SKOV-3 ovarian carcinoma cells were cultured for Ebergy hours. Serine enery. Photosynthesis is a light-dependent process Black pepper extract for wound healing converts light energy into chemical energy, stored in the form of ATP and other energy-rich molecules metabollsm NADPH. GI weight loss to the decreased relevance of OXPHOS to the metabolic process in cancer, it may be represented by either ordered or disordered OXPHOS networks. Keto- lysis. It plays a critical role in facilitating various cellular activities, ranging from basic maintenance and growth to specialized functions like muscle contraction, nerve transmission, and cellular signaling. The turnover of ATP is rapid, as cells continuously consume and regenerate this vital molecule to meet their energy demands.

Cellular energy metabolism -

Retrieved 26 March From Friedrich Wöhler to Hans A. Rose S, Mileusnic R The Chemistry of Life. Penguin Press Science. Schneider EC, Sagan D Into the Cool: Energy Flow, Thermodynamics, and Life. University of Chicago Press. Lane N Oxygen: The Molecule that Made the World.

USA: Oxford University Press. Price N, Stevens L Fundamentals of Enzymology: Cell and Molecular Biology of Catalytic Proteins. Oxford University Press. ISBN X. Berg J, Tymoczko J, Stryer L Freeman and Company. Cox M, Nelson DL Palgrave Macmillan. Brock TD , Madigan MR, Martinko J, Parker J Brock's Biology of Microorganisms.

Benjamin Cummings. Da Silva JJ, Williams RJ The Biological Chemistry of the Elements: The Inorganic Chemistry of Life. Clarendon Press. Nicholls DG, Ferguson SJ Academic Press Inc. Wood HG February Wikiversity has learning resources about Topic:Biochemistry.

Wikibooks has more on the topic of: Metabolism. Look up metabolism in Wiktionary, the free dictionary. Wikimedia Commons has media related to Metabolism. Articles related to Metabolism. Metabolism map. Carbon fixation. Photo- respiration.

Pentose phosphate pathway. Citric acid cycle. Glyoxylate cycle. Urea cycle. Fatty acid synthesis. Fatty acid elongation. Beta oxidation. beta oxidation. Glyco- genolysis. Glyco- genesis. Glyco- lysis. Gluconeo- genesis. Pyruvate decarb- oxylation. Keto- lysis.

Keto- genesis. feeders to gluconeo- genesis. Light reaction. Oxidative phosphorylation. Amino acid deamination. Citrate shuttle. MVA pathway. MEP pathway. Shikimate pathway. Glycosyl- ation. Sugar acids. Simple sugars. Nucleotide sugars. Propionyl -CoA. Acetyl -CoA. Oxalo- acetate. Succinyl -CoA.

α-Keto- glutarate. Ketone bodies. Respiratory chain. Serine group. Branched-chain amino acids. Aspartate group. Amino acids. Ascorbate vitamin C.

Bile pigments. Cobalamins vitamin B Various vitamin Bs. Calciferols vitamin D. Retinoids vitamin A. Nucleic acids. Terpenoid backbones. Bile acids. Glycero- phospholipids.

Fatty acids. Glyco- sphingolipids. Polyunsaturated fatty acids. Endo- cannabinoids. Metabolism , catabolism , anabolism.

Metabolic pathway Metabolic network Primary nutritional groups. Purine metabolism Nucleotide salvage Pyrimidine metabolism Purine nucleotide cycle. Pentose phosphate pathway Fructolysis Polyol pathway Galactolysis Leloir pathway. Glycosylation N-linked O-linked. Photosynthesis Anoxygenic photosynthesis Chemosynthesis Carbon fixation DeLey-Doudoroff pathway Entner-Doudoroff pathway.

Xylose metabolism Radiotrophism. Fatty acid degradation Beta oxidation Fatty acid synthesis. Steroid metabolism Sphingolipid metabolism Eicosanoid metabolism Ketosis Reverse cholesterol transport.

Metal metabolism Iron metabolism Ethanol metabolism Phospagen system ATP-PCr. Fructose-bisphosphate aldolase Aldolase A , B , C Triosephosphate isomerase. Glyceraldehyde 3-phosphate dehydrogenase Phosphoglycerate kinase Phosphoglycerate mutase Enolase Pyruvate kinase PKLR , PKM2.

Pyruvate carboxylase Phosphoenolpyruvate carboxykinase. Lactate dehydrogenase. Alanine transaminase. Glycerol kinase Glycerol dehydrogenase. Fructose 6-P,2-kinase:fructose 2,6-bisphosphatase PFKFB1 , PFKFB2 , PFKFB3 , PFKFB4 Bisphosphoglycerate mutase. Metabolism : carbohydrate metabolism fructose and galactose enzymes.

Hepatic fructokinase Aldolase B Triokinase. Sorbitol dehydrogenase Aldose reductase. Lactose synthase Lactase. Mannose phosphate isomerase. Metabolism : carbohydrate metabolism proteoglycan enzymes.

L-xylulose reductase L-gulonolactone oxidase UDP-glucuronate 5'-epimerase Xylosyltransferase Sulfotransferase Heparan sulfate EXT1 EXT2 Chondroitin sulfate PAPSS1 PAPSS2. Iduronatesulfatase Iduronidase. Heparan sulfamidase N-acetyltransferase Alpha-N-acetylglucosaminidase Glucuronidase N-acetylglucosaminesulfatase.

Arylsulfatase B Galactosamine-6 sulfatase Beta-galactosidase GLB1. Metabolism : carbohydrate metabolism · glycoprotein enzymes. Dolichol kinase GCS1 Oligosaccharyltransferase. Neuraminidase Beta-galactosidase Hexosaminidase mannosidase alpha-Mannosidase beta-mannosidase Aspartylglucosaminidase Fucosidase NAGA.

N-acetylglucosaminephosphate transferase. Metabolism , lipid metabolism , glycolipid enzymes. Glycosyltransferase Sulfotransferase. From ganglioside Beta-galactosidase Hexosaminidase A Neuraminidase Glucocerebrosidase From globoside Hexosaminidase B Alpha-galactosidase Beta-galactosidase Glucocerebrosidase From sphingomyelin Sphingomyelin phosphodiesterase Sphingomyelin phosphodiesterase 1 From sulfatide Arylsulfatase A Galactosylceramidase.

Ceramidase ACER1 ACER2 ACER3 ASAH1 ASAH2 ASAH2B ASAH2C. Sphingosine kinase. Palmitoyl protein thioesterase Tripeptidyl peptidase I CLN3 CLN5 CLN6 CLN8.

Serine C-palmitoyltransferase SPTLC1 Ceramide glucosyltransferase UGCG. Metabolism : lipid metabolism — eicosanoid metabolism enzymes. Phospholipase A2 Phospholipase C Diacylglycerol lipase. Cyclooxygenase PTGS1 PTGS2 PGD2 synthase PGE synthase Prostaglandin-E2 9-reductase PGI2 synthase TXA synthase.

ATP citrate lyase Acetyl-CoA carboxylase. Beta-ketoacyl-ACP synthase Β-Ketoacyl ACP reductase 3-Hydroxyacyl ACP dehydrase Enoyl ACP reductase.

Stearoyl-CoA desaturase Glycerolphosphate dehydrogenase Thiokinase. Carnitine palmitoyltransferase I Carnitine-acylcarnitine translocase Carnitine palmitoyltransferase II.

Acyl CoA dehydrogenase ACADL ACADM ACADS ACADVL ACADSB Enoyl-CoA hydratase MTP : HADH HADHA HADHB Acetyl-CoA C-acyltransferase. Enoyl CoA isomerase 2,4 Dienoyl-CoA reductase. Propionyl-CoA carboxylase.

Hydroxyacyl-Coenzyme A dehydrogenase. Malonyl-CoA decarboxylase. Long-chain-aldehyde dehydrogenase. Metabolism : amino acid metabolism - urea cycle enzymes. Carbamoyl phosphate synthetase I Ornithine transcarbamylase. Argininosuccinate synthetase Argininosuccinate lyase Arginase.

