This envelope curve de nes the best performance of the biochemical network, i. The efficiency is represented by color in Fig. We investigated how efficiency depends on the three concentrations. In SI we show in a simple model of chemical reaction cycles that with a constant energy dissipation, i. All the points lie above an envelope curve, which follows Eq.
Points in square indicate the points shown in Fig. Colors of the points indicate the efficiency. Colors of the points indicate the efficiency as in a. Points with high efficiency are clustered.
This result indicates that high efficiency is achieved when the kinetic rates in the two halves of the PdP cycle phosphorylation and dephosphorylation are matched. We analysed the data according to Eq. S4 in SI for details. Experimental evidence from Ref. Error bars represent standard deviations.
The data can be fitted by Eq. To understand the relationship between phase accuracy and energy dissipation, we consider the noisy Stuart-Landau equation for x,y :. The corresponding equation in polar coordinates are:. It is clear from Eq. To compute the free energy dissipation, we rst determine the phase-space probability distribution function P x, y, t. We compute the system's entropy production rate [ 37 , 38 ], from which we obtain the minimum free energy dissipation see SI for details : W.
The energy dissipated in one cycle is:. Eliminating a from Eq. The region where Eq. The noise strength affects the lower bound. The phase-amplitude correlation parameter d controls the upper bound see SI and Fig. S5 for details. S6 for a direct simulation demonstration. In a system with multiple variables, large energy dissipation may completely suppress fluctuations in some of the variables. However, other variables, which are not subject to the energy-assisted noise control mechanism, can introduce a finite contribution to the phase diffusion resulting in a finite C.
Analysis of the noisy Stuart-Landau equation clearly shows that free energy dissipation is used to suppress phase diffusion. The parameters in this relationship and the range of its validity depend on the details of the system. However, the inverse dependence of phase diffusion on energy dissipation appears to be general.
Oscillations are critical for many biological functions that require accurate time control, such as circadian clock, cell cycle, and development. However, biological systems are inherently noisy. The phase of a noisy oscillator uctuates diffuses without bound and eventually destroys the coherence accuracy of the oscillation.
Here, our study shows that free energy dissipation can be used to reduce phase diffusion and thus prolong the coherence of the oscillation. A general relationship between the phase diffusion constant and the minimum free energy cost, as given in Eq. The amplitude fluctuations also decrease with free energy dissipation see Fig. S7 in SI for details , as fluctuations in phase and amplitude are coupled in realistic systems. Our study thus establishes a cost-performance tradeoff for noisy biochemical oscillations.
How do biological systems use their free energy sources e. As illustrated in Fig. These chemical states are characterized by the conformational and chemical modification e. The forward transition from one state to the next is coupled to a PdP cycle blue arrowed circle driven by hydrolysis of one ATP molecule.
For each forward step, the reverse transition introduces a large error in the clock. The system suppresses these backward transitions by utilizing the ATP hydrolysis free energy. However, this is just one half of the story. Even in the absence of the reverse transition, the time duration between two consecutive states is highly variable due to the stochastic nature Poisson process of the chemical transitions.
A general strategy of increasing accuracy is averaging [ 39 ]. In the case of biochemical oscillations, each period may consist of multiple steps, each powered by at least one ATP molecule. As a result of averaging, the error in the period should go down as the number of steps increases. This result reveals a general strategy for oscillatory biochemical networks to enhance their phase coherence by coupling to multiple energy consuming cycles in each period.
For the cyanobacteria circadian clock, each KaiC molecule has two phosphorylation sites. Our study suggests that the extra ATP molecules may be used to increase the phase coherence of the circadian clock. Oscillation coherence increases with the number of ATP hydrolyzed per period. The transition from one tick to the next is coupled with a ATP hydrolysis cycle. Biological systems need to function robustly against variations in its underlying biochemical parameters rates, concentrations [ 41 , 42 ]. For oscillatory networks, the free energy dissipation needs to reach a critical value W c to drive the system to oscillate.
