Collective rhythms of gene regulatory networks have been a subject of considerable interest for biologists and theoreticians, in particular the synchronization of dynamic cells mediated by intercellular communication. Synchronization of a population of synthetic genetic oscillators is an important design in practical applications, because such a population distributed over different host cells needs to exploit molecular phenomena simultaneously in order to emerge a biological phenomenon. However, this synchronization may be corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Therefore, robust synchronization is an important design topic in nonlinear stochastic coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations and extrinsic molecular noise.Initially, the condition for robust synchronization of synthetic genetic oscillators was derived based on Hamilton Jacobi inequality (HJI). We found that if the synchronization robustness can confer enough intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then robust synchronization of coupled synthetic genetic oscillators is guaranteed. If the synchronization robustness of a population of nonlinear stochastic coupled synthetic genetic oscillators distributed over different host cells could not be maintained, then robust synchronization could be enhanced by external control input through quorum sensing molecules. In order to simplify the analysis and design of robust synchronization of nonlinear stochastic synthetic genetic oscillators, the fuzzy interpolation method was employed to interpolate several local linear stochastic coupled systems to approximate the nonlinear stochastic coupled system so that the HJI-based synchronization design problem could be replaced by a simple linear matrix inequality (LMI)-based design problem, which could be solved with the help of LMI toolbox in MATLAB easily.If the synchronization robustness criterion, i.e. the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the stochastic coupled synthetic oscillators can be robustly synchronized in spite of intrinsic parameter fluctuation and extrinsic noise. If the synchronization robustness criterion is violated, external control scheme by adding inducer can be designed to improve synchronization robustness of coupled synthetic genetic oscillators. The investigated robust synchronization criteria and proposed external control method are useful for a population of coupled synthetic networks with emergent synchronization behavior, especially for multi-cellular, engineered networks. Many biochemically dynamical systems are controlled by intrinsic rhythms generated by specialized cellular clocks within the organism itself. These rhythm generators are composed of thousands of clock cells that are intrinsically diverse, but nevertheless manage to function in a coherent oscillatory state -6. HowevEscherichia coli from a network of three transcriptional repressors that inhibit one another in a cyclic way [Recently designed synthetic genetic oscillators can offer an alternative approach, and provide a relatively well-controlled test bed in which the function and behavior of natural genetic oscillators can be isolated and characterized in detail ,3,7,8,17clic way -21. Sponclic way . RecentlIn previous studies -16,23-26Generally, biological systems or organisms are subject to time-varying uncertainties, assumed to be in the form of both internal noise resulting from the birth and death of biochemical molecules, and external noise resulting from environmental perturbations ,19,30-34H∞ noise-filtering perspective. The robust ability to tolerate stochastic kinetic parameter fluctuations and the filtering ability to attenuate extrinsic environmental molecular noise to maintain synchronization of nonlinear stochastic coupled synthetic genetic oscillations was measured from the nonlinear stochastic system theory point of view. In the case where robust synchronization cannot be maintained or is corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise, some robust synchronization design methods are discussed to enhance synchronization. Both the physical insight into the robust synchronization mechanisms and the designs to improve these mechanisms require solving nonlinear HJIs, which cannot be easily achieved by any analytical or numerical method at present. In order to simplify this analysis and design, the Takagi-Sugeno (T-S) fuzzy model [Based on nonlinear stochastic equations of coupled synthetic genetic oscillators distributed in different host cells, the robust synchronization mechanism is discussed from the zy model -40 is emzy model , via theet al. constructed an impulse control system to model the process of periodical injection of coupling substances with constant or random impulse control into a common extracellular medium and studied its effect on the dynamics of collective rhythms [If the robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators could not be achieved spontaneously, an external control input was developed to synchronize the coupled synthetic genetic oscillators. External stimulation inputs are known to play an important role in the synchronization of biological rhythms. For