N-Acetylglutamate synthase Ornithine translocase. Enzymes involved in neurotransmission. Histidine decarboxylase. Histamine N-methyltransferase Diamine oxidase.

Tyrosine hydroxylase Aromatic L-amino acid decarboxylase Dopamine beta-hydroxylase Phenylethanolamine N-methyltransferase. Catechol-O-methyl transferase Monoamine oxidase A B.

Glutamate decarboxylase. Tryptophan hydroxylase Aromatic L-amino acid decarboxylase Aralkylamine N-acetyltransferase Acetylserotonin O-methyltransferase. Nitric oxide synthase NOS1 , NOS2 , NOS3. Choline acetyltransferase. Cholinesterase Acetylcholinesterase , Butyrylcholinesterase.

Enzymes involved in the metabolism of heme and porphyrin. Aminolevulinic acid synthase ALAS1 ALAS2. Porphobilinogen synthase Porphobilinogen deaminase Uroporphyrinogen III synthase Uroporphyrinogen III decarboxylase. Coproporphyrinogen III oxidase Protoporphyrinogen oxidase Ferrochelatase.

Heme oxygenase Biliverdin reductase. glucuronosyltransferase UGT1A1. Metabolism of vitamins , coenzymes, and cofactors. Retinol binding protein.

Alpha-tocopherol transfer protein. liver Sterol hydroxylase or CYP27A1 renal Hydroxyvitamin D 3 1-alpha-hydroxylase or CYP27B1 degradation 1,Dihydroxyvitamin D 3 hydroxylase or CYP24A1.

Vitamin K epoxide reductase. Thiamine diphosphokinase. Indoleamine 2,3-dioxygenase Formamidase. Pantothenate kinase. Dihydropteroate synthase Dihydrofolate reductase Serine hydroxymethyltransferase.

Methylenetetrahydrofolate reductase. MMAA MMAB MMACHC MMADHC. L-gulonolactone oxidase. Riboflavin kinase.

GTP cyclohydrolase I 6-pyruvoyltetrahydropterin synthase Sepiapterin reductase. PCBD1 PTS QDPR. MOCS1 MOCS2 MOCS3 Gephyrin. Metabolism : Protein metabolism , synthesis and catabolism enzymes.

Essential amino acids are in Capitals. Saccharopine dehydrogenase Glutaryl-CoA dehydrogenase. D-cysteine desulfhydrase.

L-threonine dehydrogenase. Histidine ammonia-lyase Urocanate hydratase Formiminotransferase cyclodeaminase. Ornithine aminotransferase Ornithine decarboxylase Agmatinase.

Glutamate dehydrogenase. Branched-chain amino acid aminotransferase Branched-chain alpha-keto acid dehydrogenase complex Enoyl-CoA hydratase 3-hydroxyisobutyryl-CoA hydrolase 3-hydroxyisobutyrate dehydrogenase Methylmalonate semialdehyde dehydrogenase.

Branched-chain amino acid aminotransferase Branched-chain alpha-keto acid dehydrogenase complex 3-hydroxymethylbutyryl-CoA dehydrogenase. Threonine aldolase. Propionyl-CoA carboxylase Methylmalonyl CoA epimerase Methylmalonyl-CoA mutase.

Metabolism : amino acid metabolism nucleotide enzymes. Ribose-phosphate diphosphokinase Amidophosphoribosyltransferase Phosphoribosylglycinamide formyltransferase AIR synthetase FGAM cyclase Phosphoribosylaminoimidazole carboxylase Phosphoribosylaminoimidazolesuccinocarboxamide synthase IMP synthase.

Adenylosuccinate synthase Adenylosuccinate lyase reverse AMP deaminase. IMP dehydrogenase GMP synthase reverse GMP reductase. Hypoxanthine-guanine phosphoribosyltransferase Adenine phosphoribosyltransferase. Adenosine deaminase Purine nucleoside phosphorylase Guanine deaminase Xanthine oxidase Urate oxidase.

CAD Carbamoyl phosphate synthase II Aspartate carbamoyltransferase Dihydroorotase. CTP synthetase. Ribonucleotide reductase Nucleoside-diphosphate kinase DCMP deaminase Thymidylate synthase Dihydrofolate reductase. Acetyl-Coenzyme A acetyltransferase HMG-CoA synthase regulated step.

HMG-CoA lyase 3-hydroxybutyrate dehydrogenase Thiophorase. HMG-CoA reductase. Mevalonate kinase Phosphomevalonate kinase Pyrophosphomevalonate decarboxylase Isopentenyl-diphosphate delta isomerase. Dimethylallyltranstransferase Geranyl pyrophosphate.

Farnesyl-diphosphate farnesyltransferase Squalene monooxygenase Lanosterol synthase. Lanosterol 14α-demethylase Sterol-C5-desaturase-like 7-Dehydrocholesterol reductase. Cholesterol 7α-hydroxylase Sterol hydroxylase. Cholesterol side-chain cleavage. Aromatase 17β- HSD. Steroid metabolism : sulfatase Steroid sulfatase sulfotransferase SULT1A1 SULT2A1 Steroidogenic acute regulatory protein Cholesterol total synthesis Reverse cholesterol transport.

Metabolism : carbohydrate metabolism · pentose phosphate pathway enzymes. Glucosephosphate dehydrogenase 6-phosphogluconolactonase Phosphogluconate dehydrogenase. Phosphopentose isomerase Phosphopentose epimerase Transketolase Transaldolase.

Metabolism - non-mevalonate pathway enzymes. DXP synthase DXP reductoisomerase 2-C-methyl-D-erythritol 4-phosphate cytidylyltransferase 2-C-methyl-D-erythritol 2,4-cyclodiphosphate synthase 4- cytidine 5'-diphospho C-methyl-D-erythritol kinase 4-hydroxymethylbutenyl diphosphate synthase 4-hydroxymethylbutenyl diphosphate reductase.

Food science. Allergy Engineering Microbiology Nutrition Diet clinical Processing Processing aids Psychology Quality Sensory analysis Discrimination testing Rheology Storage Technology.

Food chemistry. Additives Carbohydrates Coloring Enzymes Essential fatty acids Flavors Fortification Lipids "Minerals" Chemical elements Proteins Vitamins Water. Food preservation. Biopreservation Canning Cold chain Curing Drying Fermentation Freeze-drying Freezing Hurdle technology Irradiation Jamming Jellying Jugging Modified atmosphere Pascalization Pickling Potting Confit Potjevleesch Salting Smoking Sugaring Tyndallization Vacuum packing.

Food portal Category: Food preservation. Manufacturing Packaging Marketing Foodservice Fortification. Consumer food safety. Flavorings Monosodium glutamate MSG Salt Sugar High-fructose corn syrup. Amoebiasis Anisakiasis Cryptosporidiosis Cyclosporiasis Diphyllobothriasis Enterobiasis Fasciolopsiasis Fasciolosis Giardiasis Gnathostomiasis Paragonimiasis Toxocariasis Toxoplasmosis Trichinosis Trichuriasis.

Botulism Campylobacter jejuni Clostridium perfringens Cronobacter Enterovirus Escherichia coli OH4 Escherichia coli OH7 Hepatitis A Hepatitis E Listeria Norovirus Rotavirus Salmonella Vibrio cholerae. Chlorpyrifos DDT Lindane Malathion Methamidophos.

Benzoic acid Ethylenediaminetetraacetic acid EDTA Sodium benzoate. Acesulfame potassium Aspartame controversy Saccharin Sodium cyclamate Sorbitol Sucralose.