We showed here that additional free energy cost in excess of W c is needed to make the oscillation more accurate, as demonstrated explicitly in Eq. In addition to this accuracy-energy tradeoff, we found that larger energy dissipation can also enhance the system's robustness against its parameter variations.
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Take the activator-inhibitor model, for example: the concentrations of enzyme M M T and phosphatase K K T may vary from cell to cell. Robustness is de ned as the area of the parameter space where oscillation exists. S8 in the SI , the robustness increases as the system becomes more irreversible, i. This suggests a possible general tradeoff between the functional robustness and energy dissipation in biological networks. The Gillespie algorithm [ 43 ] is used for the stochastic simulations of the reaction kinetics. The parameters in Fig. Parameters for the other three models are given in SI.
For given kinetic rates and the volume V , we simulated trajectories starting with the same initial condition. For the j th trajectory, we obtained its i th peak time t ij from the trajectory x j t after smoothing smooth function in MATLAB was used. The peak positions for two trajectories are shown in Fig.
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Nat Phys. Author manuscript; available in PMC Mar 1. PMID: Copyright notice. See other articles in PMC that cite the published article. Abstract Oscillation is an important cellular process that regulates timing of different vital life cycles. Keywords: Biochemical oscillations, energy dissipation, noise, phase diffusion, network motif. Open in a separate window. Figure 1. Embryonic stem cell-derived cardiocytes ESCs , however, often exhibit dysrhythmic excitations. The impact of physical loading is a potent stimulus required to maintain stem cell homeostasis, but the mechanism s by which cells can sense a biophysical force and translate that into a developmental decision mechanotransduction remains poorly understood.
The primary cilium is a single sensory cellular extension, which has recently been shown to demonstrate an important role in mechanotransduction and lineage commitment in human mesenchymal stem cells hMSCs. It is intriguing that even this sort of cellular antenna has been fashioned to act as a sensor of oscillatory patterns In fact, it has been shown that short periods of mechanical stimulation in the form of oscillatory fluid flow OFF is sufficient to enhance osteogenic gene expression and proliferation of human mesenchymal stem cells hMSCs.
Furthermore, the cilium mediates fluid flow mechanotransduction in hMSCs by maintaining OFF-induced increases in osteogenic gene expression and, surprisingly, to limit OFF-induced increases in proliferation. The general relevance of the primary cilium as an oscillatory mechanosensor is underlined by its ability to coordinate early cardiogenesis and hedgehog signaling during cardiomyocyte differentiation. It has been shown that the pluripotent P CL6 mouse stem cell line, which can differentiate into beating cardiomyocytes, forms primary cilia that contain essential components of the hedgehog pathway, including Smoothened, Patched-1 and Gli2 Knockdown of the primary cilium by Ift88 and Ift20 siRNA or treatment with cyclopamine, an inhibitor of Smoothened, blocked hedgehog signaling, as well as differentiation into beating cardiomyocytes.
Very recently, a tight interplay between surface contact topography within the cell culture environment and the structure and function of the primary cilium was found to play an essential role in the growth and differentiation of hMSCs These findings may have remarkable implication in the development of novel cell therapy strategies. In fact, the application of OFF period with defined oscillatory patterning coupled with targeted surface topographies may be beneficial components of bioreactor-based approaches to form stem cell-derived specific tissues suitable for regenerative medicine.
Moreover the primary cilium may become itself a potential therapeutic target for efforts to mimic loading with the aim of optimizing stem cell commitment and differentiation. In the absence of Hes7, somites fuse, which results in fused vertebrae and ribs Interestingly, a lack of oscillation due to persistent Hes7 expression also leads to somite fusion Hes7 oscillation, like that of Hes1, is regulated by negative feedback Lfng oscillation is also crucial for segmentation, as both the loss of and persistent expression of Lfng has been shown to lead to severe somite fusion 34 , 35 , Defects in the assembly or rhythmic function of primary cilia, which are oscillatory sensory organelles, are tightly coupled to developmental defects and diseases in mammals.
