Modelling of Metabolic Processes: Bridging the Gap between Data and

Process Understanding

Aleš BeličLaboratory for modelling, simulation and control

Faculty of Electrical EngineeringUniversity of Ljubljana

Since we became aware of ourselves, we try to describe our functioning in abstract terms in order to answer the basic questions of our existence ...

The Hitchhikers Guide to the Galaxy has this to say on the topic

• ...• First, they built a computer named Deep Thought to calculate

the answer to the ultimate question of life, universe and everything.

• After 7.5 M years of computing Deep Thought announced the answer: forty-two

• Next, Deep Thought constructed another computer, to calculate what was actually the question.

• The computer was named Earth and the program should run for 10 M years, but the Earth was unfortunately destroyed 5 min before the program ended to make space for the galactic hyperspace by-pass.

• ... • Douglas Adams understood modelling better than many of the

real-life modellers

Aleš Belič 4

Understanding how human works• Motivation

– advantages in survival• knowing your limits• healing• making of tools• ...

• Methods (chronologically)– holistic approaches (psychology, body, effects of chemical substances, ...)– invention of writing enables faster gathering of knowledge and people are no

longer able to maintain the overview• development of special areas of science that cover only specific problems

– introduction of mathematics and engineering in some areas– universality of mathematical notation could again lead to systemic overview

• systems biology• systems medicine

Holistic approach!!

All processes in an organism are always connected in integral system that enables optimal functioning. When only a part of the system is damaged it causes global adaptation of the system.• Feedback loops prevent simple detection of the disturbance origin

– feedback loops are hierarchically nested with aim to maintain the most vital functions at all costs, therefore, the phenomena may be detected far away from the real origin.

– dynamic nature of the processes!!!• Without understanding of the integral system we cannot correctly

understand the functioning of the sub-systems• Modelling and simulation can be efficiently applied

Metabolic networks

Monitoring of metabolic processes

• Important for healthcare!• Metabolites

– chromatography, mass spectrometry,– tedious work– mostly static data– metabolic fluxes

• Proteins/Enzymes– adapted methods of chromatography and mass spectrometry– tedious, expensive, poor precision (concentration ≠ activity)– indirect measurement of activity– static data, if any at all ...

• Genome/expression– large selection of methods– static and occasionally dynamic data

Modelling of metabolic processes• Enzyme reactions

– Michaelis-Menten (1916)– still basic equations for describing enzyme reactions dynamics

• Gene expression– transcription of mRNA from DNA

• Composition of enzymes– proteins with special characteristics– mRNA is translated into amino acids and they are combined into protein

• Disturbed balance between concentrations of the key molecules can reduce or elevate:– gene expression– enzyme stability/activity

• All the molecules have limited stability so they are subject to constant decay– metabolic processes must have some non-zero steady state flux

Michaelis-Menten model

ES P EES+ +kf

kr

kcat

𝑑 [𝑃 ]𝑑𝑡 =

𝑉𝑚[𝑆 ]𝐾𝑚+[𝑆 ]

;𝐾𝑚=𝑘𝑟+𝑘𝑐𝑎𝑡

𝑘 𝑓;V m=k cat [E ]0

ES P EES+ +kf1

kr1

kf2

kr2

reversible reactions

metabolic process

ES P EES+ +kC

kCR

kP

kPR

M-M model in metabolic conditions

𝑑 [𝑆 ]𝑑𝑡 =Φ𝑖𝑛−𝑘𝐶 [𝐸 ] [𝑆 ]+𝑘𝐶𝑅 [𝐸𝑆 ]

𝑑 [𝐸𝑆 ]𝑑𝑡 =𝑘𝐶 [𝐸 ] [𝑆 ]+𝑘𝑃𝑅 [𝐸 ] [𝑃 ]−𝑘𝑃 [𝐸𝑆 ]−𝑘𝐶𝑅 [𝐸𝑆 ]

𝑑 [𝑃 ]𝑑𝑡 =𝑘𝑃 [𝐸𝑆 ]−Φ𝑜𝑢𝑡−𝑘𝑃𝑅 [𝐸 ] [𝑃 ]

𝑑 [𝐸 ]𝑑𝑡 =Φ𝐸𝑖𝑛+𝑘𝑃 [𝐸𝑆 ]+𝑘𝐶𝑅 [𝐸𝑆 ]−𝑘𝐶 [𝐸 ] [𝑆 ]−𝑘𝑃𝑅 [𝐸 ] [𝑃 ]−Φ𝐸𝑜𝑢𝑡

S ES PΦ𝑖𝑛 Φ𝑜𝑢𝑡𝑘𝐶 𝑘𝑃𝑘𝐶𝑅 𝑘𝑃𝑅

EΦ𝐸𝑖𝑛 Φ𝐸𝑜𝑢𝑡

Steady-state

0=Φ𝑖𝑛−𝑘𝐶 [𝐸 ] [𝑆 ]+𝑘𝐶𝑅 [𝐸𝑆 ]