instance, many organisms display a circadian rhythm of 24-hours periodicity entrained to the light–dark cycle . Other e rhythms . Synchro rhythms ,43. In o rhythms .in vivo; (2) The conditions of robust synchronization are developed from the nonlinear stochastic dynamical system point of view, i.e. if the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the intrinsic parameter fluctuation can be tolerated and the extrinsic noise can be buffered so that the robust synchronization of coupled oscillators can be guaranteed. Therefore, we could obtain better insight into synchronization mechanisms of coupled synthetic molecular systems distributed in host cells, and to provide further systematic analysis and control design to improve the synchronization robustness. Further, if the robust synchronization conditions cannot be guaranteed, some synchronization control schemes are also developed to improve the synchronization robustness of coupled synthetic gene networks; (3) The fuzzy interpolation method is introduced to simplify the analysis and design procedure of robust synchronization of coupled nonlinear stochastic synthetic gene networks; and (4) An external inducer input control design method is also developed to guarantee robust synchronization of the nonlinear stochastic synthetic genetic oscillators, when spontaneous synchronization cannot be achieved under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Finally, a design example is provided in silico to illustrate the design procedure and to confirm the performance of the proposed design methods.The contributions of this paper are fourfold: (1) A nonlinear stochastic system is introduced to model a population of nonlinear stochastic coupled synthetic genetic oscillators under random intrinsic kinetic parameter fluctuations and extrinsic molecule noise lacI (from E. coli.) codes for the protein LacI, which inhibits the transcription of the gene tetR. The product of the latter, TetR, inhibits the transcription of gene cI (from λ phage); the protein product CI in turn inhibits the expression of lacI, thus completing the cycle. Garcia-Ojalvo et al. proposed a modular addition to the repressilator, with the aim of coupling a population of cells containing this network [Vibrio fisceri. These bacteria exhibit collective behaviors, mainly using two proteins . Recent [E. coli,9,15-17.i = 1, 2, …, N are governed by repressible transcription of all three genes of the repressilator plus transcription activation of the additional copy of the lacI gene, and by mRNA degradation [Based on the synchronization scheme of coupled repressilators via quorum sensing mechanism Figure , the mRNradation .xai , xbi and xci are the concentration of mRNA transcribed from tetR, cI and lacI in cell i, respectively; concentrations of the corresponding proteins are represented by xAi , xBi and xCi , respectively. The concentration of AI inside each cell is denoted by xSi. αa, αb, and αc are the dimensionless transcription rate in the absence of the repressor, and μ is the repression coefficient. αS is the maximal contribution to lacI transcription in the presence of saturating amounts of AI, and μS is the activation coefficient. γm is the respective dimensionless degradation rate of mRNA for tetR, cI and lacI in the cell, and n is the Hill coefficient [where fficient .The dynamics of proteins TetR, CI and LacI are given respectively as βA, βB and βC are the translation rates of the proteins from their mRNA, and γP represents the dimensionless degradation rate of proteins TetR, CI, and LacI in the cell. The intercellular protein AI in cell i is synthesized by the protein LuxI and diffuses through the cell wall where it undergoes degradation, leading to the following equation [where parameters equation .ηs measures the diffusion rate of AI across the cell membrane, βs is the synthesis rate of AI, and γs gives the rate of decay of AI. Consequently, the whole coupled synthetic system is expressed by (1)-(3), which can be represented by the following more generalized nonlinear dynamic equationwhere xi(t) = (xai(t), xbi(t), xci(t), xAi(t), xBi(t), xCi(t), xSi(t))T ∈ Rm is the state vector of the ith synthetic oscillator; f(·) : Rm → Rm is a smooth nonlinear function that characterizes the behavior of the synthetic oscillator; g(·) : Rm → Rm is a smooth nonlinear inner-coupling function; and C = (cij)N × N ∈ RN × N is the coupling configuration matrix, where cij>0 means that the ith synthetic oscillator is coupled with the jth synthetic oscillator directly, otherwise cij=0. Assume that the diagonal elements of C satisfy where In general, the synthetic genetic oscillators suffer kinetic parameter fluctuations from transcription control, alternative splicing, translation, genetic mutation and diffusion to biological modification of transcription factors ,31,34,44Δαj, j ∈ {a, b, c, S}, Δβj, j ∈ {A, B, C, s}, Δγj, j ∈ {m, p, s} and Δηs denote the amplitudes of the kinetic parameter fluctuations and ni(t) is random white noise with zero-mean and unit variance, i.e., Δαj, Δβj, Δγj and Δηs denote the deterministic part of the stochastic kinetic parameter fluctuations Δαjni(t), Δβjni(t), Δγjni(t) and Δηsni(t), respectively, and