Primary ciliary dyskinesia most often arises from loss of the dynein motors that power ciliary beating. In this regard, it has been shown that DNAAF3 also known as PF22 , a previously uncharacterized protein, is essential for the preassembly of dyneins into complexes before their transport into cilia Loss-of-function mutations in the human DNAAF3 gene were identified in individuals from families with situs inversus and defects in the assembly of inner and outer dynein arms Knockdown of DNAAF3 in zebrafish likewise disrupts dynein arm assembly and ciliary motility, causing primary ciliary dyskinesia phenotypes that include hydrocephalus and laterality malformations These results support the existence of a conserved, multistep pathway in the cytoplasmic assembly of competent ciliary dynein complexes.
The primary cilium coordinates early cardiogenesis and hedgehog signaling in cardiomyocyte differentiation. J Cell Sci ; Pt 17 : These data support the conclusion that cardiac primary cilia are crucial in early heart development, where they partly coordinate hedgehog signaling. In general, malfunction of the circadian timing system may result in cardiovascular and metabolic diseases, and conversely, these diseases can impair the circadian system. Moreover, the output rhythm was dampened. In the colon of SHR the clock function was severely altered, whereas the differences are only marginal in the liver These changes may likely result in a mutual desynchrony of circadian oscillators within the circadian system of SHR, thereby potentially contributing to metabolic pathology of the strain.
As discussed above, oscillatory patterns and their synchronization into resonating rhythms identify dynamic contexts that transcend the boundary of a single cell level, extending to and specifying the long-range assembly of tissues, organs, and of the entire individual. There is compelling evidence that cytoskeleton, which forms the physical and biochemical interface for a large variety of cellular processes, plays an essential role in the orchestration of mechanical and signaling patterning of living cells.
Microtubules are important structures in the cytoskeleton, and have long been shown to display intrinsic oscillatory patterns. Moreover, microtubules are electrically polar. Further deployment of microtubular oscillatory dynamics and polarity has recently placed the role of the cytoskeleton into a novel perspective. In fact, it is conceivable that certain microtubule normal vibration modes efficiently generate oscillating electric field. This oscillating field may be important for the intracellular organization and intercellular networking.
Worthy to note there is strong experimental evidence which indicates electrodynamic activity of multiple cell types in the frequency region from kHz to GHz, expecting the microtubules to be the source of this activity 42 In a recent and comprehensive review, Zhao Y and Zhan Q 43 integrated a large body of studies reporting the existence of dielectrophoretic forces and electromagnetic interaction around and between cells in different experimental conditions, discussing how cellular dynamics may be regulated by electric fields generated by synchronized oscillations of microtubules, centrosomes and chromosomes.
The intensities of electric field and of radiated electromagnetic power from the whole cellular microtubule network have been calculated The subunits of microtubule tubulin heterodimers can be approximated by elementary electric dipoles. Mechanical oscillation of microtubule can be represented by the spatial function which modulates the dipole momentum of subunits.
In these studies, the field around oscillating microtubules has been calculated as a vector superposition of contributions from all modulated elementary electric dipoles which comprise the cellular microtubule network Theoretical analysis of the electromagnetic radiation and field characteristics of the whole cellular microtubule network indicate that a macroscopic detection system antenna is not suitable for measurement of cellular electrodynamic activity in the radiofrequency region since the radiation rate from single cells is very low lower than 10 W.
Low noise nanoscopic detection methods with high spatial resolution which enable measurement in the cell vicinity are desirable in order to measure cellular electrodynamic activity reliably. Nevertheless, these studies suggest the possibility that cell behavior and fate may be influenced by magnetic fields. In this regard, we have previously shown that exposure of mouse embryonic stem ES cells to extremely low frequency pulsed magnetic fields 50 Hz, 0.
The effect was mediated by an increase in the transcription rate of a number of cardiogenic and cardiac-specific genes, involving the expression of their biologically active end-products For decades Scientists have attempted to drive stem cell fate through the use of chemistry. These findings shown for the first time the possibility to use magnetic energy to drive stem cell growth and differentiation and lead to a number of interrelated considerations: i electromagnetic resonance modes between the endogenous cellular radioelectric electromagnetic oscillations are very likely occurring, although hardly detectable; ii every particular level of cellular hierarchy conceivably possesses a characteristic spectrum of endogenous electromagnetic oscillations originating from various processes; iii intra- and inter-level resonances should occur and provide a networking landscape between these processes.