Aflatoxin Arsenic contamination of groundwater Benzene in soft drinks Bisphenol A Dieldrin Diethylstilbestrol Dioxin Mycotoxins Nonylphenol Shellfish poisoning. Devon colic Swill milk scandal Esing Bakery incident Bradford sweets poisoning English beer poisoning Morinaga Milk arsenic poisoning incident Minamata disease Iraq poison grain disaster Toxic oil syndrome Austrian diethylene glycol wine scandal United Kingdom BSE outbreak Australian meat substitution scandal Jack in the Box E.

coli outbreak Odwalla E. coli outbreak North American E. coli outbreaks ICA meat repackaging controversy Canada listeriosis outbreak Chinese milk scandal Irish pork crisis United States salmonellosis outbreak Germany E.

coli outbreak United States listeriosis outbreak Bihar school meal poisoning incident horse meat scandal Mozambique funeral beer poisoning Brazil Operation Weak Meat — South African listeriosis outbreak Australian rockmelon listeriosis outbreak Australian strawberry contamination Food safety incidents in China Food safety incidents in Taiwan Foodborne illness outbreaks death toll United States.

Acceptable daily intake E number Food labeling regulations Food libel laws Food safety in Australia International Food Safety Network ISO Nutrition facts label Organic certification Quality Assurance International United Kingdom food information regulations.

The origins of our cellular metabolism model lie in the work of Lancaster et al. where glycolysis and OXPHOS are each represented by bi-directionally coupled non-autonomous Kuramoto oscillators Kuramoto, , and each driven by a non-autonomous oscillator depicting the supply of glucose and oxygen, respectively Lancaster et al.

This model was built on the theory of chronotaxicity Suprunenko et al. However, like most biological processes, neither glycolysis nor OXPHOS are a single process, but many Kurz et al. Glycolysis occurs distributed throughout the cytosol, while OXPHOS is localized within the many mitochondria of the cell.

These processes further communicate between themselves as well as one another. Glycolysis was found to signal inter- and intra-cellularly through the exchange of acetaldehyde Richard, ; Madsen et al.

Here, we extend the Lancaster et al. These networks are furthermore weighted such that oscillators closer to each other around a ring are connected more strongly than those further from one another, to reflect the nature of molecular exchange over a range of distances.

We also draw from the work of Petkoski and Stefanovska ; Petkoski et al. who introduced a method of phase coupling through mean fields of ensembles of oscillators.

We present here a summary of the Lancaster et al. We will discuss further the analysis that had and can be done on these models, and what they can reveal about the biology of the cellular production of ATP and its role in wider processes.

Our modeling approach consists of four main principles, which are summarized in Table 1. We consider the cell to be the minimal functioning biological unit: processes within the cell cannot be isolated and still function and more macroscopic functions can be built from a cellular level, but the cell itself can survive provided the appropriate molecular supply in its environment.

It is crucial however that the cell is able to expel waste and absorb needed molecules. This makes the cell a thermodynamically open system: matter and energy must cross its boundaries in order for the cell to survive. One of the principles of our approach is therefore to treat the cell and its internal processes as open, constructing a model that does not impose a constant mass on the system.

While many models make mass their subject, it is much easier to achieve the aim of an open system by focusing the phase of the processes instead, and so in our model we consider the phase of oscillations.

Table 1. Summary of the principles informing our modeling approach contrasted to those of mainstream approaches. Our second principle is to treat oscillating systems as not just a temporary perturbation from a steady state, but as fundamentally defined by their oscillations. We therefore do not construct our model as a non-oscillating set of processes and subsequently find sets of parameters that induce oscillations, but set oscillations as the foundation of the model by representing each process with a phase oscillator.

Cellular processes are also inevitably characterized by their non-linearity Carballido-Landeira and Escribano, , and modeling these non-linearities is essential to understanding their dynamics.

We therefore use Kuramoto oscillations to model these interactions. Unlike theories that assume variations in the features of these oscillations, in particular frequency, are due solely to noise endemic to the complexity of biological systems, we treat much of these observable variations as deterministic.

Our modeling approach to these systems is to represent them as non-autonomous Kuramoto phase oscillators. The biological system as considered in this model is summarized in Figure 1A , and represented in the model's format in Figure 1B. It is constituted by four key processes: glycolysis, converting glucose, ATP and ADP into NADH, pyruvate and ATP, OXPHOS, converting oxygen, NADH and pyruvate into ATP, and the supplies of glucose and oxygen.

The main purpose of this mechanism is the creation of ATP, which is primarily used to fuel ion pumps. Ion pumps actively transport ions across the cell's boundary against the electrochemical gradient, without which the cell would be forced to maintain an ionic equilibrium with its surroundings.

Instead, the cell is able to accept the ions it needs for survival, and prevent itself from being flooded with an unhealthy quantity. Neuronal firing also relies on the ability of ion pumps to dramatically and rapidly change the balance of ions between the cell interior and exterior: the process is triggered only once the cell's membrane potential crosses the action potential threshold, typically requiring a change of some mV Catterall et al.

Figure 1. A The cellular energy metabolism considered in the model, reprinted with permission from Lancaster et al. B An oscillator model diagram of A , where each circle represents an oscillator, and each line a coupling. MO denotes the mitochondrial oscillator, GO the glycolysis, G the glucose driving, and O the oxygen.

Communication between the metabolic processes is also well-established Richard, ; Madsen et al. Glycolysis enzymes exist all around the cytosol, each facilitating an element of the wider glycolytic reaction.

Not only do these distributed enzymes rely on regulation and supply common to them all, but the exchange of acetaldehyde molecules has been observed to drive coherence between glycolytic processes. Mitochondria, housing the OXPHOS process, exist in a more fixed state than the glycolysis enzymes of the cytosol, but are similarly thought to mutually organize their processes for the efficient running of the cell.

Mapping precisely the exact positions and connections of these processes however would be challenging, if not impossible.

In our model we therefore focus on the importance of molecular exchanges in their communication, and the diffusive nature of these exchanges making distance a key consideration. Hence, we have assumed all-to-all coupled networks, but weighted these connections such that if were they considered around a ring, coupling strength would decrease the further apart any two given oscillators were.

Each of these four metabolic processes is represented by a Kuramoto oscillator. Kuramoto oscillators are a type of non-linearly interacting phase oscillators, which are a reduction of ordinary differential equations featuring self-sustaining oscillations from many degrees of freedom to just one: the phase of the oscillation.

The phase of an oscillator is defined as its position along its cycle at a given time. This cycle can be represented in phase space, as shown in Figure 2A , where the meaning of any particular phase value can easily be seen.

Figure 2. A An oscillatory cycle in phase space, at a phase value of θ. B A point perturbed from an oscillatory cycle, returning along isochron I to the cycle at a point with phase φ.

The perturbed point is therefore also assigned the phase φ. Here phase has only been defined on the cycle of the oscillator equation. However, when oscillators interact or are driven by external forces, they will be perturbed away from this cycle. The phase in the vicinity of the cycle must therefore also be defined, which can be done for stable oscillators using isochrons.

When a stable oscillator is perturbed its phase will initially leave its cycle, but will return to it over time if not further perturbed. Isochrons connect the point to which a phase is perturbed to the point on the cycle it will first return to after the decay of the perturbation, assigning both the same phase value.

This is demonstrated in Figure 2B. In order to remain in this region of attraction of the cycle, where isochrons can be used, the perturbations must be sufficiently weak, placing constraints on the strength of couplings between oscillators and drivers Pikovsky et al.

This definition requires further extension to allow for the phases of non-autonomous oscillators. As the frequency, also known as the velocity of the phase, of the oscillator changes at each moment in time the system is transformed from one autonomous system to another.

To maintain a consistent definition of phase across these systems, we must require that each system resides in the region of attraction of the one proceeding it, in order to use the same reasoning as the isochrons of perturbations.

As with the weak coupling requirements of interactions, this definition constrains the system to only small changes in the frequency of oscillation from second to second Kloeden and Rasmussen, This theory was applied to the biology of cellular ATP production by Lancaster et al.

where the subscript GO represents the glycolytic oscillator, MO the OXPHOS, G the glucose driving and O the oxygen. ω X is the frequency of oscillator X , ϵ the relevant oscillator coupling strength, θ X the phase, t time, η t a noise term and σ the scaling parameter of the noise.

These are hence two oscillators as described above, coupled to one another and their respective metabolic drivers, with their frequency rendered non-autonomous by the addition of a time-dependent noise parameter. We convert this model to now consist of networks of oscillators, weighted such that neighbors around a ring interact with a maximal coupling strength, and those opposite with a minimal strength.