0=𝑘𝐶 [𝐸 ] [𝑆 ]+𝑘𝑃𝑅 [𝐸 ] [𝑃 ]−𝑘𝑃 [𝐸𝑆 ]−𝑘𝐶𝑅 [𝐸𝑆 ]

0=𝑘𝑃 [𝐸𝑆 ]−Φ𝑜𝑢𝑡 −𝑘𝑃𝑅 [𝐸 ] [𝑃 ]

0=Φ𝐸𝑖𝑛+𝑘𝑃 [𝐸𝑆 ]+𝑘𝐶𝑅 [𝐸𝑆 ]−𝑘𝐶 [𝐸 ] [𝑆 ]−𝑘𝑃𝑅 [𝐸 ] [𝑃 ]−Φ𝐸𝑜𝑢𝑡

Normalisation introduces relative concentration values

0=𝑘𝐶𝑁 [𝐸𝑁 ] [𝑆𝑁 ]+𝑘𝑃𝑅𝑁 [𝐸𝑁 ] [𝑃𝑁 ]−𝑘𝑃𝑁 [𝐸𝑆𝑁 ]−𝑘𝐶𝑅𝑁 [𝐸𝑆𝑁 ]

[𝑆𝑁 ]= [𝑆][𝑆𝑠𝑠 ]

; [𝐸𝑁 ]= [𝐸 ][𝐸𝑠𝑠]

; [𝑃𝑁 ]= [𝑃 ][𝑃𝑠𝑠 ]

; [𝐸 𝑆𝑁 ]= [𝐸𝑆 ][𝐸 𝑆𝑠𝑠]

Alternative introduction of parameters𝑘𝑃𝑅𝑁

𝑘𝐶𝑁=𝑟1

𝑘𝐶𝑅𝑁

𝑘𝑃𝑁=𝑟2

0=𝑘𝐶𝑁 [𝐸𝑁 ] [𝑆𝑁 ]+𝑘𝐶𝑁𝑟1 [𝐸𝑁 ] [𝑃𝑁 ]−𝑘𝑃𝑁 [𝐸𝑆𝑁 ]−𝑘𝑃𝑁𝑟 2 [𝐸𝑆𝑁 ]0=𝑘𝐶𝑁 [𝐸𝑁 ] ([𝑆𝑁 ]+𝑟1 [𝑃𝑁 ] )−𝑘𝑃𝑁 [𝐸𝑆𝑁 ] (1+𝑟2 )

0=𝑘𝐶𝑁 (1+𝑟1 )−𝑘𝑃𝑁 (1+𝑟 2 )Normalised concentrations are in steady-state = 1

Since high reversibility of reactions makes sense only in special cases we can assumethat r1 and r2 are small and equal

0=𝑘𝐶𝑁 (1+𝑟 )−𝑘𝑃𝑁 (1+𝑟 )0=𝑘𝐶𝑁 −𝑘𝑃𝑁

𝑘𝐶𝑁=𝑘𝑃𝑁

S ES PΦ𝑖𝑛 Φ𝑜𝑢𝑡𝑘𝐶𝑁 𝑘𝑃𝑁𝑘𝐶𝑅𝑁 𝑘𝑃𝑅𝑁

EΦ𝐸𝑖𝑛 Φ𝐸𝑜𝑢𝑡

Basic parameters as functions of new ones

𝑘𝐶𝑁=∅𝑖𝑛

(1−𝑟 )

𝑘𝑃𝑁=∅ 𝑖𝑛

(1−𝑟 )

𝑘𝐶𝑅𝑁=∅ 𝑖𝑛𝑟(1−𝑟 )

𝑘𝑃𝑅𝑁=∅ 𝑖𝑛𝑟

(1−𝑟 )

Substrate to product ratio in steady-state

[𝑆¿¿𝑁 ]=1−𝑟 2

[𝐸¿¿𝑁 ]+𝑟 2 [𝑃𝑁 ]¿¿

[𝑆¿¿𝑁 ][𝑃𝑁 ]

=1−𝑟 2

[𝐸¿¿ 𝑁 ][𝑃𝑁 ]+𝑟2¿¿

Description of the basic parameters with reversibility and metabolic fluxin steady-state enables studies of enzyme activity on

product and substrate concentrations in steady-state without knowing the realmetabolic flux through the system!!!!

