- Email: [email protected]
A cybernetic model for the dynamics of blood glucose and its regulating hormones
140 A CYBERNBTIC MODEL FOIl THE DYllAMICS or BLOOO GLUCOSE D.G.Cremp. ·P.R.lI:dvard".
AND
ITS REGULATING IIOIUIOMES
E.R.Cars~
Department of Chemic:al Pathology. Royal rree Hospital. London MW3 2QG. UK. and *DepartlMnt of Syst... Sdenee. The City Univeraity. London EC1V 4PB. UK S·. .·ryl A c:ybernetiC: .adel for the study of blood gluc:ose dynamic:s is presented. It is a non-linear . . thematic:al representation of unit proc:ess dynaadc:s. making use Of Ialown ozy.oloaic:al data. that peraita study of the metabolic: and endoc:rine proc:e.ses involved in the ay.t. .•• response. The liver play. a c:entral role in maintaining the blood gluc:ose c:okc:entration at or near ita opti_. by releasing or taking up gluc:ose. Tbis homeostatic: system. in whic:h the liver is the . .in effec:tor organ. is primarily c:ontrolled by the hormones insulid and gluc:agon. Tbis paper desc:ribes a c:ybernetic: model for the dynaadc:s of blood gluc:ose and its regulating hormones. whic:h .is suitable for the study of short-term regulation. The .add is a non-linear mathematic:al representation based .. on the enzyme dynamic:s oc:c:urring in the liver. and their modulation by substrate and hormanes.
The model. whic:h extends a previous model Ill. enables the c:omplete system behaviour of the metabolic: system and its endoc:rine c:ontrol to be analysed. Tbis is made possible by simulating either oral or intravenous administration of gluc:ose and observing the response of the system over the subsequent 3-4 hours. A better understanding of ·the liver gluc:ose system should have c:onsiderable c:linic:al implic:ations for c:haOsea in the liver enzyme .systems and c:ontrol loops c:ould lead to the disorders of c:arbohydrate metaboliam c:ommonly enc:ountered in c:linic:al prac:tic:e. At the molec:ular le.v el. there is muc:h information available. relating to the physiologic:al and bioc:hemic:al proc:esses assec:iated with liver metaboliam. to provide the building bloc:ks for .aggregation into overall metabolic: pathway dynamic:s. Equally. the struc:tural and behavioural modes of the shortterm c:ontrollers that affec:t liver gluc:ose metaboliam. insulin. gluc:agon and adrenal in are better understood than those of c:hronic: adaptation. Model Formulation: Tbe c:ompartlMntal model for liver gluc:ose metabolism is ahown in Figure 1.
In addition to c:ompartments for plasma gluc:ose. gluc:ose-6-phosphate
and ~patic: glyc:ogen. c:ompartmentation is provided for hepatic: portal gluc:ose. Tbis not only allows the system's response to intravenous test stimuli to be examined. but with the addition of a simple gut transport model also enables the responses observed following an oral gluc:ose load to be examined. Portal and plasma c:ompartments are provided for insulin and gluc:agon with a single c:ompartment for adrenaUn.
141
"&pAr... gI:,~CI\ t--z::"[7':"...-.t
obl;~~
Iou 10
~~~ A~~ ~---f
Fig. 1.
Compartmental model for liver glucose metabolism and its associated features.
The model incorporates the dominant stages of the relevant liver enzyme dynamics, extensive use being made of currently available data drawn from enzymological studies /2/.
These non-linear enzymic reactions are modelled using a
Briggs-Haldane approximation /3/ except where, over the normal range, linearisation resul ts in no signi.ficant, loss of accuracy (for example, the incorporation of glucose-6-phosphate into glucose).
For each compartment a mass balance type
equation is written, adopting units of molar
c~ncentration.
The following nomenclature is adopted: C p
- plasma glucose (M)
I
Cl
- portal glucose
11 - portal insulin (mU)
Ll
- liver glycogen
J
p - plasma glucagon (l1g)
J
l - portal glucagon (l1g)
G6Pl - liver glucose-6-phosphate
(mU) p - plasma insulin
Ep - plasma adr,e nalin (mg) This leads to the following mathematical description: df,: clt
-+
Cl'lo~"'Oo>S
(" t
91..CooSQ. doS&.