This is shown diagrammatically in Figure 3. Figure 3. The network cellular metabolism model, with each circle representing an oscillator and each line a coupling. We also consider, instead of the stochastic non-autonomicity in Lancaster et al. This gives the glycolysis and OXPHOS phase equations as.

respectively, where θ GOni is the phase of the oscillator i due to network interactions, N is the number of glycolytic oscillators, M the number of OXPHOS oscillators, K X the relevant network coupling strength and W ij the weighting function between oscillators i and j. The oscillators are organized into all-to-all couple networks, with a certain weight applied to each coupling.

The weight of the coupling between these oscillators is determined by their indices, such that the larger the difference between the indices, the smaller the weighting of their coupling. where θ GOGi is the phase of glycolysis oscillator i due to glucose coupling.

The inter-network interactions arise through coupling each network to the mean field of the other Strogatz and Mirollo, ; Petkoski and Stefanovska, ; Petkoski et al. This mean field arises as the average of each individual oscillation, characterizing their collective state.

The inter-network equations therefore are. In this paper we use the deterministic variation formulation for these frequencies, but any other time varying formulation, such as random noise, are also valid methods provided that the variation is slow.

ω G is the mean frequency around which the non-autonomous frequency is modulated, A G is the amplitude of modulation of the frequency, ω Gm is the frequency of modulation and t i is a perturbation of the modulation in time, taking a random number between 0 and 1 ω G m s. This perturbation ensures a distribution of frequencies within each element, while assigning the oscillators the same mean frequency and deterministic cycle of modulation.

The phenomenon of synchronization between oscillators is a key part of understanding their dynamics. Oscillators can be considered synchronized when the difference between their phases remains constant. This is well-established in the context of permanent synchronization, where the phase difference between two oscillators does not ever change unless the parameters of the system change or a new influence is introduced Pikovsky et al.

This phenomenon has only been observed for non-autonomous oscillators, and only when examined over finite time periods. When observed in an asymptotic, averaging time scale, it can easily be mistaken for complete desynchronization.

For living systems, synchronization between oscillators represents a state of stability and cooperative working between oscillators.

Synchronized oscillators are, to an extent, able to resist perturbation away from this state and coordinate their oscillations for a variety of ends, including temporally compartmentalizing conflicting processes Tu et al.

As in, for example, Lancaster et al. We will apply these methods of synchronization analysis to our cellular metabolism model. We conducted analysis of the model to determine the impacts on the dynamics made by the additions of weighted networks and deterministic non-autonomicity to the Lancaster et al.

These simulations involved numerical integration of the differential phase equations, defined in Equation 7. This was conducted using the inbuilt Matlab ode15s algorithm, which is a partially implicit numerical integration scheme using a variable integration step and evaluates errors through interpolated backwards differences Shampine and Reichelt, The equations were integrated for a period of 10, s at a sampling frequency of 0.

The first 5, s were discarded, assuming they were dominated by transient dynamics, and then the final 5, s analyzed to determine what, if any, modes of synchronization were present. This analysis involved calculating the phase coherence, as defined in Bandrivskyy et al. The phase difference between these components was also calculated, as was the Kuramoto order parameter of each network.

For autonomous systems, time series are defined as coherent at a phase coherence value of or close to 1. However in non-autonomous systems, series may be coherent yet exhibit a time-averaged phase coherence of significantly less than 1 due to their modulation in time away from their coherent mean.

Additionally, slight numerical simulation errors and noise can make it impossible to attain a numerical phase coherence of precisely 1.

Through observations of numerical simulations, we have therefore defined coherence greater than 0. If the coherence value was greater than 0. Networks were considered synchronized when their time-averaged Kuramoto order parameter exceeded 0.

This was considered permanent if the parameter varied by less than 0. Similarly to phase coherence, the Kuramoto order parameter of non-autonomous oscillations will naturally vary in time due to frequency modulation, even in highly ordered networks, and so simulations indicated that only variations of greater than 0.

We have also analyzed data collected by Amemiya et al. In this experiment, the optical NADH fluorescence of numerous cells was measured over time after glucose was added to their environment. We calculated the group phase coherence, as defined by Sheppard et al.

This coherence was further tested against 19 WIAAFT surrogates, as defined in Lancaster et al. We analyzed both groups near to one another and far from one another, to identify any significant differences between the two. The culture was μ m by μ m in area, and near groups were defined as having —μ m between their average positions, while the average positions of far groups were —1, μ m apart.

Simulations of this experiment were also conducted, using some of the results of the group coherence analysis and the general numerical simulations.

This was done by numerically integrating a realization of the system at a certain parameter set using a four step Runge-Kutta algorithm. The results of this and all the above methods are presented in the following section. There are six possible modes of synchronization within our cellular metabolism model: glycolysis to glucose, glycolysis network, glycolysis to OXPHOS, OXPHOS network, and OXPHOS to oxygen.

While it would not be possible for glycolysis to be synchronized to OXPHOS, but OXPHOS to not synchronize to glycolysis in an individual oscillator model, it is possible for a network to become synchronized to a mean field driving, without the network from which that mean field arises becoming synchronized to the network it is driving.

We examined whether each of these synchronizations occurred, and whether they were permanent or intermittent, at 2, different combinations of the parameters F GO and F MO , as defined in Equation 2. This is similar to the analysis conducted in Lancaster et al. The parameters for which these simulations were conducted are given in Table 2.

Most of these parameters, ϵ G , ϵ O , F GO , F MO , ω G , ω GO , ω MO , ω O , are identical to those used in Lancaster et al.

K GO and K MO did not exist in the Lancaster et al. model, and they have been set to be equal to the other non-varied coupling parameters. The frequencies and amplitudes of modulation were determined by their ratio to the mean frequencies, as studied by Lucas et al.

W may be set to 1 as the relevance of the weighted coupling is in the relative weighings between different oscillator pairs. N and M cannot be determined purely biologically: the glycolysis oscillators represent a collection of often-distributed glycolytic enzymes that are not realistically quantifiable, while the number of a mitochondria in a cell type can vary significantly Wilson, ; Chaudhry and Varacallo, Instead, the network sizes are chosen such that there a sufficiently many oscillators to validate the mean field approximation Strogatz and Mirollo, , and not so many as to make computational simulation infeasible.

We present first the analysis of the individual oscillator model of Lancaster et al. Figure 4. Analysis of the synchronization regimes at different parameter values, at parameter steps of 0.

Specifically, we study the molecular basis for how cells synthesize TGs, how cells form LDs, how proteins target to LDs, and how TGs are utilized to meet energetic demands. Our laboratory studies these fundamental questions using a wide range of cutting-edge interdisciplinary approaches, including biophysical, biochemical, and cell biological methods.

We also investigate the physiological importance of these lipid storage mechanisms in various cellular and animal models, including models of metabolic diseases or cancer. Identification of two pathways mediating protein targeting from ER to lipid droplets. Nat Cell Biol. Arlt H, Sui X, Folger B.

A, DiMaio F, Liao M, Goodman J.

Thank eneryg for visiting nature. You metabolisn using a Meatbolism version with limited enegy for CSS. To Cellular energy metabolism the best Cwllular, we Meal planning tips you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Altered cellular energy metabolism is a hallmark of many diseases, one notable example being cancer. Here, we focus on the identification of the transition from healthy to abnormal metabolic states. To do this, we study the dynamics of energy production in a cell.

Cellular energy metabolism -

The culture was μ m by μ m in area, and near groups were defined as having —μ m between their average positions, while the average positions of far groups were —1, μ m apart. Simulations of this experiment were also conducted, using some of the results of the group coherence analysis and the general numerical simulations.

This was done by numerically integrating a realization of the system at a certain parameter set using a four step Runge-Kutta algorithm. The results of this and all the above methods are presented in the following section. There are six possible modes of synchronization within our cellular metabolism model: glycolysis to glucose, glycolysis network, glycolysis to OXPHOS, OXPHOS network, and OXPHOS to oxygen.