ni(t) absorbs the stochastic property of random intrinsic kinetic parameter fluctuations. The covariances of the stochastic parameter fluctuations Δαjni(t), Δβjni(t), Δγjni(t) and Δηsni(t) are given as cov(ξni(t), ξni(t)) = ξ2Δt,τ, with ξ ∈ {Δαj, Δβj, Δγj, Δηs}, respectively, where Δt,τ denotes the delta function, that is, Δt,τ = 1 if t = τ and Δt,τ = 0 if t ≠ τ, i.e., Δαj, Δβj, Δγj and Δηs denote the corresponding standard deviations of the stochastic parameter fluctuations Δαjni(t), Δβjni(t), Δγjni(t) and Δηsni(t), respectively. Thus, the nonlinear stochastic coupled synthetic genetic oscillators under intrinsic kinetic parameter fluctuations in the host cell i can be represented bywhere i. For the convenience of analysis and control design of the synchronization in synthetic genetic oscillators inserted into different host cells, the nonlinear stochastic equation in (6) can be represented by the following Ito’s stochastic differential equationwhere wi(t) is a standard Wiener process or Brownian motion with dwi(t) = ni(t)dt to represent the random kinetic fluctuations of synthetic gene circuits.where in vivo also suffers from extrinsic environmental molecular noise, such as transmitted noise from upstream and global noise affecting all cells. Therefore, the nonlinear stochastic equation of coupled oscillators in (7) should be modified to mimic realistic dynamic behavior, as follows:In general, a synthetic genetic oscillator vi(t) denotes extrinsic environmental molecular noise from the environment and hi denotes the noise-coupling matrix in cell i.where the signal vector x1(t) = x2(t) = ⋯ = xN(t) = s(t) in probability as t → tf orThe nonlinear stochastic coupled synthetic genetic oscillators in (8) are said to reach synchronization asymptotically if s(t) ∈ Rm is the synchronization solution satisfyingin which Let us denote the synchronization error signal for synthetic genetic oscillators asi are then described byAccording to (8), the synchronization error dynamics for cell i = 1, 2, …, N which can be augmented asfor whereandρ of coupled synthetic oscillators [Suppose the influence of extrinsic environmental molecular noise on the synchronization error can be bounded by the following noise-filtering level illators ,31,44v(t), where ρ2 denotes the upper bound of the effect of v(t) on synchronization error from the mean energy point of view, i.e., the noise filtering level ρ in (14) denotes the upper bound of the noise-filtering ability ρ0 of coupled synthetic genetic oscillators, and R is a symmetric weighting matrix to be specified by designer.for all possible environmental noises ρ from the mean energy point of view, i.e., the value of the noise-filtering ability ρ0 is the lower bound of ρ. Because the statistics of extrinsic molecular noise may be unavailable or uncertain, it is very difficult to obtain the noise filtering ability ρ0 for all possible extrinsic noises v(t) directly and only the upper bound ρ of the noise-filtering ability ρ0 can be given in (14) at first. Then, we will decrease the upper bound ρ to as small a value as possible to approach its lower bound for the noise-filtering ability ρ0, i.e., to get ρ0 by minimizing ρ (or ρ0 = min ρ) indirectly. If the noise filtering ability ρ0 is small, it means that the environmental molecular noise has less influence on synchronization and vice versa. It will be further discussed in the sequel.(i) The inequality in (14) means that the effect of extrinsic environmental molecular noise on the synchronization error is less than v is deterministic, then the expectation on v(t) in (14) should be neglected. If the initial condition e(0) is considered, then the noise-filtering level in (14) should be modified as follows [(ii) If the extrinsic environmental molecular noise  follows ,16:V(e(0)), i.e., the energy due to the initial condition e(0) should be considered in the influence of noise on synchronization [For some Lyapunov function nization ,45.H∞ noise filtering performance in (14) or (15), we obtain the following robust synchronization result for nonlinear stochastic coupled synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.Based on the synchronization error dynamic equation in (13) and the V(e) > 0 with V(0) > 0 solving the following HJI,If there exists a positive function v(t) on the synchronization of the nonlinear stochastic coupled synthetic oscillation systems in (8) is less than or equal to a prescribed filtering level ρ, i.e., the inequality in (14) holds.then the stochastic intrinsic noise can be tolerated (i.e. the synchronization of coupled synthetic genetic networks cannot destroyed by intrinsic noise) and the influence of extrinsic environmental molecular noise ρ denotes an upper bound of the effect of v(t) on synchronization, the real effect can be obtained by minimizing ρ to as small a value as possible. Therefore, the noise-filtering ability of synchronized oscillators on v(t) can be obtained by solving the following constrained optimization:Since V(e) > 0subject to HJI in (16) with ρ0 on v(t) for the synchronized synthetic oscillators can be evaluated by solving the constrained optimization in (17). The noise-filtering ability ρ0 of the