--Uc.. _
.OOOI!o, _ .1S'C p
. 2.se,
·tl.av...
'lAooo.a,,1. (".11.(;,",,,
~.M (t;.&..C,.)
11.'1 + IlSoJ:,.
l ~ ,-·s <.~,~oo,
'l..~,. U'l + h$'O.C,
lit c.. ~ <. !to "100
'
1. 11IIIU.1;~~ (tr"~l/7
n'1 + "'.~pltr"~ )
o
1,. " ."11.) ::J, ,. ...'1.)
l~t,.
> 100)
bf It ~c..s)
142
(.·'i1.t;"'L 1. ... 1000. ~"I\.
~
•• 0oS' -
I".U~~
,,~
..
4·1.1(fl n·l+"~o.a.t.
~n.C"'"
{
1. ••
,
\
I.1.,"C ..
'.&
·1o~u.""'t.~6.
'~~)/l
l~ If H",,)
C!~'I ."'1. ~ .C,-.,.~C::"P.&
'-.l~.~.'+ ~'-'''O·<:",.&,,"·'/t....... .:c.P.I''')
. (il i't ~.V _ tip ~.h)
".I" .\S')
o
'OOO.a ....L (Xf
(rtlt<,,..I..L>.S'.:r.&>.U)
0
~l~<."'.ir< " ~)
143
A plasma volume and hormonal distribution volumi of 3.2 1 were adopte~ for a 70 kg subject in formulating the equations. Table 1 Ip(mU)
f( I p HmU/min),
Ip(mU)
f(IpHmu/min)
0.0 86,.4 112.0 128.0 150.4
4.0 6.3 I 7.3 15.7 39.8
176.0 217.6 230.6 272.0 291.0
61.1 93.7 115.0 174.0 227.2
Situlation: The model was simulated on a CDC 7600 computer using FORTRAN for the following five test input conditions: i) Intravenous glucose injection: 0.5 g glucose/kg bodyweight administered over . a 2 minute period ii) Primed glucose infusion: priming dose of 15 ml of 50 g/100 ml concentration glucose solution applied over 3 minutes, followed by a 50 g/100 ml glucose 'solution infused at the rate of 30 ml/h iii}
~al
glucose load : 50 g, representing an input rate to the portal glucose
compara.nt of:
.oo... ""/.......
(t..
'30 ...;,,)
. 004- ·llt.Jo)~.....(t: >'30 ...... ) iv) Intravenous insulin injectiOn: 0.1 U insulin/kg bodyweight administered over
{
a 1 minute period v) Intravenous glucagon injection: 0.25 mg/min over a 1 minute period Results ad ec-ntary: Typical results of si..tlations are shown in Figures 2-8. They yield generally good conformational agreement with clinical teat data. No . attempt has been made at this stage to quantify the error between data &ad mode! respOD.e. It is ~re importat to strive for a model structure and parameter values which, aa far ' as possible, mirror the appropriate physiology and biocbemi.try and lead to good qualitative agreement, vlth the test data than merely to mint.! .. a cost function. ' Physiological reali. outweighs purely mathematical considerations of goodness ef fit.
The model developed bere offers a number of advantages over previous fonaulUnlike many previous studies in which there was an implicit relationahip to the physiology, here the model structure is explicitly baaed upon current phy.iological knowledge and incorpotatea the relevant unit proces. dynaadcs ./4/. \ It also provides a repreaentati'on which tends towards iSOlllOrphi., ao hypotheses ations.
concerning relation.hips between system structure and parameter value. and known
144
o
0
,....-.
L-__________________~__~_____
•
~
le
IS.
...
So
..
). .....
Fig. 2. Simulated responses of glucose insulin and glucagon following an intravenous glucose injection (0.0304 M/min over 2 min)
o
..
10
:a. . . . . ..
,.toIo.
Pig . 4. Simulated responaes of clucose insulin and glucagon following an intravenous injection of glucagon (0.25 mg over 1 min)
Fig. 3. Simulated responses of glucose insulin and glucagon following an intravenous insulin injection (7 mU over 1 min)
o
"~.~,,,ClDM"."'"
