547
0.034627
∑
0.967841
0.9677
0.930064
0.93014
Notes:
N is the number of flux samples.
∑ implies fractional contribution of first five eigenvectors to the description of the matrix of flux vector.
16
Figure S1.
17
Figure S2.
18
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