synchronized synthetic oscillators in (17) can be obtained by decreasing ρ until no positive solution V(e) > 0 exists for HJI in (16) again.i.e., the noise-filtering ability ρ0 cannot satisfy the designer’s specification, in order to enhance the noise filtering of extrinsic noise, we need to specify the design parameters of nonlinear stochastic coupled synthetic genetic oscillators, for example, the kinetic parameters αa, αb, αc, βA, βB, βC, γm, and γp in (5) to solve the constrained optimization in (17) to enhance the noise-filtering ability, i.e.,If the noise-filtering ability V(e) > 0subject to HIJ in (16) with Before the discussion on the synchronization robustness criterion of coupled synthetic genetic oscillators, some definitions on synchronization robustness, intrinsic robustness and extrinsic robustness are given as follows:(1) Synchronization robustness: The ability of coupled synthetic genetic oscillators to resist both intrinsic noise and extrinsic noise so that the synchronization can be maintained.(2) Intrinsic robustness: The ability of coupled synthetic genetic oscillator to tolerate intrinsic parameter fluctuation to maintain synchronization.(3) Extrinsic robustness: The filtering ability to attenuate the effect of environmental noise on the synchronization of coupled synthetic genetic network.ρ0 of (17) into (16) in Proposition 1, we get the following equivalent synchronization robustness criterionSubstituting the noise-filtering ability ρ0, and the term on the right hand side of (19) indicates the synchronization robustness of the coupled synthetic gene networks. The biological meaning of synchronization robustness criterion in (19) is that if the synchronization robustness can confer both the intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then the coupled synthetic networks will synchronize with a noise filtering ability ρ0. If the synchronization robustness criterion in (19) is violated, then the synchronization of coupled synthetic gene networks may not be achieved due to the intrinsic parameter fluctuation and extrinsic noise.The first term on the left hand side of (19) indicates the intrinsic robustness to tolerate the intrinsic parameter fluctuation in (13) because this term is induced by intrinsic noise (or random parameter fluctuation), the second and third term on the left hand side are due to the noise filtering in (14) and indicate the extrinsic robustness to filter the extrinsic noise with the noise filtering ability V(e) > 0 and V(0) = 0 to guarantee robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators with a prescribed attenuation level ρ under intrinsic kinetic parameters fluctuations and extrinsic environmental molecular noise or to solve the constrained minimization in (18) for robust synchronization design to achieve the optimal molecular noise filtering of the synchronized coupled synthetic oscillators. Recently, the fuzzy dynamic model has been widely used to interpolate several local dynamic models to efficiently approximate a nonlinear dynamic system [In general, it is still very difficult to solve the second-order HJI in (16) with c system ,38,39. Hkth rule of the fuzzy model for the synchronization error dynamics for cell i in (12) is proposed in the following form [In this study, the T-S fuzzy method is employed to simplify the analysis and design procedure for robust synchronization of nonlinear stochastic coupled synthetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. The T-S fuzzy model for the synchronization error dynamics is described by fuzzy if-then rules. The ing form ,32,38,39Rulek: If zi1,(t) is Fk1 and zi2,(t) is Fk2 … and zg,i(t) is Fkg, thenk = 1, 2, ⋯, L, where zg,i is the element of premise variables of the ith coupled oscillation system, i.e., zi = zi1,, …, zg,iT; Fkg is the fuzzy set; Ak, Bk, AWk, and BWk are the fuzzy system matrices; L is the number of if-then rules; and g is the number of premise variables. The physical meaning of fuzzy rule k is that if the premise variables zi1,(t), zi2,(t), ⋯, zg,i(t) are with the fuzzy sets Fk1, Fk2, ⋯, Fkg, then the synchronization error dynamics in (12) can be represented by interpolating the linearized synchronization error dynamics in (20) via the fuzzy basis. The fuzzy synchronization error dynamics in (20) is referred as follows [for  follows ,40,46:Fkj is the grade of membership of zj,i(t) in Fkj or the possibility function of zj,i(t) in Fkj, and μk(zi) is called fuzzy basis function for k = 1, 2, …, L. The denominator or L local linear stochastic systems through nonlinear basis μk(zi) to approximate the nonlinear stochastic system in (13). In this situation, the nonlinear stochastic coupled oscillation systems in (13) can be represented by the fuzzy interpolation system as follows:where μk(z) = diag, …, μk,N(zN)) and z = z1, …, zNT.where Ak, Bk, AWk, and BWk in (21) or (22) can be identified by least square estimation method. On the other hand, many studies have proved that the T-S fuzzy model can approximate a continuous function with any degree of accuracy. Actually, there is still some fuzzy approximation error in (22). In the robust synchronization control design, for simplicity, the fuzzy approximation