While it would not be possible for glycolysis to be synchronized to OXPHOS, but OXPHOS to not synchronize to glycolysis in an individual oscillator model, it is possible for a network to become synchronized to a mean field driving, without the network from which that mean field arises becoming synchronized to the network it is driving.

We examined whether each of these synchronizations occurred, and whether they were permanent or intermittent, at 2, different combinations of the parameters F GO and F MO , as defined in Equation 2.

This is similar to the analysis conducted in Lancaster et al. The parameters for which these simulations were conducted are given in Table 2.

Most of these parameters, ϵ G , ϵ O , F GO , F MO , ω G , ω GO , ω MO , ω O , are identical to those used in Lancaster et al. K GO and K MO did not exist in the Lancaster et al. model, and they have been set to be equal to the other non-varied coupling parameters. The frequencies and amplitudes of modulation were determined by their ratio to the mean frequencies, as studied by Lucas et al.

W may be set to 1 as the relevance of the weighted coupling is in the relative weighings between different oscillator pairs. N and M cannot be determined purely biologically: the glycolysis oscillators represent a collection of often-distributed glycolytic enzymes that are not realistically quantifiable, while the number of a mitochondria in a cell type can vary significantly Wilson, ; Chaudhry and Varacallo, Instead, the network sizes are chosen such that there a sufficiently many oscillators to validate the mean field approximation Strogatz and Mirollo, , and not so many as to make computational simulation infeasible.

We present first the analysis of the individual oscillator model of Lancaster et al. Figure 4. Analysis of the synchronization regimes at different parameter values, at parameter steps of 0.

B for the Lancaster et al. individual oscillator model with added deterministic non-autonomous frequencies and intermittent synchronization analysis.

C for the unweighted network model. D for the weighted network model. Regimes are defined in Table 3. Introducing each new element of our model in turn to examine this same parameter space, we first include our deterministic variation of the frequency and analyse for intermittent synchronization, as well as permanent, but otherwise maintain the Lancaster et al.

The results are in Figure 4B , and the main regimes described in Table 3. This results in the splitting of the red region in the Lancaster et al.

The dark blue region, where only glycolysis and OXPHOS are synchronized, is also made significantly larger, and there are spots of intermittently synchronized regimes that appear only briefly throughout the parameter space.

The next step is to introduce unweighted networks of glycolysis and OXPHOS oscillators. The result is in Figure 4C. This introduces a new regime, where only the networks are internally synchronized, and converts the dark green regime, where there is no synchronization, into the even further increased dark blue regime.

Once again, intermittent regimes are spotted briefly throughout the parameter space. The final step in constructing our full model, is to weight the glycolysis and OXPHOS networks according to Equations 3 and 4.

Figure 4D shows the results of this final simulation. This splits the new regime observed in the previous simulation into the purple, green and cyan regimes: the purple representing the same permanent synchronization within each network, the green a new intermittent synchronization of the OXPHOS network, and the cyan a new intermittent synchronization of the glycolysis network.

The weighting reduces the size of the dark blue region, giving more space to the blue and light blue, and as in the previous simulations produces small regimes of intermittent synchronization. In Amemiya et al. This model adopted an approach more similar to the mainstream discussed in the previous section.

We therefore offer a comparison between this model and the one we have presented here, to help illuminate further the differences between our approach and ones more characteristic of the cellular modeling mainstream, applied in the context of this experiment.

The Amemiya et al. The former is modeled as the first step, converting glucose and ATP into intermediaries, while the second is the last reaction, converting these intermediaries into ATP and pools of NADH and other products.

The model focuses on the masses of the metabolites required for these reactions, from their entry into the cell to their consumption in the metabolic process. This technique consists of seven autonomous linear differential equations and twenty two parameters to model the glycolysis metabolic branch only, which contrasts to the four non-autonomous non-linear oscillator equations of Equation 7 and the thirteen parameters of Table 2 to model both the glycolysis and OXPHOS branches.

In addition to a measure of coherence within a network, the order parameter may also be considered the amplitude of the network's mean field. We can therefore consider it both an indication of the amplitude of our system, and the degree to which the glycolysis and OXPHOS networks are operating effectively.

We introduce a modified Kuramoto order parameter s , where. which takes into account both networks. This parameter can be compared to the time series of NADH fluorescence from a single cell in the Amemiya et al. We provide this comparison in Figure 6 , and this can be further compared to an equivalent output of the model in Figure 2 of Amemiya et al.

The modulation frequency of the glycolysis oscillations was extracted from group coherence analysis of the Amemiya et al.

data, which found that for both cell groups close to and far from one another there was significantly coherent oscillations in the range 0. This analysis is presented in Figure 5. The other frequencies were selected to maintain the same ratio with the extracted glycolysis modulation as discussed in Lancaster et al.

Over the next This results in a trending decrease in the networks' amplitude and the emergence of oscillations. After For the final s of the simulation F GO and F MO have reached 0 as the cells begin to die, their oscillations continue to diminish, and their NADH production dries up.

Figure 5. Surrogate tested coherence between groups of cells examined in Amemiya et al. Red coloring indicates groups of cells far from one another, —1, μ m distance between their average positions, and blue close to one another, —μ m distance between their average positions.

The dimensions of the culture were 1, by 1, μ m. The solid colored lines are the median coherence of each pair of groups, and the shaded regions the range from the minimum to maximum coherence. Figure 6. A Simulation of the HeLa experiment using a modified order parameter.

B The time series of NADH fluorescence in a single cell in the Amemiya et al. While the curve presented in Figure 6A depends on the initial phases of each oscillator, which are randomized, and therefore will not be identical from simulation to simulation, its oscillator features and overall trend are indicative of the parameters in Table 4.

And while this simulation is not an identical reflection of the experiment in every feature, it is an indication of the capacity of our model to reproduce the oscillating nature of biological processes, and the ease with which it can be adapted to a plethora of different cells and circumstances.

The conversion of established metabolic models, such as that of Lancaster et al. The step from Figures 4B,C for example overhauls the parameter space, introducing entirely new regimes and destroying once-firm fixtures of the non-network model.

It is clear from all of these results that networks result in an even greater area of the parameter space featuring synchronization, with the only regime of total desynchronization disappearing once networks are introduced, and the networks themselves never being desynchronized.

This aligns well with the imperative of such biological processes to remain robust against significant external perturbations, and the expectation that these parameter values do not represent catastrophic departure from the healthy state of the system.

More significant perturbations of the coupling parameters, to both higher values and the entire elimination of more coupling modes, are likely required to completely desynchronize the networks, which would represent even further departures from the healthy parameter states of the cell.

In healthy human cells, ATP is produced primarily through OXPHOS, with support from glycolysis. In our model, this may be represented by synchronization between the networks, and between the OXPHOS network and its oxygen driving Lancaster et al.

Internal synchronization of both networks is also required to characterize a healthy condition: disregulation within the metabolic processes is a key indicator of a malfunctioning cell. This state is represented in the bottom right of each graph in Figure 4 , but is significantly diminished in area with the addition of deterministic frequency modulation from Figure Figures 4A,B.

A cancerous state, may be indicated by an opposite state: a mode switch to the dominance of glycolysis, known as the Warburg effect, is reflected by synchronization between the networks and between glycolysis and glucose, but not OXPHOS and oxygen Lancaster et al.

Due to the decreased relevance of OXPHOS to the metabolic process in cancer, it may be represented by either ordered or disordered OXPHOS networks.

This regime is found in the top left of each of Figure 4 , similarly decreasing in area between Figures 4A,B as with the bottom right regime. Network models also offer greater potential for oscillator systems: while reducing oscillating differential equations to just their phase provides a much simpler system that still contains the key dynamics, only at the mesoscopic level of networks of many oscillators can the system amplitude be rebuilt.

Further work on this model could therefore provide not just an order parameter of the network indicative of its activity, but an amplitude of its production. The turn to deterministic non-autonomous frequencies and finite time synchronization analysis similarly promises a significant change to the dynamics of metabolic models.

