Journal of Theoretical Biology

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Excitation states of metabolic networks predict dose- response fingerprinting and ligand pulse phase signalling.

Item Type Article

Authors Coggan, Jay S;Keller, Daniel;Markram, Henry;Schürmann, Felix;Magistretti, Pierre J.

Citation Coggan, J. S., Keller, D., Markram, H., Schürmann, F., &

Magistretti, P. J. (2020). Excitation states of metabolic networks predict dose-response fingerprinting and ligand pulse phase signalling. Journal of Theoretical Biology, 487, 110123.

doi:10.1016/j.jtbi.2019.110123 Eprint version Publisher's Version/PDF

DOI 10.1016/j.jtbi.2019.110123

Publisher Elsevier BV

Journal Journal of theoretical biology

Rights © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://

creativecommons.org/licenses/by-nc-nd/4.0/) Download date 2023-11-29 20:55:45

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ContentslistsavailableatScienceDirect

Journal of Theoretical Biology

journalhomepage:www.elsevier.com/locate/jtb

Excitation states of metabolic networks predict dose-response ﬁngerprinting and ligand pulse phase signalling

Jay S Coggan

^a^,^∗

, Daniel Keller

^a

, Henry Markram

^a

, Felix Schürmann

^a

, Pierre J Magistretti

^b

aBlue Brain Project, École Polytechnique Fédérale de Lausanne (EPFL), Geneva CH-1202, Switzerland

bBiological and Environmental Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia

a rt i c l e i n f o

Article history:

Received 6 September 2019 Revised 8 November 2019 Accepted 16 December 2019 Available online 19 December 2019 Keywords:

Enzyme cascade Phase signalling Ligand pulse

Dose-response ﬁngerprinting Synthetic biology

a b s t r a c t

Withacomputationalmodelofenergymetabolisminanastrocyte,weshowhowasystemofenzymesin acascadecanactasafunctionalunitofinterdependentreactions,ratherthanmerelyaseriesofindepen- dentreactions.Thesesystemsmayexistinmultiplestates,dependingonthelevelofstimulation,andthe effectsofsubstratesatanypointwilldependonthosestates.Responsetrajectoriesofmetabolitesdown- streamfromcAMP-stimulatedglycogenolysisexhibitahostofnon-lineardynamicalresponsecharacter- isticsincludinghysteresisandresponseenvelopes.Dose-dependentphasetransitionspredictanovelin- tracellularsignallingmechanismandsuggestatheoreticalframeworkthatcouldberelevanttosinglecell informationprocessing,drugdiscoveryorsyntheticbiology.Ligandsmayproduceuniquedose-response ﬁngerprintsdependingonthestateofthesystem,allowingselectiveoutputtuning.Weconcludewith theobservation that state-and dose-dependentphase transitions,what wedub“ligand pulses” (LPs), maycarryinformationandresembleactionpotentials(APs)generatedfromexcitatorypostsynapticpo- tentials.Inourmodel,therelevantinformationfromacAMP-dependentglycolyticcascadeinastrocytes couldreﬂectthelevelofneuromodulatoryinputthatsignalsanenergydemandthreshold.Wepropose thatbothAPsandLPsrepresentspecializedcasesofmolecularphasesignallingwithacommonevolu- tionaryroot.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The long-standing inadequacy of our understanding of cells, from biochemistry to behaviour, from bacteria to neurons, has vexed the development of more effective treatments for many medicaldisorders, mostnotablythose thatlead toneurodegener- ation. Variouslevels of biological complexityare usually ﬁngered as the culprits, with expansive systems of interlocked biochem- ical cascades and cellular networks at the root of their nature.

For example, ubiquitous intracellular signalling pathways in sin- glecellsworkinparallelatmultiplescalesofspaceandtime.The challenge ofunderstanding phenomenasuch asagingin orderto delay orcircumvent senescence orany numberof neuropatholo- gieswilllikelyrequirealeapforwardinsystemsbiologyconcepts (Kirkwood, 2011,Kowald andKirkwood, 1996, Kriete etal., 2011, McAuleyetal.,2017).