error can be merged into the external noise, which could be efficiently attenuated by the proposed H∞ robust synchronization control design in the sequel.In , Takagi After investigating the approximation of nonlinear stochastic coupled synthetic oscillators by the fuzzy interpolation method, in order to avoid solving the nonlinear constrained optimization problem in (18) for the robust synchronization design problem of coupled synthetic oscillators under intrinsic kinetic parameter fluctuation and extrinsic environmental molecular noise, the measurement procedure for the noise-filtering ability of synchronized synthetic genetic oscillators could also be simplified by the fuzzy approximation method. Then, we get the following result.P>0 solving the following LMIs,If there exists a positive definite symmetric matrix k = 1, 2, …, L, then the noise-filtering level ρ in (14) holds, or the intrinsic kinetic parameter fluctuations are tolerated by the synchronized synthetic genetic oscillators, and the influence of extrinsic environmental molecular noise v(t) on the synchronized synthetic oscillation systems in (13) is less than or equal to a prescribed filtering level ρ.for Therefore, the optimal noise-filtering design of synchronized oscillation systems obtained by solving the HJI-constrained optimization problem in (18) could be replaced by solving the following constrained optimizations, respectivelyP>0 and LMIs in (23)subject to ρ is prescribed by a biological engineer, a robust synchronization design would involve specifying the design parameters αa, αb, αc, βA, βB, βC, γm, and γp of the synthetic gene oscillators in Ak, so that the LMIs in (23) have a positive solution P>0 in Proposition 2. If we want to achieve optimal filtering of extrinsic noise for the synchronized synthetic oscillators, some design parameters need to be specified for coupled synthetic oscillators to achieve the constrained optimization in (24).(i)  If the prescribed noise-filtering level P>0 for LMIs in (23) instead of solving V(e) > 0 for HJI in (16) directly. Further, based on the fuzzy interpolation of local linear systems, i.e., replacing F, G, FW and GW by the fuzzy approximations in (22), in Proposition 2, V(e) = eTPe is employed to solve the HJI (16) in Proposition 1. The HJI in Proposition 1 is replaced with a set of LMIs in Proposition 2 and we only need to solve P>0 for LMIs to guarantee the coupled synthetic genetic oscillators have a noise filtering level ρ.(ii)  In this study, the fuzzy approximation method in (21) or (22) is only employed to simplify the analysis and design procedure via solving (iii)  In general, the constrained optimization problems in (24) are called eigenvalue problem , which c(iv)  In addition to the robust oscillation synchronization, the proposed method can be applied to robust synchronization design of coupled synthetic gene networks with any kind of dynamic behavior.(v)  In the fuzzy approximation case, the synchronization robustness criterion in (19) is equivalent to the followingk = 1, 2, …, L which is equivalent to (23) with ρ being replace by ρ0. The biological meaning of synchronization robustness criterion in (25) is that if the local synchronization robustness of local coupled synthetic genetic oscillators can confer local intrinsic robustness to tolerate local intrinsic parameter fluctuation and local extrinsic robustness to filter external noise, then the coupled synthetic genetic oscillators can be synchronized with a noise filtering ability ρ0. If the synchronization robustness criterion in (25) is violated, then the synchronization of coupled synthetic genetic oscillators may not be achieved due to intrinsic parameter fluctuation and extrinsic noise. In general, if the design parameters of coupled synthetic genetic oscillators are specified so that the eigenvalues of local coupled system matrix IN ⊗ AWk + C ⊗ BWk are far in the left hand side of complex s-domain , then the coupled synthetic genetic networks are more easy to synchronize in spite of intrinsic parameter fluctuation and extrinsic noise.for H∞ stochastic control theory, an input control strategy is introduced to enhance the robust synchronization. If AI is injected into a common medium to increase the average concentration of AI protein in the exteracellular environment, which in turn increases the cellular communication of coupled oscillation systems, then the dynamics of the signaling molecule AI in the cellular environment, as shown in (3), should be modified asIf robust synchronizations of coupled synthetic genetic oscillators cannot be achieved spontaneously via the parameter design in the above sections, then a control strategy is needed from external stimulation inputs to improve the robust synchronization of coupled synthetic genetic oscillators. External stimulation inputs are known to play an important role in the synchronization of biological rhythms. Recently, several methods of periodic stimulation for synchronization of nonlinear oscillators have been introduced -9,17. Houe = Q represents an extracellular control input, which can be implemented via the injection of inducer AI.where ue = Q is employed to improve the robust