These considerations may harbor an important implication: modulation of these resonance modes may be conceived as a new strategy to handle stem cell homeostasis and fate. We have recently accomplished this task by exposing isolated cells to a Radio Electric Asymmetric Conveyer REAC , an innovative device based upon the interaction of two oscillating magnetic fields. One is generated by the entire organism or cultured cells, and the other is a weak 2 mW radiated power electromagnetic field of 2. Cellular responses were investigated by real-time PCR, Western blot and confocal microscopy.
Albert Goldbeter (Author of Biochemical Oscillations and Cellular Rhythms)
REAC exposure enhanced the expression of cardiac, skeletal and neuronal lineage-restricted marker proteins. The number of spontaneously beating ES-derived myocardial cells was also increased Based on these results, we exposed to REAC human mesenchymal stem cells isolated from the adipose tissue hASCs with a novel non-enzymatic method 52 and provided evidence that this treatment remarkably enhanced the transcription of a program of multilineage, tissue restricted genes 53 , including: i the cardiogenic genes prodynorphin, GATA-4 and Nkx REAC exposure had no detrimental effects on hASCs, since it did not affect significantly the amount of apoptotic, or necrotic cells.
Stem cell exposure to REAC also finely tuned the expression of stemness-related genes, inducing an early increase in Nanog, Sox-2 and Oct-4 transcription during the first hours, followed by a significant down-regulation of transcript levels below the control value after 24 hours of treatment Such a transcriptional profile was mirrored at the protein expression level.
The finding that REAC exposure elicited a concomitant, early expression of both stemness-related and lineage-restricted genes together with a high-throughput of cellular differentiation indicate that its action may result from the optimization of stem cell expression of multipotency.
In this regard, the downregulation in stemness gene and protein expression observed after 12 h of REAC exposure may be worthy of consideration, since it is now evident that the downregulation of Sox-2, Nanog and Oct-4, after their initial induction, is a critical step in cell progression towards a differentiated state 56 , 57 , 58 , The outcomes of the synergistic interplay between REAC asymmetrically conveyed radio electric fields and hASCs were achieved after extremely brief exposure pulses, showing long-lasting persistence of cell commitment upon the cessation of REAC treatment.
Ongoing studies will dissect the electrophysiological and functional properties of REAC-committed hASCs, and assess whether improved tissue healing may result from their transplantation in defined animal models of disease, including heart failure, neurodegeneration and skeletal muscle dystrophy. In the affirmative, the proper use of electromagnetic energy may represent the underpinning for future cell therapy approaches. Recently, we shown that REAC conveyed radioelectric fields afforded a direct reprogramming of human dermal skin fibroblasts into cardiac, neuronal and skeletal muscle lineages Similar to the effects yielded in hASCs, REAC exposure of human fibroblasts induced a biphasic response on pluripotency genes, enhancing the expression of Nanog, Sox2, Oct4, and cMyc during the first hours, while producing a stable downregulation in their transcription after 24 hours For the first time, human non-stem somatic adult cells were rapidly committed to a high-yield of critical fates in Regenerative Medicine, only proceeding through a transient pluripotent state, without being plastered in such intermediate condition.
Importantly, these results were achieved without the aid of potentially harmful viral or protein transduction. On the whole, REAC acted as a time machine capable to reset the clock of somatic human adult cells to an ancestral time where they may behave as pluripotent elements capable of multifaceted phenotypic decisions. A major problem in the clinical application of cell therapy is the progressive decline of stem cell multipotency with age. We have recently shown that geroconversion of hASCs and loss of their multilineage commitment after multiple passages in culture a method to induce cell aging in vitro were efficaciously counteracted by hASC exposure to REAC delivered radioelectric fields.