Enzyme concentration affects the substrate to product ratio because of constant flux

=

A. Belič J. Ačimovič, A. Naik, M. Goličnik. Analysis of the steady-state relations and control-algorithmcharacterisation in a mathematical model of cholesterol biosynthesis. Simulation Modelling Practice and Theory 33 (2013) 18–27

Inspecificity of enzymes with respect to substrate

• Enzymes recognise 3D structures of substrates– distribution and atom types– vibrations of molecular structure of enzyme and substrate

• Different molecules may have the same key structures (domains)– enzymes may operate on more than one molecule!!!– extremely large interconnected metabolic networks!!!

• Domains and enzyme effect may be described by a binary code– presence or absence of:

• bonds (single, double, triple)• molecular groups or atoms (hydrogen, methyl group, amino group, ...)

– a relatively simple algorithm can be constructed for prediction of possible networks based on known enzyme-metabolite interactions!

Late part of the cholesterol biosynthesis metabolic network

A, Belič, D. Pompon, K. Monostory, D. Kelly, S. Kelly, D. Rozman. An algorithm for rapid computational construction of metabolic networks: A cholesterol biosynthesis example. Computers in Biology and Medicine 43 (2013) 471–480

Liver metabolism modelling

• Aim: better understanding of NAFLD (Non-Alcocholic Fatty Liver Disease)

• Framework: a network of enzyme reactions for transport and metabolism for control of:– body energy– basic metabolites (cholesterol, glucoze, ...)

• Enzyme activity is controlled by– enzyme stability– gene expression

• Chemical communication with other organs is important• M-M model of enzyme reactions• Simple piece-wise linear models for expression and stability• Static data

Model structure

Metabolic pathways:• glycolisis/gluconeogenesis• penthose phosphate pathway• synthesis of fatty acids/oxidative pathway• citric acid cycle• cholesterol metabolism • amino acids metabolism• chormonal regulation (insulin, glucagon)• adipokine (adiponectin, leptin) & citokine regulation (TNFa)• expression regulation with transcription factors (PPAR, LXR,

FXR, SREBP-1c,-2; FOXO1)• exchange between liver, blood flow, adipocites and periferl tissues

(LDL-R, CD36, itd.)

145 metabolites259 enzymes60 proteins

Cholesterol biosynthesis

• One of the most important pathways in the liver– present in all types of cells– the liver covers most of the body requirements for cholesterol

• growth• tissue repair• ...

Cholesterol biosynthesis

Detailed analysis of cholesterol biosynthesis

• Interesting experimental and simulation results:– exogenous substances that influence cholesterol synthesis can either

reduce cholesterol concentration to zero or have no effect on the concentration while gene expression is altered in both cases

• Modelling purpose: – to understand basic regulation structure of cholesterol biosynthesis

through SREBF2 transcription factor

Model of cholesterol biosynthesis path

DNASREBF2

The character of SREBF2 regulator

Simplifications

DNA

SREBF2

The flux through the pathway is regulated by all the enzymes simultaneously, therefore the simplification is sensible!!

Standard control scheme

regulatorUE

process

-

R Y

Description of the bio-controller

act. nonact. SREBF2

cholesterol

DNA

mRNA

E

controllerUE

E

U

From the scheme to the equations𝐸=𝑘𝑟 𝑃 −𝑀𝑃=𝑃𝑎+𝑃𝑖

act. nonact.

cholesterol

E

E

U

P

𝑑𝑃 𝑎

𝑑𝑡 =𝑘𝑎𝑐𝑡 𝐸+𝑘 𝑓 𝐺𝑒−𝑘𝑒𝑃𝑎

𝑑𝑃 𝑖

𝑑𝑡 =−𝑘𝑎𝑐𝑡𝐸

Ge

𝑑𝐺𝑒

𝑑𝑡 =𝑘𝑒𝑥𝑃𝑎−𝑘𝐺𝐺𝑒

𝑃𝑎=𝑈𝑃 𝑖≈const .

𝑑𝑈𝑑𝑡 =𝑘𝑎𝑐𝑡𝐸+𝑘𝑓 𝐺𝑒−𝑘𝑒𝑈

𝑑𝐺𝑒

𝑑𝑡 =𝑘𝑒𝑥𝑈 −𝑘𝐺𝐺𝑒

Evolving the equations𝑑𝑈𝑑𝑡 =𝑘𝑎𝑐𝑡𝐸+𝑘𝑓 𝐺𝑒−𝑘𝑒𝑈

𝑑𝐺𝑒

𝑑𝑡 =𝑘𝑒𝑥𝑈 −𝑘𝐺𝐺𝑒

act. nonact.