∗ Corresponding author.

E-mail addresses: jay.coggan@epﬂ.ch (J.S. Coggan), daniel.keller@epﬂ.ch (D.

Keller), henry.markram@epﬂ.ch (H. Markram), felix.schuermann@epﬂ.ch (F. Schür- mann), [email protected] (P.J. Magistretti).

Second-messenger signalling can involve any number of cas- cading, enzyme-catalyzed steps whereby stimuli such as neuro- modulators are transduced from one metabolite to another, each inturn capable offeeding another cascade. Elusivedisease cures requireanunderstanding that currentlyeludestraditionalexperi- mentaldesignsandcapabilities.Computationalmethodsthatcom- plementlaboratorytechniquesmayprovidethisunderstanding.

Computational modelling of intracellular enzymatic cascades is a valuable tool for understanding single cell functions (Schoeberl et al., 2002). As a case in point, here we describe a novel analysis of a previously described model of energy metabolism and neuromodulation in glia (Coggan et al., 2018, Jolivetetal.,2015).Weshowhowthisapproachcanrevealinsights notonlyaboutthisparticularpathway,butalsomake predictions about the behavior and previously unsuspected roles of second- messengersystemsingeneral,sincemetaboliccascadesarecentral toalmostallcellularfunctionsanddysfunctions(DeBerardinisand Thompson,2012).

With the simple concept of excitability states of an enzy- maticcascadestimulated,forexample,by aneuromodulatorsuch as norepinephrine, we are able to make predictions about their https://doi.org/10.1016/j.jtbi.2019.110123

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Fig. 1. Model summary and response trajectories of selected metabolites in 5 excitation levels of a metabolic enzyme cascade stimulated by 5 doses of cAMP. A) The metabolic cascade system, from cAMP to the pyruvate (PYR) bifurcation to lactate and ATP, is stimulated to 5 excitability states by 5 concentrations of cAMP. Trajectories of primary metabolites of interest B) cAMP, C) pyruvate (PYR), D) lactate (LAC), E) ATP. The phase transitions occur in the concentration step between states 3 and 4. F, G) Derivative phase plots show phase transition between lower (cyan, blue) and higher (green, red, black) excitability states, corresponding to concentrations of cAMP. Examples made of LAC and ATP.

propertiesandbehavior.Weobservedthat,dependingonthestate ofmetabolic excitation, each metabolite concentration may cycle throughasetofstereotypedresponsestotheriseandfallofcAMP concentrations.Theseresponsesmayincludepatternsofhysteresis, responseenvelopesandphasetransitions.

Phasetransitions(deﬁnedhereasathermodynamicshiftinthe state or properties of a system of reactions) among intracellular reactions play a role in normal and pathological cell physiology (Nedelsky and Taylor, 2019, Tsuruyama, 2014), including energy productionpathwayssuchasglycolysis(Mulukutlaetal.,2015).We ﬁndthat excitation state-dependent, hysteretic, phase plots yield richinformationaboutthebehaviorofanenzymaticcascade.Fur- thermore,wemakethetheoreticalpropositionthatunderstanding theeffectofadrugona biologicalsystemrequiresknowledge of thestateoftheentiresystem.

We further hypothesize that both ligand pulses (LPs) and ac- tionpotentials(APs)representspecializedcasesofmolecularphase signalling (MPS), and suggest that APs might be late evolution- aryadaptationsofmoreancientandestablishedintracellularMPS mechanismsforthepurposeoflong-rangecommunicationandin- formationprocessingin nervoussystems (Boyer and Wisniewski- Dyé,2009).

Our results exemplify how a large-scale biomolecularsimula- tionsuch astheone beingbuiltatthe BlueBrain Project canbe

employed as a platform for theoretical advancements in understanding complexmolecular systems. Beyondunderstanding normal physiology, simulation science will also play an increasingly fundamentalroleintheburgeoningdisciplineofsyntheticbiology, aﬁeldthat seekstounderstandandmanipulatecatabolicandan- abolicpathwaysthatmightimprovetheyieldofusefulmetabolites (Mulukutlaetal.,2016,Erbetal.,2017),orengineerdenovopath- ways forthe production ofhigh-value compounds (Martinet al., 2009).