Fig. S. Si.ulated reapoa... of clueoae following an intravenous clueoae injection (0.0304 NI_in over 2 min) with illpaired glucokinase activity
145
·os
\
I
, \ \
I
\ , / 10.$..1.... ~..s""';t; ~!. I
\
I
. _L-------~~~~~~ " 10 a. h 4h So Coo lD ...... Fig. 6.
J}
Simulated responses of glucose following an intravenous glucose injection the effect of varying insulin sensitivity
:"J
l-
•
l, "" ~
l
15.~
04-
...... ?
~
:J
j
l Fig. 7.
Si~lated respon.e~of glucose insulin and Clucagon following a 50 g oral load of glucose with impaired glucose-6-phosphatase activity
146
.-
,
.t .f. ."
t
J.,
1L
.('
l'
,;J
j
061f' .~
"
.~
.!
o Fig. 8.
10
\
•
00
' • •,.
00"': ..
Simulated responses of glucose insulin and glucagon following a 50 g oral load of glucose
pathological states can be tested. Since short-term control effects are under investigation, the model focuses upon hepatic function with incorporation of the dominant enzyme controlled chemical
re.ctions , dyn~ics.
The glucose-glucose-6-phosphate-glycogen axis and its
relation to the 'hepatic portal system is considered to be 'of prime importance. Other physiological features of the model include: glucagon control of glycogen production, uecessary to inhibit the glucose-6-phosphate' to glycogen con, ' version if the necessary glucose elevation in response to a glucagon stimulus is to be achieved;- and hexokinase mediated transport of glucose in peripheral tissues. \
In the short-term period, coarse control appears to be provided by the horaoaes which are predominantly involved in ,threshold effects, but are also permi88ive so if a switching condition is satisfied, the appropriate reaction then pro-
ceeds. Variation in the hormonal controller sensitivities to glucose, whilst re'sultiDg in Sc.M steady state changes, does not grossly perturb conformational relationships, for example those between glucose and insulin. Not all the hormonal effects are permissive and in this model the multiplier effect of insulin at high levels provides an example of direct involvement in the mediation of peripheral glucose trausport. The gener'a l evidence available, however, does suggest that i t is the hormones which are providing the coarse cont~ol of glucose metabolism. The involvement of insulin and its autagonist glucagon, both responding to the s _
stimuli, provides au interesting example of two channel permissive/inhibitory cOlltrol.
Fine control of glucose metabolism appears to be found in small ·cbanges of tbe enzyme controlled chemical dynamics.
The examples of glucokinase and hexo-
kinase indicate how comparatively small changes in enzyme activity can produce substantial changes in the glucose ,teady state. Many of the examples of steady state switching which have been revealed can be analysed in terms of finite state machine behaviour.
The applicability of
automata theory to biochemical situations and. in particular, enzyme dynamics is receiving considerable attention.
In . the glucose system the pattern of enzymic
activation and inhibition of reactions gives rise to a variety of control characteristics.
Hexokinase, phosphofructokinase, and glycogen synthetase respectively
exemplify end product inhibition, end product activation and substrate activation. Each has its own type of dynamic behaviour.
Even in the simplified representation
given here there are instances of alternative pathways being avaUable to a substrate with each having · a different pattern of enzyme dynamics.
In general, for a
given ' substrate concentration, the flux will be directed along one rise to a particular set of steady state conditions. of transitions from one state to another is seen.
pa~hway,
giving
In the tests here triggering
Regarding the .test ·stimulus as
an environmentsl control variable and the sub's trate as a state variable, in s~ situations the control variable is not Increased above a critical threahold and, remove~
. when returned to . its original value - that is with pulse load
- the sub-
strate returns to its original value. Under other conditions of reaction dynamics, #uch as with normal hexokinase activity, state transitions can occur.
In this case the intravenous glucose load
raises the controlling variable above its critical threshold for sufficient time to trigger the process, the system making the transitfon to a new high state (0.0062 M glucose concentration), where it remains even although the pulse · load has been removed.
The system has thus been switched with the. pa'ttern of substrate \
utilisation in the metabolic pathways being altered.
In effect it exhibits bi-
stable characteristics. References
.1-
Carson,E.R. , Cramp,D.G. : Int. J.
Bi~dic.l
lli.,
286 (1966)
Computing
2.
London,W.P. :. J. BioI. Chem.
3.
Brigs,G.E. , Haldane,J.B.S.i Biochem. J;
4.