Moreover, unlike unexposed cells, REAC treated hASCs maintained their typical fibroblast-like morphology, as well as a multilineage potential along osteogenic, adipogenic, chondrogenic, and vasculogenic fates, even at late passages, both at morphologic and gene expression levels. So far, reversing of stem cell aging could only be achieved by the aid of viral vector-mediated gene delivery For the first time, age-associated molecular and morphological alterations were reversed in stem cells by exposure to an innovative system of electromagnetic energy delivery, suggesting future perspectives in the clinical application of the REAC technology for the treatment of pathologic aging and degenerative diseases.
Accordingly, we found that the REAC treatment was able to reestablish the normal cellular homeostasis in human chondrocytes obtained from the femoral head articular cartilage of patients with osteoarthritis OA 67 , the most common articular degenerative disease of the hyaline cartilage and subchondral bone. Chondrocytes are responsible for the synthesis and maintenance of extracellular matrix within articular cartilage, balancing both synthetic and degradative processes. The constitutive role of spontaneous and modulated oscillatory patterning in cellular homeostasis, the intrinsic oscillatory features of the cytoskeleton, and the ability of cells of fashioning their temporal and spatial chromatin organization through amplitude and frequency dependent tuning of their intrinsic oscillators, form a solid rationale for the deployment of nano mechanical vibrations as a strategy to reprogram cellular decisions and fate.
On these bases, it is also becoming clear that mechanical dynamics and the mechanosensing apparatus are deeply involved in intra- and inter-cellular cross-talk and may be recruited to enhance the healing potential from tissue-resident stem cells. Mechanical boundary conditions drive transcriptional programs, homing, engraftment and fate of stem cells in living tissues, also providing relevant cues for the development of engineered tissues 71 , Based on the different behavior of the mechanosensing apparatus in undifferentiated and terminally committed stem cells 73 , innovative strategies for tissue repair may be envisioned through the modulation of stem cell mechanosensors.
Stem cell engraftment after transplantation, an extremely low-yield process limiting tissue repair in animal models and patients 74 , may also be rewardingly improved by the synergistic interplay between biochemical signals, including growth factors, and adhesion molecules similar to those engaged by leukocytes for recruitment to inflammatory sites 75 , 76 , Mechanistically, physical forces can coordinate entailed signals between defined tissue elements, regulating stem cell growth and differentiation, especially in tissues chronically exposed to mechanical loading, such as the myocardium, and the vessel wall.
In vivo , stem cells reside within highly dynamic microenvironments, referred to as niches, where they receive complex anisotropic physical stimuli, occurring as time-scaled, as well as synchronous events. Within the niche, extracellular matrix displays hierarchical nanotopographies embedding stem cells with other non-stem elements 79 : in this multifaceted domain, multiple signals self-organize as oscillatory patterns, undergoing synchronization and phase coherent organization which associate with remarkable structural and functional changes. In living cells, biological processes deeply rely on the nanomechanical properties of subcellular structures and of the cell plasma membrane or wall.
By the aid of atomic force microscopy AFM it is now possible to gain information on the integrity and local nanomechanical properties of mammalian and microbial cellular membranes under normal or pathological conditions. The scanning probe of the AFM, usually a carbon nanotube, can sense local traits at a quasi atomic level, affording a dynamic reconstruction of mechanical, topographic, electric and thermal features, also providing important cues on optical absorption or magnetism. Importantly, AFM imaging of biological samples can be performed at sub-nanometer resolution in their natural aqueous environment, therefore allowing unprecedented characterization analyses in living cells, detecting and applying small forces with high sensitivity under physiologic conditions.
This approach can also offer the chance to analyze the nanomechanics of dynamic cellular processes, including stem cell commitment and differentiation. In yeasts and bacteria, cellular activity, metabolism, growth and morphogentic changes are associated with defined nanomechanical activity, merging to the cell surface up to the generation of defined patterns of vibration 81 , Figure 1.