However, with the introduction of this non-autonomicity comes greater challenges for numerical simulations: the numerical integration of non-linear oscillating differential equations is an already delicate task, and the addition of another dimension of time sensitivity requires alternative methods.

Further work with more sensitive numerical integration algorithms and more sophisticated methods for identifying intermittent synchronization would be likely to find a far greater role of the phenomenon in the model's parameter spaces, and further clarify exactly which dynamic we can expect to find at each parameter combination.

Non-autonomous oscillations pose a particular challenge to numerical integration schemes due to their two highly distinct frequency modes. Schemes designed to adapt to this situation may be able to provide greater clarity on our model, with which we may be able to further identify parameters leading to pathological states and more complex dynamics within the model.

The measured data analyzed in this paper were originally collected and presented by Amemiya et al. They are available at doi: The MatLab codes used for numerical modeling and analyses of the numerical and measured data can be found at doi: JRA proposed the inclusion of weighted networks, designed and ran the numerical simulations of the model, performed thorough investigation of the model behavior including the intermittent synchronization analysis, and drafted the manuscript.

AS conceived the study, and proposed to incorporate networks instead of individual oscillators used in the earlier version of the model. She oversaw the development of the work, provided regular guidance at every stage, and advised about the structure and the content of the manuscript.

All authors edited the manuscript and approved the submitted version. This work was funded by the EU's Horizon research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank Maxime Lucas, Yevhen Suprunenko, Julian Newman, Gemma Lancaster, Peter Kloeden, Sam McCormack, Juliane Bjerkan, Lars F. Olsen, Takashi Amemiya, Kenichi Shibata and Lawrence Sheppard for useful discussions of this work. Additionally we thank Takashi Amemiya, Kenichi Shibata, Yoshihiro Itoh, Kiminori Itoh, Masatoshi Watanabe and Tomohiko Yamaguchi for sharing their experimental data with us, and Lawrence Sheppard for sharing his group coherence algorithm.

Numerical simulations were conducted using the High End Computing facility at Lancaster University. Akter, K. Diabetes mellitus and Alzheimer's disease: shared pathology and treatment? doi: PubMed Abstract CrossRef Full Text Google Scholar.

Amemiya, T. Modeling studies of heterogeneities in glycolytic oscillations in hela cervical cancer cells. Chaos Primordial oscillations in life: Direct observation of glycolytic oscillations in individual hela cervical cancer cells.

CrossRef Full Text Google Scholar. Bandrivskyy, A. Wavelet phase coherence analysis: application to skin temperature and blood flow. Battle, C. Broken detailed balance at mesoscopic scales in active biological systems. Science , — Bechtel, W.

Complex Biological Mechanisms: Cyclic, Oscillatory, and Autonomous. Betz, A. Phase relationship of glycolytic intermediates in yeast cells with oscillatory metabolic control. Bosco, D. Possible implications of insulin resistance and glucose metabolism in Alzheimer's disease pathogenesis.

Carballido-Landeira, J. Nonlinear Dynamics in Biological Systems. New York, NY: Springer International Publishing. Catterall, W. The Hodgkin-Huxley heritage: from channels to circuits. Chaudhry, R. Biochemistry, Glycolysis. Treasure Island, FL: StatPearls Publishing.

Google Scholar. Cortassa, S. Computational modeling of mitochondrial function from a systems biology perspective. Mitochondrial Bioenerget. In fact, the Sun is the ultimate source of energy for almost all cells, because photosynthetic prokaryotes, algae, and plant cells harness solar energy and use it to make the complex organic food molecules that other cells rely on for the energy required to sustain growth, metabolism, and reproduction Figure 1.

Cellular nutrients come in many forms, including sugars and fats. In order to provide a cell with energy, these molecules have to pass across the cell membrane, which functions as a barrier — but not an impassable one. Like the exterior walls of a house, the plasma membrane is semi-permeable.

In much the same way that doors and windows allow necessities to enter the house, various proteins that span the cell membrane permit specific molecules into the cell, although they may require some energy input to accomplish this task Figure 2.

Figure 2: Cells can incorporate nutrients by phagocytosis. This amoeba, a single-celled organism, acquires energy by engulfing nutrients in the form of a yeast cell red.

Through a process called phagocytosis, the amoeba encloses the yeast cell with its membrane and draws it inside. Specialized plasma membrane proteins in the amoeba in green are involved in this act of phagocytosis, and they are later recycled back into the amoeba after the nutrients are engulfed.

Figure Detail. Complex organic food molecules such as sugars, fats, and proteins are rich sources of energy for cells because much of the energy used to form these molecules is literally stored within the chemical bonds that hold them together. Scientists can measure the amount of energy stored in foods using a device called a bomb calorimeter.

With this technique, food is placed inside the calorimeter and heated until it burns. The excess heat released by the reaction is directly proportional to the amount of energy contained in the food.

Figure 3: The release of energy from sugar Compare the stepwise oxidation left with the direct burning of sugar right. Through a series if small steps, free energy is released from sugar and stored in carrier molecules in the cell ATP and NADH, not shown.

On the right, the direct burning of sugar requires a larger activation energy. In this reaction, the same total free energy is released as in stepwise oxidation, but none is stored in carrier molecules, so most of it will be lost as heat free energy.

This direct burning is therefore very inefficient, as it does not harness energy for later use. In reality, of course, cells don't work quite like calorimeters. Rather than burning all their energy in one large reaction, cells release the energy stored in their food molecules through a series of oxidation reactions.

Oxidation describes a type of chemical reaction in which electrons are transferred from one molecule to another, changing the composition and energy content of both the donor and acceptor molecules. Food molecules act as electron donors. During each oxidation reaction involved in food breakdown, the product of the reaction has a lower energy content than the donor molecule that preceded it in the pathway.

At the same time, electron acceptor molecules capture some of the energy lost from the food molecule during each oxidation reaction and store it for later use. Eventually, when the carbon atoms from a complex organic food molecule are fully oxidized at the end of the reaction chain, they are released as waste in the form of carbon dioxide Figure 3.

Cells do not use the energy from oxidation reactions as soon as it is released. Instead, they convert it into small, energy-rich molecules such as ATP and nicotinamide adenine dinucleotide NADH , which can be used throughout the cell to power metabolism and construct new cellular components.

In addition, workhorse proteins called enzymes use this chemical energy to catalyze, or accelerate, chemical reactions within the cell that would otherwise proceed very slowly. Enzymes do not force a reaction to proceed if it wouldn't do so without the catalyst; rather, they simply lower the energy barrier required for the reaction to begin Figure 4.

Figure 4: Enzymes allow activation energies to be lowered. Enzymes lower the activation energy necessary to transform a reactant into a product. On the left is a reaction that is not catalyzed by an enzyme red , and on the right is one that is green.

In the enzyme-catalyzed reaction, an enzyme will bind to a reactant and facilitate its transformation into a product.

Consequently, an enzyme-catalyzed reaction pathway has a smaller energy barrier activation energy to overcome before the reaction can proceed.

Figure 5: An ATP molecule ATP consists of an adenosine base blue , a ribose sugar pink and a phosphate chain.

The high-energy phosphate bond in this phosphate chain is the key to ATP's energy storage potential. Figure Detail The particular energy pathway that a cell employs depends in large part on whether that cell is a eukaryote or a prokaryote.

Eukaryotic cells use three major processes to transform the energy held in the chemical bonds of food molecules into more readily usable forms — often energy-rich carrier molecules.

Adenosine 5'-triphosphate, or ATP, is the most abundant energy carrier molecule in cells. This molecule is made of a nitrogen base adenine , a ribose sugar, and three phosphate groups. The word adenosine refers to the adenine plus the ribose sugar. The bond between the second and third phosphates is a high-energy bond Figure 5.

The first process in the eukaryotic energy pathway is glycolysis , which literally means "sugar splitting. Glycolysis is actually a series of ten chemical reactions that requires the input of two ATP molecules. This input is used to generate four new ATP molecules, which means that glycolysis results in a net gain of two ATPs.