Carson,E.R •• Finkelstein,L.: Measureaient and Control
!2.,
L,
21 (1976)
338 (1925)
1.
157 (1970)
AND
ITS REGULATING IIOIUIOMES
E.R.Cars~
Department of Chemic:al Pathology. Royal rree Hospital. London MW3 2QG. UK. and *DepartlMnt of Syst... Sdenee. The City Univeraity. London EC1V 4PB. UK S·. .·ryl A c:ybernetiC: .adel for the study of blood gluc:ose dynamic:s is presented. It is a non-linear . . thematic:al representation of unit proc:ess dynaadc:s. making use Of Ialown ozy.oloaic:al data. that peraita study of the metabolic: and endoc:rine proc:e.ses involved in the ay.t. .•• response. The liver play. a c:entral role in maintaining the blood gluc:ose c:okc:entration at or near ita opti_. by releasing or taking up gluc:ose. Tbis homeostatic: system. in whic:h the liver is the . .in effec:tor organ. is primarily c:ontrolled by the hormones insulid and gluc:agon. Tbis paper desc:ribes a c:ybernetic: model for the dynaadc:s of blood gluc:ose and its regulating hormones. whic:h .is suitable for the study of short-term regulation. The .add is a non-linear mathematic:al representation based .. on the enzyme dynamic:s oc:c:urring in the liver. and their modulation by substrate and hormanes.
The model. whic:h extends a previous model Ill. enables the c:omplete system behaviour of the metabolic: system and its endoc:rine c:ontrol to be analysed. Tbis is made possible by simulating either oral or intravenous administration of gluc:ose and observing the response of the system over the subsequent 3-4 hours. A better understanding of ·the liver gluc:ose system should have c:onsiderable c:linic:al implic:ations for c:haOsea in the liver enzyme .systems and c:ontrol loops c:ould lead to the disorders of c:arbohydrate metaboliam c:ommonly enc:ountered in c:linic:al prac:tic:e. At the molec:ular le.v el. there is muc:h information available. relating to the physiologic:al and bioc:hemic:al proc:esses assec:iated with liver metaboliam. to provide the building bloc:ks for .aggregation into overall metabolic: pathway dynamic:s. Equally. the struc:tural and behavioural modes of the shortterm c:ontrollers that affec:t liver gluc:ose metaboliam. insulin. gluc:agon and adrenal in are better understood than those of c:hronic: adaptation. Model Formulation: Tbe c:ompartlMntal model for liver gluc:ose metabolism is ahown in Figure 1.
In addition to c:ompartments for plasma gluc:ose. gluc:ose-6-phosphate
and ~patic: glyc:ogen. c:ompartmentation is provided for hepatic: portal gluc:ose. Tbis not only allows the system's response to intravenous test stimuli to be examined. but with the addition of a simple gut transport model also enables the responses observed following an oral gluc:ose load to be examined. Portal and plasma c:ompartments are provided for insulin and gluc:agon with a single c:ompartment for adrenaUn.
141
"&pAr... gI:,~CI\ t--z::"[7':"...-.t
obl;~~
Iou 10
~~~ A~~ ~---f
Fig. 1.
Compartmental model for liver glucose metabolism and its associated features.
The model incorporates the dominant stages of the relevant liver enzyme dynamics, extensive use being made of currently available data drawn from enzymological studies /2/.
These non-linear enzymic reactions are modelled using a
Briggs-Haldane approximation /3/ except where, over the normal range, linearisation resul ts in no signi.ficant, loss of accuracy (for example, the incorporation of glucose-6-phosphate into glucose).
For each compartment a mass balance type
equation is written, adopting units of molar
c~ncentration.
The following nomenclature is adopted: C p
- plasma glucose (M)
I
Cl
- portal glucose
11 - portal insulin (mU)
Ll
- liver glycogen
J
p - plasma glucagon (l1g)
J
l - portal glucagon (l1g)
G6Pl - liver glucose-6-phosphate
(mU) p - plasma insulin
Ep - plasma adr,e nalin (mg) This leads to the following mathematical description: df,: clt
-+
Cl'lo~"'Oo>S
(" t
91..CooSQ. doS&.
--Uc.. _
.OOOI!o, _ .1S'C p
. 2.se,
·tl.av...