2. Methods

Themodelusedinthispaperisbasedonthat previouslypub- lished(Cogganetal., 2018,Jolivetetal., 2015).Here, wefocuson asegmentofthedeterministiccascadebetweencAMPproduction (as stimulated by 5 levels ofnoradrenergic activation ofthe

β

^2-

adrenergicreceptorasin(Coggan etal.,2018))witheach colored plotcomingfromanindependentsimulationwithadifferentcon- centrationofcAMPandoutputs ofthepyruvate(PYR) bifurcation, lactate(LAC)andATP(cAMP→… PYR→LAC,ATP).Theﬁverela- tivelyequallyspacedstimulationlevelsofcAMPwerechosenasa dose-responseregimen withwhich tostimulate the enzyme system to a range of levels of excitation roughly corresponding to thedose-responserangeforcAMPinthissystem.Theselevelsare

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Table 1

Governing equations

Variable Value at rest Equation

Intracellular sodium 8/15 mM _dt^dNa ⁺_x= J ^x_leak_,_Na−3 J ^x_pump+ J ^x_stim(t) ^∗(1)

Neuronal glucose 1.2 mM _dt^dGL C n = J ^en_GLC−J ⁿ_HKPFK (2)

Astrocytic glucose 1.19 mM _dt^dGL C g= J ^cg_GLC+ J ^eg_GLC−J ^g_HKPFK (3)

Glyceraldehyde-3-phosphate 0.0046 mM _dt^dGA P x= 2 J ^x_HKPFK−J ^x_PGK+ vL 3 − v _ 3 (4)

Phosphoenolpyruvate 0.015 mM _dt^dPE P x= J ^x_PGK−J ^x_PK (5)

Pyruvate 0.17 mM _dt^dPY R x= J ^x_PK−J ^x_LDH−J ^x_mito_,_in (6)

Neuronal lactate 0.6 mM _dt^dLA C n= J ⁿ_LDH−J ^ne_LAC (7)

Astrocytic lactate 0.6 mM _dt^dLA C g= J ^g_LDH−J ^ge_LAC−J ^gc_LAC (8)

Cytosolic NADH † 0.006/0.1 mM _dt^dNADH ^cytox = (1 −ξ)⁻¹(J ^x_PGK−J ^x_LDH−J ^x_shuttle) (9) Mitochondrial NADH † 0.12 mM _dt^dNADH ^mitox = ξ⁻¹(4 J ^x_mito_,_in−J ^x_mito_,_out+ J ^x_shuttle) (10) Neuronal ATP ‡ 2.2 mM _dt^dATP _n= (−2 J ⁿ_HKPFK+ J ⁿ_PGK+ J ⁿ_PK− J ⁿ_ATPases− J ⁿ_pump+ 3 . 6 J ⁿ_mito_,_out+ J ⁿ_CK)(1 −^d_d^AM_AT_P^P_nⁿ)⁻¹ (11) Astrocytic ATP ‡ 2.2 mM _dt^dATP g=

(−2 J ^g_HKPFK+ J ^g_PGK+ J ^g_PK− J ^g_ATPases− ⁷₄J ^gpump+ ³₄J ^g_pump_,₀+ 3 . 6 J ^g_mito_,_out+ J ^g_CK)(1 −^d_d^AM_AT_P^P_g^g)⁻¹ (12)

Phosphocreatine 4.9 mM _dt^dPCr x= −J ^x_CK (13)

Neuronal oxygen 0.028 mM _dt^dO 2n= J ^cn_O₂_m−0 . 6 J ⁿ_mito_,_out (14)

Astrocytic oxygen 0.028 mM _dt^dO 2g = J ^cg_O₂_m−0 . 6 J ^g_mito_,_out (15)