Two NADH molecules are also produced; these molecules serve as electron carriers for other biochemical reactions in the cell. Glycolysis is an ancient, major ATP-producing pathway that occurs in almost all cells, eukaryotes and prokaryotes alike.

The external influences on the GO and MO are represented as unidirectionally coupled drivers, with coupling strengths ε 3 and ε 4 for O and G respectively, which may also vary. The chronotaxic dynamics of this model was tested numerically.

The system 1 was integrated with varying parameters. Numerically, for an oscillator to be chronotaxic, it was required that the observed oscillator was synchronized to one of the unidirectionally coupled drivers G or O.

Using this test, 7 different types of dynamics were revealed, the most relevant 5 regions shown in Fig. Example phase trajectories for all types are shown in Supplementary Fig.

Approximate frequencies of these oscillations in different regions are summarised in Table 1. In real data, utilising chronotaxicity as the defining parameter of the system is superior to the consideration of synchronization alone, as it can be identified experimentally from any single time series, whereas synchronization requires measurements of all interacting oscillators.

Therefore, in this case, it would be sufficient to have measurements of only ATP GO or ATP MO to determine their chronotaxicity. To demonstrate this, phase fluctuation analysis was applied to the generated dynamics of GO and MO using parameters from each region shown in Fig.

The chronotaxicity of each metabolic oscillator was tested separately. PFA was used to characterise the phase fluctuations for each oscillator in each region see Fig. Excellent agreement is shown between the chronotaxicity as calculated by the model using synchronization conditions and chronotaxicity as calculated via the inverse approach with no prior knowledge.

This illustrates that the method is applicable to the observation and identification of chronotaxicity in real systems, where the dynamics are unknown beforehand.

a Examples of dynamics from regions A—E, as defined in Fig. In region C the exponent α changes too fast and the DFA method is not applicable, however such dynamics of α suggest that system is not chronotaxic.

This inverse approach shows very good agreement with the chronotaxicity tests directly from the model, revealing that metabolic dynamics in regions A, B and D are chronotaxic, while regions C and E are not.

Thus, it may be used to identify chronotaxicity in real data, using a single time series. To demonstrate the method, the time series used here contain at least cycles of oscillation. In reality, this number of cycles is not always feasible. However, this method may still be applied on shorter time series, with reliable results, see Supplementary Fig.

The applicability of the model is tested on the data of NADH recorded by Gustavsson et al. For this we utilise the closely linked dynamics of intracellular ATP and NADH arising from glycolysis. This means that measurements of NADH in yeast cells may be used to provide an approximation of ATP dynamics.

Although the amplitude of these parameters will differ, their phase relationship will remain the same and can thus be represented by our phase oscillator model. Using this information, the model can be tested for the case of the glycolytic oscillator based on NADH measurements, which are more readily available.

a The yeast data recorded from individual yeast cells by Gustavsson et al. The addition of glucose followed by cyanide to glucose starved yeast cells induces glycolytic oscillations by preventing respiration.

Therefore, glycolysis is upregulated by the reversal of PFK inhibition in response to the reduction in ATP MO.

b The situation described in a can be represented by our model as the dynamics of the system being driven by the glucose driver, causing the system to be chronotaxic.

c Example NADH time series from an isolated yeast cell exhibiting glycolytic oscillations. d Comparison of NADH and ATP oscillations in yeast glycolysis shows that they oscillate out of phase 48 , Modified from In the experiments by Gustavsson et al.

NADH in individual yeast cells was shown to oscillate following starvation and the addition of cyanide. In this state, glycolysis can be the only means of energy production, as cyanide halts respiration, effectively removing the effects of the mitochondrial oscillator MO. region B in Fig. To make the system chronotaxic, the extracted frequency was used as the driver ω G of the glycolytic oscillator.

This provided a chronotaxic oscillator with the same oscillation frequency as the experimental data, allowing the DFA exponent α to be compared between cases see Fig. Due to the relatively short recording time causing variation between simulations, they were repeated 3 times and the average value of α taken.

The mean value of α for the chronotaxic simulations was 0. This shows that our model, although simple, incorporates enough features to allow the calculation of the presented characteristic, chronotaxicity and that evidence of this characteristic appears to be present in yeast glycolytic dynamics.

This verifies the applicability of our model to the glycolytic oscillator. Further investigation is required into the mitochondrial oscillator and other metabolic states.

To identify metabolic states, we consider an altered state, such as the glycolysis dependent and potentially carcinogenic case discussed above, to correspond to the dynamics of the model in which the phase of a chronotaxic GO entrains the phase of MO, i.

In contrast, the normal state may correspond to the case where the phase of MO entrains the phase of GO, as discussed in ref. Therefore, the normal state corresponds to region D where oscillators are chronotaxic with oxygen as a driving system.

However, it may also be possible that in the normal state both oscillators are chronotaxic due to only their external influences from oxygen and glucose , with the interactions between MO and GO not strong enough for entrainment, as shown in region A.

In all chronotaxic cases, phase fluctuation analysis will return a DFA exponent around 0. This work is based on experimental evidence of metabolic oscillators 3 , 13 , 15 , 19 , 22 , 23 and their interactions 20 , 21 , 27 , In experimental studies oscillations are often overlooked 24 , as several oscillations may contribute to the same signal thus almost cancelling each other despite existing separately.

Alternatively, due to their highly complex nature they are often treated as stochastic Even in cases when oscillations and the interactions between them have been studied, the exact characteristics of their amplitude and phase relations have not been considered.

Several studies of mitochondrial and glycolytic oscillations have shown that fluctuations of ATP production in a cell are not fully stochastic, but have nonautonomous and deterministic oscillatory components 3 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 26 , 27 , 33 , Therefore, these oscillations reflect underlying deterministic processes.

Here we have proposed a measurable characteristic of metabolic oscillations, chronotaxicity, which is expected to change during the transition of a healthy cell to a state with altered energy metabolism. This was demonstrated using a qualitative phase oscillator model of metabolic oscillations.

The model captures only the most general and universal oscillatory dynamics and interactions and thus provides the advantage that it is applicable to oscillatory metabolism in general, independent of cell type.

The model explicitly takes into account the openness of cellular energy metabolism and therefore its nonautonomous dynamics. The model, based on the theory of chronotaxic systems, introduces a new approach in which the complex metabolic system is considered as a set of functional rather than structural units, these functional units being interacting glycolytic and mitochondrial oscillators with chronotaxic properties.

As a concept within the theory of dynamical systems, describing oscillators which are driven by other oscillators, it is not trivial that chronotaxicity can be applied to biological oscillators and that the driving in biological oscillators represents itself in the same way as in dynamical systems.

In this work, we investigated the applicability of the concept of chronotaxicity to real biological oscillations observed in real experimental studies of cellular metabolism. We proved our assumptions that biological oscillators can be chronotaxic and that their chronotaxicity can be identified from real experimental observations and developed a model of cellular energy metabolism.

The interactions of oscillatory processes can be between amplitude and amplitude, phase and phase, or phase and amplitude. These possibilities show how many scenarios of regulations of physiological functions may occur. In this paper, for the sake of simplicity, we have selected phase oscillators and restricted the discussion to phase-phase interactions.

Nevertheless, in observations of real life systems the chronotaxicity can be identified from phase dynamics alone, even when amplitude dynamics is complex 37 , thus making our model verifiable by experimental observations. Moreover, by identifying the characteristics of chronotaxicity we have shown that one in principle can detect a transition to a carcinogenic state of cellular functioning, provided metabolic oscillations are present in both states of the cell.

The described inverse approach methods found evidence of chronotaxicity in this case, as expected from the numerical simulations of the model. Cellular metabolism may be affected by many more processes and interactions than those considered here.

The simplicity of the model could easily facilitate the inclusion of further couplings, for example the consideration of calcium dynamics or genetic factors in energy production. Calcium has been shown to directly influence mitochondrial dynamics via many pathways 50 , 51 , while genetic mutations can have a direct effect on mitochondrial function These effects could be included in the model as influences to couplings, extra oscillators, or adaptations of the external drivers.