'lAooo.a,,1. (".11.(;,",,,
~.M (t;.&..C,.)
11.'1 + IlSoJ:,.
l ~ ,-·s <.~,~oo,
'l..~,. U'l + h$'O.C,
lit c.. ~ <. !to "100
'
1. 11IIIU.1;~~ (tr"~l/7
n'1 + "'.~pltr"~ )
o
1,. " ."11.) ::J, ,. ...'1.)
l~t,.
> 100)
bf It ~c..s)
142
(.·'i1.t;"'L 1. ... 1000. ~"I\.
~
•• 0oS' -
I".U~~
,,~
..
4·1.1(fl n·l+"~o.a.t.
~n.C"'"
{
1. ••
,
\
I.1.,"C ..
'.&
·1o~u.""'t.~6.
'~~)/l
l~ If H",,)
C!~'I ."'1. ~ .C,-.,.~C::"P.&
'-.l~.~.'+ ~'-'''O·<:",.&,,"·'/t....... .:c.P.I''')
. (il i't ~.V _ tip ~.h)
".I" .\S')
o
'OOO.a ....L (Xf
(rtlt<,,..I..L>.S'.:r.&>.U)
0
~l~<."'.ir< " ~)
143
A plasma volume and hormonal distribution volumi of 3.2 1 were adopte~ for a 70 kg subject in formulating the equations. Table 1 Ip(mU)
f( I p HmU/min),
Ip(mU)
f(IpHmu/min)
0.0 86,.4 112.0 128.0 150.4
4.0 6.3 I 7.3 15.7 39.8
176.0 217.6 230.6 272.0 291.0
61.1 93.7 115.0 174.0 227.2
Situlation: The model was simulated on a CDC 7600 computer using FORTRAN for the following five test input conditions: i) Intravenous glucose injection: 0.5 g glucose/kg bodyweight administered over . a 2 minute period ii) Primed glucose infusion: priming dose of 15 ml of 50 g/100 ml concentration glucose solution applied over 3 minutes, followed by a 50 g/100 ml glucose 'solution infused at the rate of 30 ml/h iii}
~al
glucose load : 50 g, representing an input rate to the portal glucose
compara.nt of:
.oo... ""/.......
(t..
'30 ...;,,)
. 004- ·llt.Jo)~.....(t: >'30 ...... ) iv) Intravenous insulin injectiOn: 0.1 U insulin/kg bodyweight administered over
{
a 1 minute period v) Intravenous glucagon injection: 0.25 mg/min over a 1 minute period Results ad ec-ntary: Typical results of si..tlations are shown in Figures 2-8. They yield generally good conformational agreement with clinical teat data. No . attempt has been made at this stage to quantify the error between data &ad mode! respOD.e. It is ~re importat to strive for a model structure and parameter values which, aa far ' as possible, mirror the appropriate physiology and biocbemi.try and lead to good qualitative agreement, vlth the test data than merely to mint.! .. a cost function. ' Physiological reali. outweighs purely mathematical considerations of goodness ef fit.
The model developed bere offers a number of advantages over previous fonaulUnlike many previous studies in which there was an implicit relationahip to the physiology, here the model structure is explicitly baaed upon current phy.iological knowledge and incorpotatea the relevant unit proces. dynaadcs ./4/. \ It also provides a repreaentati'on which tends towards iSOlllOrphi., ao hypotheses ations.
concerning relation.hips between system structure and parameter value. and known
144
o
0
,....-.
L-__________________~__~_____
•
~
le
IS.
...
So
..
). .....
Fig. 2. Simulated responses of glucose insulin and glucagon following an intravenous glucose injection (0.0304 M/min over 2 min)
o
..
10
:a. . . . . ..
,.toIo.
Pig . 4. Simulated responaes of clucose insulin and glucagon following an intravenous injection of glucagon (0.25 mg over 1 min)
Fig. 3. Simulated responses of glucose insulin and glucagon following an intravenous insulin injection (7 mU over 1 min)
o
"~.~,,,ClDM"."'"
Fig. S. Si.ulated reapoa... of clueoae following an intravenous clueoae injection (0.0304 NI_in over 2 min) with illpaired glucokinase activity
145
·os
\
I
, \ \
I
\ , / 10.$..1.... ~..s""';t; ~!. I
\
I
. _L-------~~~~~~ " 10 a. h 4h So Coo lD ...... Fig. 6.