Capillary oxygen 7 mM _dt^dO 2c= J ^c_O₂−1 / r cnJ ^cn_O₂_m−1 / r cgJ ^cg_O₂_m (16) Capillary glucose 4.5 mM _dt^dGL C c= J ^c_GLC−1 / r ceJ ^ce_GLC−1 / r cgJ ^cg_GLC (17) Capillary lactate 0.55 mM _dt^dLA C c= J ^c_LAC+ 1 / r ceJ ^ec_LAC+ 1 / r cgJ ^gc_LAC (18)

Venous volume 0.02 _dt^dV v= F in(t)−F out ∗(19)

Deoxyhemoglobin 0.058 mM _dt^ddH b = F in(t)(O 2a−O 2c¯)−F out dHb Vv

∗(20) Extracellular glucose 2.48 mM _dt^dGL C e= J ^ce_GLC−1 / r egJ êg_GLC−1 / r enJ ên_GLC (21) Extracellular lactate 0.6 mM _dt^dLA C e= 1 / r enJ ^ne_LAC+ 1 / r egJ ^ge_LAC− J êc_LAC (22) Neuronal membrane voltage −73 mV _dt^dψn= C m⁻¹(−I L−I Na−I K−I Ca−I mAHP−I pump+ I syn(t)) ^∗(23)

h gating variable 0.99 _dt^dh = ^φ_τ_h^h(h ∞−h ) (24)

n gating variable 0.02 _dt^dn = ^φ_τ_nⁿ(n ∞−n ) (25)

Neuronal calcium 5 10 ⁻⁵mM _dt^dC a ²⁺= −^S^m_F^VⁿI Ca−1 / τCa(C a ²⁺−Ca ²⁺0 ) (26) Glycogen Module

Glycogen-G6P equilibria vL 1 = ⁽_k^k_m⁻₋^L¹_L₁₊⁾⁽^gluc_gluc⁾ (27)

v−L 1 = ⁽_k^k_m⁻₋^L¹_L₁₊⁾⁽^G_G⁶₆^P_P⁾ (28)

vL 2 = ⁽^kL²_kmL⁾⁽^GSa₂₊⁾⁽_G₆^G_P⁶^P⁾ (29)

v₋L 2 = ⁽_k^k_m⁻₋^L²_L₂₊⁾⁽^glyc_glyc⁾ (30)

Glycogenolytic glucose _dt^dgluc = vtL 1 − vL 1 + v−L 1 (31)

Glycogen _dt^dglyc = vL 2 − v−L 2 (32)

Glucose 6-phosphate _dt^dG 6 P = vL 1 − v_ L 1 −vL 2 + v−L 2 (33)

Blood glucose inﬂux rate vtL 1 = ktL 1(B gluc−gluc )= 0 (34)

Blood glucose derivative _dt^dB gluc= 0 (35)

cAMP —— _dt^dcAMP = (x (_kDne^ne₊_ne)+ y ((_τ_cAMP¹ )( ^hb−ha

1+(^hchk)^hd)+ (_τ_cAMP^ha ))−2 ×kcg1 ×R 2 C2 ×(cAMP )² + 2 ×k −cg1 ×R 2 CcAMP ×C − 2 ×kgc2 ×R 2 CcAM P2 × (cAM P )² + 2 ×k −cg2 × R 2 cAMP4 ×C)−cAMP/ τcAMP{^x⁼^≥⁰^;^y⁼^≥⁰}

(36)

Glycogen Phosphorylase _dt^dGPa =

(_kmg5⁽⁽^kg5₍₁₊⁾⁽(s^PKa1)(G6⁾⁽P)^pt−GPa⁾⁾

kg2 )+(pt−GPa))− (⁽⁽^kg6⁾⁽^Pkmg6^P¹⁺^P^P¹^GPa⁾⁽^GPa⁾⁾ (1+(s2)(gluc)

kgi )+(GPa) )−(k a)(P P 1 )(GPa )+ (k _−a)(P P 1 GPa)

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Glycogen Synthase _dt^dGSa = (⁽⁽^kg8kmg8⁾⁽^P^P¹⁾⁽^st−GPa⁾⁾ (1+(s1)(G6P)

kg2 )+st−(GSa))−(_kmg7⁽⁽^kg7₍₁₊⁾⁽^PKa(s1)(⁺G6^CP⁾⁽)^GSa⁾⁾

kg2 )+(GSa)) (38)