Results presented in this paper set up bases for experimental verification of the hypothesis that chronotaxicity can be used to identify transitions between metabolic states in a cell, for example the metabolic switch observed in cancer cells.

We have demonstrated evidence of chronotaxicity in real metabolic oscillations, which led to a new way of studying metabolic processes inside a cell. The model provides a framework within which the existing understanding of biochemical reactions involved in metabolic processes along with new observations based on recently introduced functional imaging methods 2 , 3 , 4 , 5 can be unified in a single picture.

Furthermore, focusing on the transitions between metabolic states could facilitate the development of new therapeutic strategies.

As a real life example of driven metabolic oscillations we use glycolytic oscillations in individual isolated yeast cells recorded by Gustavsson et al. The brief description of the experiments setup is presented below, for more details see the original work by Gustavsson and co-authors The class of chronotaxic systems identifies oscillatory dynamical systems with dynamics ordered in time chronos — time, taxis — order Such ordering is typical for driven oscillators, where the drive system determines the dynamics of a response system.

Chronotaxic systems can sustain their dynamics even with continuous external perturbations. Introduced for low-dimensional and high-dimensional dynamical systems 34 , 35 , 36 , chronotaxic systems have been shown to be useful in studies of living systems, one example being the application to the cardiorespiratory system In addition to living and open systems, which are in continuous contact with the external environment, chronotaxic systems are nonautonomous dynamical systems 29 , i.

their dynamics explicitly depends on time, as shown in the equation,. Alternatively, chronotaxic systems can be described by drive and response systems, as follows. The main defining feature of chronotaxic systems is a time-dependent steady state of point attractor x A t see Fig.

The trajectory x A t can be viewed as a uniformly hyperbolic trajectory 53 which is linearly attracting in such a way that the distance between a neighboring trajectory and x A t can only contract in an unperturbed chronotaxic system.

For more details and for relations between chronotaxic and other dynamical systems see ref. A simple example is given by unidirectionally coupled phase oscillators with phase φ X driven by a phase φ P as shown in the equation,.

where and ω 0 is the natural frequency of the observed oscillator, is coupling strength and ω is the frequency of the driving oscillator. The time-dependent point attractor will exist if the condition of chronotaxicity 35 is fulfilled, i.

if and if the coupling strength ε t does not change its sign. Taking into account that the dynamical system x is chronotaxic due to the influence from the driver p , chronotaxicity will change when the external influence from p changes.

This makes chronotaxicity a perfect candidate for investigation in the study of metabolic oscillators under changing driving influences. One of main advantages of chronotaxicity is that it can be identified experimentally from a single time series, whereas otherwise the identification of drive-response relationship would require measurements of the driver as well as the response system.

To identify chronotaxicity in a time series, a method named phase fluctuation analysis PFA was recently developed where Ψ s , t is the mother wavelet which is time-shifted according to t and scaled according to the parameter s. The oscillation can then be traced in W T s , t.

The instantaneous frequency of the oscillation at each time point can be estimated using either the synchrosqueezed wavelet transform 55 or a ridge-extraction method The phase is then calculated by integrating over the instantaneous frequency in time.

The estimation of angular velocity can be found by smoothing over the frequency extracted from the wavelet transform.

In this approach, perturbations are assumed to be due to an uncorrelated Gaussian process. In chronotaxic systems, perturbations decay due to the influence of the point attractor and the divergence from the attractor is similar to the original Gaussian process In contrast, in non-chronotaxic systems, where the phase of oscillation is neutrally stable, the perturbations are integrated over, resulting in a random walk i.

Brownian noise. To distinguish these two cases, detrended fluctuation analysis DFA 57 is performed on the phase fluctuations extracted from the time series. The DFA technique explores the fractal self-similarity of fluctuations at different timescales in Δφ.

The scaling of fluctuations is determined by the self-similarity parameter α. To estimate α the time series is integrated in time and divided into sections of length n.

The local trend is removed for each section by subtracting a fitted polynomial, usually a first order fit 57 , The root mean square fluctuation F n for the scale equal to n is then defined by the following equation,. where Y n t is the integrated and detrended time series of length N.

The self-similarity parameter is given by the gradient of the line of the plot of log F n against log n. The self-similarity parameter α for Δφ for uncorrelated Gaussian noise as expected in chronotaxic systems gives a value of 0.

In Fig. Supplementary Fig. S5 shows how this method may be applied for an example ATP signal obtained from the model. In order to reliably test for chronotaxicity, it should be noted that the time series should be sufficiently long, i. contain at least 30 cycles of oscillation may vary depending on the characteristics of the data , be evenly sampled and have a sampling frequency which is high enough to capture the dynamics at the frequency of interest How to cite this article : Lancaster, G.

et al. Modelling chronotaxicity of cellular energy metabolism to facilitate the identification of altered metabolic states. Törnroth-Horsefield, S. Opening and closing the metabolite gate. Article ADS CAS PubMed PubMed Central Google Scholar. Azarias, G. Selective ion changes during spontaneous mitochondrial transients in intact astrocytes.

PloS One 6, e, doi: Article CAS ADS PubMed PubMed Central Google Scholar. Merrins, M. Direct measurements of oscillatory glycolysis in pancreatic islet β-cells using novel fluorescence resonance energy transfer FRET biosensors for pyruvate kinase M2 activity. Article CAS PubMed PubMed Central Google Scholar.

Furuya, K. Real-time luminescence imaging of cellular ATP release. Methods 66, — Article CAS PubMed Google Scholar. Rieger, B. Lateral pH gradient between OXPHOS complex IV and F0F1 ATP-synthase in folded mitochondrial membranes.

Article CAS ADS PubMed Google Scholar. Ralph, S. Aspects Med. Cloonan, S. Mitochondria: commanders of innate immunity and disease?

Wallace, D. Mitochondria and cancer. Cancer 12, — Seyfried, T. Cancer as a metabolic disease. Article CAS Google Scholar.

Kroemer, G. Cancer Cell 13, — Vander Heiden, M. Understanding the Warburg effect: the metabolic requirements of cell proliferation. Science , — Fulda, S. Targeting mitochondria for cancer therapy. Drug Discovery 9, — Aon, M. Synchronized whole cell oscillations in mitochondrial metabolism triggered by a local release of reactive oxygen species in cardiac myocytes.

Kurz, F. Wavelet analysis reveals heterogeneous time-dependent oscillations of individual mitochondria. Heart Circ. O'Rourke, B. Oscillations of membrane current and excitability driven by metabolic oscillations in heart cells.

Goldbeter, A. Computational approaches to cellular rhythms. Nature , — Cortassa, S. A mitochondrial oscillator dependent on reactive oxygen species. Tu, B. Metabolic cycles as an underlying basis of biological oscillations. Cell Biol. Mitochondrial oscillations in physiology and pathophysiology.

Article CAS MathSciNet PubMed PubMed Central Google Scholar. Olsen, L. Ganitkevich, V. Glycolytic oscillations in single ischemic cardiomyocytes at near anoxia.

Spatio-temporal oscillations of individual mitochondria in cardiac myocytes reveal modulation of synchronized mitochondrial clusters. PNAS , — Chandra, F. Glycolytic oscillations and limits on robust efficiency.

Article CAS ADS MathSciNet PubMed MATH Google Scholar. Spiller, D. Measurement of single-cell dynamics. Kiviet, D.

Your Account. To Cellular energy metabolism your privacy, your Cellukar will be locked after 6 failed attempts. After that, you will need to contact Customer Service to unlock your account. You have 4 remaining attempts. You have 3 remaining attempts. Metabolism is the set Ginseng research studies life-sustaining chemical transformations within the Cellular energy metabolism of metaboliam organisms. These enzyme-catalyzed reactions allow organisms Black pepper extract for promoting healthy digestion Cellulaf and enervy, maintain their structures, and metabilism to their environments. Most are proteins. A few ribonucleoprotein enzymes have been discovered and, for some of these, the catalytic activity is in the RNA part rather than the protein part. Link to discussion of these ribozymes. Enzymes bind temporarily to one or more of the reactants — the substrate s — of the reaction they catalyze.