synchronization of the nonlinear stochastic coupled synthetic oscillation systems. In this situation, the synchronization error dynamics in (13) should be modified as follows:For the simplicity of control design, suppose that the following control input Ce(Q) = (ceij(Q))N × N ∈ RN × N is the coupling configuration matrix, in which ceii(Q) = − ηs(1 − N− 1)(Qe + Q) if i=j, otherwise ceij(Q) = ηsN− 1(Qe + Q). Then, we can also obtain the robust synchronization control design of coupled oscillation systems under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noises as follows.where ue=Q in the terms of Ce(Q) in (27), if there exists a positive solution V(e) > 0 with V(0) = 0 to the following HJIFor the nonlinear stochastic coupled synthetic genetic oscillators with an extracellular control input ρ, then the stochastic intrinsic kinetic noise can be robustly tolerated, and the influence of extrinsic environmental molecular noise v(t) on the synchronization of the nonlinear stochastic coupled synthetic oscillation systems in (27) is less than or equal to ρ, i.e., the inequality in (14) or (15) holds.for a prescribed filtering level The inequality (28) is equivalent to synchronization robustness criterionQ to improve the synchronization robustness to provide more intrinsic robustness and more extrinsic robustness to tolerate more intrinsic parameter fluctuation and filter more extrinsic noise, then the robust synchronization of the nonlinear stochastic coupled synthetic genetic oscillators in (27) can be guaranteed. Similarly, the optimal noise-filtering design of synchronized oscillation systems by the extracellular control input in (27) can be achieved by solving the following constrained optimization problem:The physical meaning of synchronization robustness criterion in (29) is that if we can specify control parameter V(e) > 0 and HJI in (28)subject to ue=Q to solve the HJI-constrained optimization in (30) for achieving the optimal noise filtering for synchronized synthetic genetic oscillators. Therefore, the fuzzy approximation method is again employed to simplify the control design procedure. Based on the fuzzy approximation method, the following fuzzy interpolation system is employed to approach the nonlinear stochastic coupled oscillation systems in (27):In general, it is still very difficult to specify the control parameter Applying the fuzzy approximation method, the external signal control design can be obtained as described in the following corollary, for robust filtering of synchronized oscillation systems with intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.P > 0 to the following LMIs for a prescribed noise-filtering level ρFor stochastic synchronized oscillation systems, if there exists a symmetric solution k = 1, 2, ⋯, L, then intrinsic parametric noise can be tolerated and the effect of extrinsic molecular noise v(t) on the synchronization of nonlinear stochastic coupled oscillation systems is less than or equal to a prescribed filtering level ρ.for Q, such that the LMIs in (32) have a positive definite solution P > 0, then the robust synchronization with a prescribed noise filtering level ρ on extrinsic environmental molecular noise is guaranteed for the nonlinear stochastic coupled synthetic genetic oscillators. If we specify control parameter Q so that the eigenvalues of local system matrix of IN ⊗ Ak + Ce(Q) ⊗ Bk of coupled synthetic gene oscillators have more negative real part (i.e. in far left hand complex s-domain), the coupled synthetic gene oscillators are with more robust synchronization to tolerate more intrinsic parameter fluctuations and to filter more extrinsic noise.The physical meaning of Corollary 4 is that if we can select a control parameter Similarly, based on the fuzzy approximation method, an optimal noise-filtering design of synchronized oscillation systems by using the extracellular control input in (30) can be achieved by solving the following constrained optimization problem:V(e) > 0 and LMIs in (32)subject to Q through the inducer concentration control method to solve the constrained optimization problem, we can achieve both robust synchronization against intrinsic kinetic parameter fluctuations and optimal filtering against external environmental molecular noise on the synchronization by using the external control signal in the coupled synthetic oscillation systems.The physical meaning of the constrained optimization in (33) is that if we can select a control parameter The design procedure of external inducer control for robust synchronization of the coupled network is summarized as follows.N coupled oscillators with intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.(1) Consider a synthetic genetic network of ρ.(2) Given the prescribed disturbance attenuation level (3) Represent the nonlinear stochastic synchronization error dynamic by the T-S fuzzy synchronization error dynamic model, using the interpolation of several local linear stochastic systems.Q to solve LMI in (32) with the help of LMI toolbox in MATLAB so that N coupled synthetic genetic oscillators can be synchronized with a prescribed noise filtering level ρ.