Protein Phosphatase 1 _dt^dP P 1 = (−k a)(P P 1 )(GPa )+ (k −a)(P P 1 GPa) (39) PP1-GPa _dt^dP P 1 GPa= (k a)(P P 1 )(GPa )+ (k −a)(P P 1 GPa) (40) Protein Kinase A _dt^dPKa = (⁽^kg3_kmg3+⁾⁽^C⁾⁽_kt−PKa^kt−PKa⁾)−(⁽⁽^kg4⁾⁽^P_kmg4+^P¹⁺^P^P¹_PKa^GPa⁾⁽^PKa⁾⁾) (41) cAMP-dependent kinase

cassette

d

dtR 2 C2 = (−k gc1)(R 2 C2 )(cAM P ²)+ (k −gc1)(R 2 CcAMP2 )(C) (42)

“” _dt^dC = (k gc1)(R 2 C2 )(cAM P ²)−(k −gc1)(R 2 CcAMP2 )(C)+ (k gc2)(R 2 CcAMP2 )(cAM P ²)−

(k −gc2)(R 2 CcAMP4 )(C) ⁽⁴³⁾

“” _dt^dR 2 CcAMP2 = (k gc1)(R 2 C2 )(cAM P ²)−(k −gc1)(R 2 CcAMP2 )(C)−

(k gc2)(R 2 CcAMP2 )(cAM P ²)+ (k _−gc2)(R 2 CcAMP4 )(C) ⁽⁴⁴⁾

“” _dt^dR 2 CcAMP4 = (k gc2)(R 2 CcAMP2 )(cAM P ²)−(k −gc2)(R 2 CcAMP4 )(C) (45) Dynamic equilibrium constant

for glycogen degradation

kd = (kma xd −kmind )(₁₊¹glyc kdmg

n)+ kmind (46)

Forward reaction rate for glycogen degradation

k a = ^k_kd^−a (47)

Cell energy charge CE = ₍_{AT P}⁽^{AT P+}₊_ADP^ADP₊²_AMP⁾ ₎ (48)

Cell oxidative status CO = ^[_[^NA_NADH^D⁺^]_] (49)

Neuromodulation Module

Rise time constants for NE τne1= 10 ms (50)

Decay time constant for NE τne2= 50 0 0 0 0 (51)

Norepinephrine waveform NE = e ⁻⁽^t^−t^st^im⁾^/^τ^ne²−e ⁻⁽^t^−t^st^im⁾^/^τ^ne¹ (52)

∗ When two values are indicated, the ﬁrst one corresponds to the neuronal compartment and the second one to the astrocytic compartment.

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Table 2

Rates, transports and currents

Reaction, transport or current Equation

Sodium leak J ^x_leak_,_Na= ^S^m_F^V^xg ^x_Na[ ^RT_F log (Na ⁺_e/ Na ⁺_x)−ψx] (53) Na,K-ATPase J ^xpump= S mV xk ^xpumpAT P xNa ⁺x(1 + _K_m^AT_,_pump^P^x)⁻¹ (54) Glucose transport J ^xy_GLC= T _max^xy_,_GLC(_GL_C_x^GL₊^C_K^xxy

t,GLC−_GL_C^GL_y₊^C_K^yxy

t,GLC) (55)

Hexokinase-phosphofructokinase J ^x_HKPFK= k ^x_HKPFKAT P x GLCx

GLCx+Kg[ 1 + (_K^AT_I_,_ATP^P^x)^nH] ⁻¹, GLCx = GLCg + g6p (astrocyte) (56) Phosphoglycerate kinase J ^x_PGK= k ^x_PGK GA P x AD P x(N −NADH ^cyto_x )/ NADH ^cyto_x (57)

Pyruvate kinase J ^x_PK= k ^x_PKPE P xAD P x (58)