cholesterol

E

E

U

PGe

𝑠𝑈=𝑘𝑎𝑐𝑡 𝐸+𝑘 𝑓 𝐺𝑒−𝑘𝑒𝑈𝑠𝐺𝑒=𝑘𝑒𝑥𝑈−𝑘𝐺𝐺𝑒

𝑠𝐺𝑒+𝑘𝐺𝐺𝑒=𝑘𝑒𝑥𝑈𝐺𝑒(𝑠+𝑘𝐺)=𝑘𝑒𝑥𝑈

𝐺𝑒=𝑘𝑒𝑥𝑈𝑠+𝑘𝐺

𝑠𝑈=𝑘𝑎𝑐𝑡 𝐸+𝑘 𝑓𝑘𝑒𝑥𝑈𝑠+𝑘𝐺

−𝑘𝑒𝑈

𝑠𝑈 (𝑠+𝑘𝐺)=𝑘𝑎𝑐𝑡 𝐸(𝑠+𝑘𝐺)+𝑘 𝑓 𝑘𝑒𝑥𝑈−𝑘𝑒𝑈 (𝑠+𝑘𝐺)

∕ (𝑠+𝑘𝐺)

𝑈 ¿

𝑈=𝑘𝑎𝑐𝑡 (𝑠+𝑘𝐺)

𝑠2+𝑠 (𝑘¿¿𝐺+𝑘𝑒)+𝑘𝑒𝑘𝐺−𝑘𝑓 𝑘𝑒𝑥𝐸 ¿

Re-arranging the equations

𝑈=𝑘𝑎𝑐𝑡 (𝑠+𝑘𝐺)

𝑠2+𝑠 (𝑘¿¿𝐺+𝑘𝑒)+𝑘𝑒𝑘𝐺−𝑘𝑓 𝑘𝑒𝑥𝐸 ¿

𝑑𝑈𝑑𝑡 =𝑘𝑎𝑐𝑡𝐸+𝑘𝑓 𝐺𝑒−𝑘𝑒𝑈

𝑑𝐺𝑒

𝑑𝑡 =𝑘𝑒𝑥𝑈 −𝑘𝐺𝐺𝑒

0

0

𝑘𝑒𝑘𝐺−𝑘𝑓 𝑘𝑒𝑥=0

We can always find some steady-state!

𝑈=𝑘𝑎𝑐𝑡(𝑠+𝑘𝐺)

𝑠2+𝑠 (𝑘¿¿𝐺+𝑘𝑒)𝐸 ¿

𝑈=𝑘𝑎𝑐𝑡 (𝑠+𝑘𝐺)

𝑠 ¿¿

Consequences of the control algorithm

• No error in the steady-state!!!!– until relatively large pool of

SREBF2 is depleted• Slower response because of

additional structures• Changed mRNA levels at

normal levels of cholesterol

𝑈=𝑘𝑎𝑐𝑡 (𝑠+𝑘𝐺)

𝑠 ¿¿

𝑈=𝐾 𝑃 𝐸+𝐾 𝐼∫ 𝐸𝑑𝑡𝑈=𝐾 𝑃 𝐸+

1𝑠 𝐾 𝐼

𝐸

𝑈=𝑝 𝑓 (𝐾 𝑃 𝑠+𝐾 𝐼)𝑠 (𝑠+𝑝 𝑓 )

𝐸

PI controller

Understanding of statin activity

DNASREBF2statin influence

To explain statin effect we need additional control loop, that allows cholesterol levels reduction at non-complete HMGCR blockade!

The consequences of the findings for the whole-body functioning

• Precise and robust system for cholesterol control in the cell– for long-term disturbance of normal levels very intense intervention on

major metabolic pathways is required!– reduction of cholesterol levels in the cell causes uptake of cholesterol

from blood stream (measurements on living organisms)– Intervention on the level of cell reference (SREBF2) does not result in

reduction of cholesterol levels in blood (experimentally proven)– spontaneous elevation of cholesterol blood cannot result from

biosynthesis deregulation in the liver (genetic disorder)• too many things would have to be affected simultaneously which is in

contrast with the disease prevalence• the cause for elevated cholesterol levels lies in the periferal tissues

(false cholesterol demand signals or tissue cannot reach the cholesterol in blood, interaction with other metabolic processes.)

Conclusion• Regarding our experience most effective models are the most simple

ones, since only the simplest models contain only vital informations– biological systems are not too complex for modelling, however they

require a lot of innovation and improvisation• Expert knowledge inclusion is important even if some details must be

omitted• Modelling procedure often provides more information than the final

model• Control loops are essential part of biological models

– many times they are discovered on the bases of discrepancy between model simulations and real data

• Never forget holistic approach!!!– sub-systems must be adequately placed within the context of the

integral system!!!!

If the reality does not fit the model ...

... it‘s reality‘s fault!