(4) Specify In this section, we provide a simulated example to illustrate the design procedure of robust synchronization of the nonlinear stochastic coupled synthetic oscillation systems and to confirm the performance of the robust synchronization of proposed method against intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise.N=10 coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise in (5). The simulation results are shown in Figure Q  in (26) to compensate for the inefficiency of coupling between the synthetic genetic oscillators from the quasi-steady-state point of view. The design procedure first begins with representing the nonlinear stochastic synchronization error dynamic in (22) by the T-S fuzzy synchronization error dynamic model in (31), using the interpolation of several linear stochastic systems as presented in Additional file Q (i.e. the corresponding density of inducer AI), so that the ten coupled synthetic genetic oscillators have a positive solution P > 0 with a prescribed noise filtering level ρ = 0.56 to guarantee robust synchronization under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. The LMI toolbox in MATLAB can then be used to significantly simplify the system analysis and design procedure. With Q = 0.66 solved from LMIs in (32), the outputs of the coupled gene network of ten synthetic oscillators under intrinsic parametric fluctuations and extrinsic noise are shown in Figure E ∫ 0100eT(t)Re(t)dt)/(E ∫ 0100vT(t)v(t)dt) ≈ 0.192 < 0.562. It can be clearly seen that based on our proposed design method, the coupled gene network cannot only tolerate kinetic parameter variations but also attenuate the extrinsic molecular noise below a desired level to achieve a robust synchronization.The purpose of this example is to demonstrate the effectiveness of the theoretical synchronization result of synthetic gene oscillators in mimicking real biological oscillator systems. We consider a synthetic genetic network of model in 1, using The cell is the functional unit of all living things, either unicellular or multicellular. A cell can sense many different signals from the internal or external context and can respond to the constantly changing environment via appropriate cellular processes. Also, cells can interact with each other via cell-to-cell communication and achieve specific physiological functions essential for life in a cooperative manner. However, many fundamental questions remain regarding how cellular phenomena arise from the interactions between genes and proteins, what features make the cell operate reliably in diverse conditions, and how the cell is responsible for these operations. To gain insight into these questions, one can construct the underlying mechanisms that constitute the web of interactions. This idea is useful to separate a complicated network into many simpler ones, which can work independently but also cooperate with each other. It may not only enhance our understanding of collective behavior particularly via synchronization but may also establish a foundation to design robust implementation of coupled synthetic gene networks ,48.synchronizable. Based on this synchronization performance, we propose a procedure for designing or compensating a coupled network with two or more coupled synthetic oscillator through a given connected topology toward a desired robust synchronization performance. Using the proposed method, the coupled synthetic oscillators cannot only tolerate kinetic parameter variations but also attenuate the extrinsic molecular noise below a desired level to achieve a desired robust synchronization. However, the HJI-constrained optimization problem is difficult to solve directly by any analytical or numerical method because of the complexity of nonlinear synchronization error dynamics. Hence, we employ a T–S fuzzy model to solve the HJI easily and indirectly. The T–S fuzzy model has been widely applied to approximate nonlinear systems by interpolating several local linearized systems. Here, we use the T–S fuzzy model to approximate the nonlinear stochastic synchronization error dynamics. By using the T–S fuzzy model and choosing the appropriate Lyapunov function, the HJI-constrained optimization for calculating the robust synchronization performance of a nonlinear coupled network is reduced to an equivalent LMI-constrained optimization problem, which can be solved efficiently by MATLAB’s LMI toolbox.In this paper, we consider a nonlinear stochastic coupled network with two or more coupled synthetic oscillators. By transforming nonlinear stochastic coupled network dynamics into synchronization error dynamics, we can use Lyapunov’s direct method to infer a sufficient condition required for robustness of the nonlinear synchronized network. Assuming that each synthetic oscillator suffers from intrinsic kinetic parameter fluctuations and extrinsic molecular noise, robust synchronization performance is defined as the effect of extrinsic molecular noise upon the synchronization error. Based on this definition, robust synchronization performance of a nonlinear coupled network can be calculated by solving an associated HJI-constrained optimization problem. We also show that nonlinear coupled networks with robust synchronization