Lactate dehydrogenase J ^x_LDH= k ^x+_LDHPY R xNADH ^cyto_x −k ^x_LDH⁻ PY R x(N −NADH ^cyto_x ) (59) Lactate transport J ^xy_LAC= T _max^xy_,_LAC(_LA_C_x^LA₊^C_K^x^xy

t,LAC−_LA_C^LA^C^y

y+K^xy_t_,_LAC) (60)

TCA cycle J ^x_mito_,_in= V _max^x _,_in _PY_R^PY^R^x

x+Km^mito

N−NADH^mitox

N−NADH^mitox +K_m^x_,_NAD (61)

Electron transport chain J ^x_mito_,_out= V max^x ,out O2x O2x+K_O^mito₂

ADPx ADPx+K^x_m_,_ADP

NADH^mito_x

NADH^mito_x +K_m^x_,_NADH (62) NADH shuttles J ^x_shuttle= T _NADH^x _R−^R⁻^x

x+M^cytox R⁺x

R⁺x+M^mitox (63) ^†

Creatine kinase J ^x_CK= k ^x+_CKADP xPC r x−k ^x_CK⁻AT P x(C −PCr x) (64) Oxygen exchange J ^cx_O₂_m= ^P^S_V^cap_x (K O2(^Hb.OPO2c −1 )⁻¹^/^nh−O 2n) (65)

Capillary oxygen ﬂow J ^c_O₂= ²^F_Vⁱⁿ_cap⁽^t⁾(O 2a−O 2c) (66)

Capillary glucose ﬂow J ^c_GLC= ²^F_Vⁱⁿ_cap⁽^t⁾(GL C a−GL C c) (67)

Capillary lactate ﬂow J ^c_LAC= ²^F_Vⁱⁿ_cap⁽^t⁾(LA C a−LA C c) (68)

Oxygen concentration at the end of the capillary O 2¯c= 2 O 2c−O 2a (69)

Leak current I L= g L(ψn−E L) (70)

Sodium current I Na= g Nam ³_∞h (ψn−^RT_F log (Na ⁺_e/ Na ⁺_n)) (71) ^‡

Potassium current I K= g Kn ⁴(ψn−E K) (72) ^‡

Calcium current I Ca= g Cam ²_Ca(ψn−E Ca) (73) ^‡

Calcium-dependent potassium current I mAHP= g mAHP Ca²⁺

Ca²⁺+KD(ψn−E K) (74)

Na,K-ATPase current I pump= F k ⁿpumpAT P n(Na ⁺n−Na ⁺0)(1 + _K^AT_m_,_pump^P^x )⁻¹ (75) Flow out of the venous balloon F out= F0[ (_V^V_v0^v)¹^/^α^v+ _V^τ_v0^V(_V^V_v0^v)⁻¹^/²^d_dt^V^v] , F 0 = 0 (76)

† With R ⁻x= NADH ^cytox / (N −NADH ^cytox )and R ⁺x= (N −NADH ^mitox )/ NADH ^mitox .

‡ Further equations in the Hodgkin-Huxley model are: αm= −0 . 1 (ψn+ 33 )/ (exp [ −0 . 1 {ψn+ 33 }] −1 ), βm= 4 exp [ −{ψn+ 58 }/ 12 ] , αh= 0 . 07 exp [ −{ψn+ 50 }^/¹⁰^]^, βh= 1 / (exp [ −0 . 1 {ψn+ 20 }^]⁺¹), αn= −0 . 01 (ψn+ 34 )/ (exp [ −0 . 1 {ψn+ 34 }^]⁻¹), βn= 0 . 125 exp [ −{ψn+ 44 }/ 25 ] , m ∞= αm(αm+ βm)⁻¹ , n ∞= αn(αn+ βn)⁻¹, h ∞= αh(αh+ βh)⁻¹, τn= 10 ⁻³(αn+ βn)⁻¹, τh= 10 ⁻³(αh+ βh)⁻¹, m Ca= 1 / (1 + exp [ −{ψn+ 20 }/ 9 ] ) and E L= (g ^pas_K + g ⁿ_Na)⁻¹[ g ^pas_K E K+ g ⁿ_Na^RT_F log (Na ⁺e/ Na ⁺n)] .

termed“states” andrangefrom1-5indescendingamplitude(with 1beingthehighestand5thelowest,Fig1A).