The early embryonic life: a Place harboring the information for our journey throughout the adulthood. The image is captured from a textile artwork Julia von Stietencron, Art Director of VID, Visual Institute of Developmental Sciences, Bologna, Italy representing the primordial of life as the self-assembling of forms in the presence of physical energies i. Figure 2. Tissue resident stem cells and their memory of an ancestral state The image is captured from a textile artwork Julia von Stietencron, Art Director of VID, Visual Institute of Developmental Sciences, Bologna, Italy symbolizing the tissue resident stem cells within their own tissue microenvironment, a niche that harbor features still resembling the early life morphogentic plains and ancestral blood vessels.
The installation highlights the emergence of a new informational paradigm: our cells entail a double memory, the memory of the ongoing story of the resident tissues, and that of an ancestral state that goes back to the initial clock when embryo is fashioned from a single fertilized totipotent cell.
From a singularity single fertilized cell , the blood, vessels and all tissues take origin, and they all enwomb the original information still talking as a mutant vibration. More complex eukaryotic cells can also be investigated accordingly. For instance, stem cells directed to cardiac myocyte differentiation will begin to beat at a given moment of their commitment point.
This beating motion requires a major reorganization of the cell cytoskeleton and in turn a significant change in cellular nanomechanical properties. Determination of these properties in form of measurable parameters as a function of progression throughout commitment and terminal differentiation may provide novel information on the mechanisms underlying the attainment of specific fates in individual stem cells Other examples of cellular processes that can be measured with the AFM include activation of platelets, exocytosis, movement of cells, and cell division.
Moreover, as discussed above, transferring of mechanical vibration to the subcellular environment triggers the mobilization of ionic species and the generation of ionic fluxes and induced microcurrents, ultimately ensuing in the appearance of endogenous oscillating electromagnetic fields 85 , Such vibrations although derived by acoustic induced mechanical oscillations are of electromagnetic nature, no more being associated with the mechanical component of the sound, but rather with its electromagnetic counterpart, the phonon.
Bearing in mind the results and considerations reported above, cells can be conceived as a fractal part of the entire organism and, to an extended view, of the entire Universe, being permeated by both local and non-local events, intimately connected by oscillatory patterns. It is seen that, as long as a minimum amount of KaiB is present, neither the amplitude nor the period of the oscillations are much affected by the amount of KaiB.
The dependence of the oscillations on the level of KaiA is more interesting. For low KaiA concentrations, the system does not exhibit oscillations. When the KaiA concentration is raised to about half the KaiC concentration, the system starts to oscillate via a supercritical Hopf bifurcation. The period of the oscillations decreases monotonically with increasing KaiA concentration. In contrast, the amplitude first increases, but then decreases until the oscillations ultimately disappear. The latter is a direct consequence of the mechanism of differential affinity, which relies on KaiA sequestration — when the concentration of KaiA is high enough, KaiA can no longer be sequestered, causing the synchronization mechanism to break down.
The prediction of the disappearance of oscillations when the KaiA concentration is raised has been confirmed experimentally Rust et al. It is seen that both the period and the amplitude are essentially independent of the concentration of KaiB, provided enough KaiB is present to sequester KaiA. The coupling between the protein modification and the protein synthesis cycle While the ground-breaking experiments of the Kondo group in opened the possibility that the circadian clock might be driven by a protein modification cycle only, in the same group showed that when in vivo the protein modification cycle was impeded, the concentration of KaiC still oscillated with a period of 24 hours.
This unambiguously demonstrated that in vivo the clock is driven by a combination of a protein synthesis cycle and a protein modification cycle. The natural question that arose was thus: why is the clock based on two oscillators? KaiC is a hexamer that, in our model, switches between an active conformational state circles in which its phosphorylation level tends to rise and an inactive state squares in which it tends to fall see also Fig. To address this question, we have studied the robustness of four models Fig.
In the PPC-in-vitro model Fig. In this case the PPC is highly robust against noise arising from the intrinsic stochasticity of chemical reactions. Even for reaction volumes smaller than the typical volume of a cyanobacterium, the correlation time is longer than that observed experimentally. Living cells, however, constantly grow and divide, and proteins must thus be synthesized to balance dilution.