J}
Simulated responses of glucose following an intravenous glucose injection the effect of varying insulin sensitivity
:"J
l-
•
l, "" ~
l
15.~
04-
...... ?
~
:J
j
l Fig. 7.
Si~lated respon.e~of glucose insulin and Clucagon following a 50 g oral load of glucose with impaired glucose-6-phosphatase activity
146
.-
,
.t .f. ."
t
J.,
1L
.('
l'
,;J
j
061f' .~
"
.~
.!
o Fig. 8.
10
\
•
00
' • •,.
00"': ..
Simulated responses of glucose insulin and glucagon following a 50 g oral load of glucose
pathological states can be tested. Since short-term control effects are under investigation, the model focuses upon hepatic function with incorporation of the dominant enzyme controlled chemical
re.ctions , dyn~ics.
The glucose-glucose-6-phosphate-glycogen axis and its
relation to the 'hepatic portal system is considered to be 'of prime importance. Other physiological features of the model include: glucagon control of glycogen production, uecessary to inhibit the glucose-6-phosphate' to glycogen con, ' version if the necessary glucose elevation in response to a glucagon stimulus is to be achieved;- and hexokinase mediated transport of glucose in peripheral tissues. \
In the short-term period, coarse control appears to be provided by the horaoaes which are predominantly involved in ,threshold effects, but are also permi88ive so if a switching condition is satisfied, the appropriate reaction then pro-
ceeds. Variation in the hormonal controller sensitivities to glucose, whilst re'sultiDg in Sc.M steady state changes, does not grossly perturb conformational relationships, for example those between glucose and insulin. Not all the hormonal effects are permissive and in this model the multiplier effect of insulin at high levels provides an example of direct involvement in the mediation of peripheral glucose trausport. The gener'a l evidence available, however, does suggest that i t is the hormones which are providing the coarse cont~ol of glucose metabolism. The involvement of insulin and its autagonist glucagon, both responding to the s _
stimuli, provides au interesting example of two channel permissive/inhibitory cOlltrol.
Fine control of glucose metabolism appears to be found in small ·cbanges of tbe enzyme controlled chemical dynamics.
The examples of glucokinase and hexo-
kinase indicate how comparatively small changes in enzyme activity can produce substantial changes in the glucose ,teady state. Many of the examples of steady state switching which have been revealed can be analysed in terms of finite state machine behaviour.
The applicability of
automata theory to biochemical situations and. in particular, enzyme dynamics is receiving considerable attention.
In . the glucose system the pattern of enzymic
activation and inhibition of reactions gives rise to a variety of control characteristics.
Hexokinase, phosphofructokinase, and glycogen synthetase respectively
exemplify end product inhibition, end product activation and substrate activation. Each has its own type of dynamic behaviour.
Even in the simplified representation
given here there are instances of alternative pathways being avaUable to a substrate with each having · a different pattern of enzyme dynamics.
In general, for a
given ' substrate concentration, the flux will be directed along one rise to a particular set of steady state conditions. of transitions from one state to another is seen.
pa~hway,
giving
In the tests here triggering
Regarding the .test ·stimulus as
an environmentsl control variable and the sub's trate as a state variable, in s~ situations the control variable is not Increased above a critical threahold and, remove~
. when returned to . its original value - that is with pulse load
- the sub-
strate returns to its original value. Under other conditions of reaction dynamics, #uch as with normal hexokinase activity, state transitions can occur.
In this case the intravenous glucose load
raises the controlling variable above its critical threshold for sufficient time to trigger the process, the system making the transitfon to a new high state (0.0062 M glucose concentration), where it remains even although the pulse · load has been removed.
The system has thus been switched with the. pa'ttern of substrate \
utilisation in the metabolic pathways being altered.
In effect it exhibits bi-
stable characteristics. References
.1-
Carson,E.R. , Cramp,D.G. : Int. J.
Bi~dic.l
lli.,
286 (1966)
Computing
2.
London,W.P. :. J. BioI. Chem.
3.
Brigs,G.E. , Haldane,J.B.S.i Biochem. J;
4.
Carson,E.R •• Finkelstein,L.: Measureaient and Control
!2.,
L,
21 (1976)
338 (1925)
1.
157 (1970)