All enzyme cascadesimulations were carried-out in NEURON, usinga ﬁxed time step of 3

μ

^s ^with ^Euler integration,and was runeitheronanUbuntu14.04LTSworkstationwitha3.6GHzIn- tel Core i7-4790 CPU and 15.6 GB RAM, or on the Blue Gene/Q in Lugano, Switzerland. Matlab was used for data analysis. AP andEPSPsimulations were performedon the EPFLBlue Brain IV BlueGene/Q hosted at the Swiss National Supercomputing Cen- ter (CSCS) in Lugano.Parameters andequations for the NEURON simulation environment model are listed in Tables 1 (Governing equations),2(Rates,transportsandcurrents),and3(parameters).

The 5concentrations ofcAMP corresponding to cascadestimula- tionstates 1–5were obtainedby setting x= 0.000009,0.00002, 0.00005,0.00008,or0.00011inEq.(36)inTable1.

3. Results

In this study, we made use of glycogenolysis and the subse- quentglycolyticcascadefromapreviouslypublishedmodelofen- ergymetabolism in glia(Coggan etal., 2018) to demonstrate the usefulnessofanoveltheoreticalanalysisofthecharacteristicsand behavioralpropertiesofasetofenzymaticreactions.

This segment of the metabolic energy cascade in our astrocyte model was stimulated into ﬁve excitability states by simu- latingthe production of5 concentrations of the second messen- gercAMP(representing 5levels ofnoradrenergic activationofthe

β

2-adrenergic receptor as in (Coggan et al., 2018)). Diagramat- ically (Fig. 1A1), the arrows between cAMP and PYR represent theenzyme sequenceproteinkinaseA(PKa),proteinphosphatase 1 (PP1), glycogen phosphorylase (GPa), glycogen synthase (GSa), phosphoglucokinase(PGK),pyruvatekinase(PK),andlactatedehy-

drogenase(LDH);aswellastheintermediatemetabolitesglucose- 6-phosphate(G6P),glyceraldehydephosphate(GAP),andphospho- enolpyruvate (PEP).Thetrajectoriesofselected metabolitescAMP, PYR,LACandATPareshowninFig.1,B–E,respectively.Derivative phase plotsare another wayto demonstratethe phasetransition betweenlower(cyan,blue)andhigher(green,red,black)excitabil- itystates,corresponding toconcentrationsofcAMPandexamples madeofLAC(Fig.1F)andATP(Fig.1G).

3.1. Excitation-statephaseplots

We examined the relationship between cAMP (the upstream trigger for the enzyme cascade we are analyzing) and a sample of several downstream intermediate metabolites (the “X” in the panel title).Plotting the relationshipbetween cAMPandselected downstream metabolites(cAMP→X) yields a typeof phaseplot (Fig.2A).Eachcoloredplotisgeneratedfromanindependentsim- ulation withadifferent concentration ofcAMP. Theseplots show differentkindsofrelationshipsvaryingfromessentiallynohystere- sis(forcAMPvs.G6P,meaning[G6P]closelyfollows[cAMP])anda varietyofdifferenthysteresisshapesforothermetabolites.Charac- teristics oftheseplotsincludetheclosefollowingby theconcen- trationofadownstreammetabolitetotheconcentrationexcursion ofcAMP,asinthecaseof[cAMP]vs.[G6P](theG6Pconcentration mirrors thecAMPconcentration withoutsigniﬁcant deviation),or a varietyof hysteresis types(all other responses inFig. 2A) with apparentresponse envelopesofvarious shapesintowhichthere- sponses “grow” and are contained with higher excitation states.

Theseresponseenvelopestherebyallowpredictionoffuturestates.

Itisobservedthattheshapesofthecurvesqualitativelychange between the two lowest states (cyan and blue) and the higher states(green,redandblack),justasthetransitionbetweenstates