performance are also Recently a simple synthetic device was engineered in a cell, and several cells were then combined, so that their connections allowed the construction of a more complex synthetic gene circuit, i.e., so-called multi-cellular engineered networks. This approach not only uses cellular consortia as an efficient way of engineering complex gene circuits, but also demonstrates the great potential for reutilization of small parts of the gene circuit. In such situations, our proposed evaluation framework may offer a possible guideline for the design or compensation of such coupled networks with a given connected topology toward a desired collective behavior.H∞ control scheme is developed to enhance synchronization by adding external control to increase the cell-to-cell communication through quorum sensing. This study enhances our understanding in this area in the following ways: (a) nonlinear stochastic systems are employed to model the coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations, extrinsic molecular noises on the host cells and quorum sensing molecules; (b) two robust synchronization criteria (19) and (25) of coupled synthetic oscillators are developed from the nonlinear stochastic filtering point of view, so that we can gain more insight into the robust synchronization mechanism from a systematic perspective. If these robust synchronization criterion cannot be guaranteed, robust synchronization control schemes via selecting adequate kinetic parameters and inducer concentration are also developed to improve the synchronization robustness of coupled synthetic genetic oscillators; and (c) the fuzzy approximation is employed to approximate the nonlinear stochastic synchronization error model by interpolating several linear stochastic systems, so that the powerful LMI toolbox in MATLAB can be used to significantly simplify the system analysis and design procedure for robust synchronization of coupled synthetic gene networks, which is very important for the emergent phenomenon of synthetic gene networks through quorum sensing at the molecular level. In addition, the proposed robust synchronization design and control scheme for nonlinear stochastic coupled genetic oscillators can be easily applied to the robust synchronization problem of other coupled genetic networks as a cellular consortium.In this study, several robust synchronization criteria and designs are proposed for a population of synthetic genetic oscillators in order to exploit an emergent synchronization phenomenon by quorum sensing molecules under intrinsic kinetic parameter fluctuations and extrinsic molecular noise. When the synchronization of nonlinear stochastic coupled synthetic genetic oscillators cannot be maintained, a robust The authors declare that they have no competing interests.BSC formulated the research topic and carried out some robust synchronization control designs. CYH participated in some control designs and performed simulation. Both authors read and approved the final manuscript.This file contains appendices A, B and C.Click here for file


TH-immunohistochemistry demonstrated that the non-glial-associated nerve fiber outgrowth in CD47−/− cultures reached significantly longer distances and higher density compared to nerve fibers formed in CD47+/+ cultures at 14 days in vitro. These nerve fibers often had a dotted appearance in CD47+/+ cultures. No difference in the astrocytic migration was observed. Further investigations revealed that the presence of CD47 in control culture did neither hamper non-glial-associated growth through SIRPα nor through TSP-1 since similar outgrowth was found in SIRPα mutant cultures and in CD47+/+ cultures treated with blocking antibodies against the TSP-1, respectively, as in the control cultures. In conclusion, long-distance growing nerve fiber formation is promoted by the absence of CD47, even though the presence of astrocytes is not inhibited.In ventral mesencephalic organotypic tissue cultures, two timely separated sequences of nerve fiber growth have been observed. The first appearing nerve fiber pattern is a long-distance outgrowth that occurs before astrocytes start to proliferate and migrate to form an astrocytic monolayer that finally surrounds the tissue slice. These long-distance growing nerve fibers are retracted as the astrocytes migrate, and are followed by a secondary outgrowth. The secondary outgrowth is persistent in time but reaches short distances, comparable with outgrowth seen from a dopaminergic graft implanted to the brain. The present study was focused on the interaction between the astrocytes and the long-distance growing non-glial associated nerve fibers. Cross talk between astroglia and neurite formation might occur through the integrin-associated protein CD47. CD47 serves as a ligand for signal regulatory protein (SIRP) α and as a receptor for the extracellular matrix protein thrombospondin-1 (TSP-1). Embryonic day 14 ventral mesencephalic tissue from CD47 Regeneration in the adult brain involves neurotrophic factors, neuroinhibitors, cell adhesion, and extracellular matrix molecules that may affect the regenerative process. These molecules are often produced by the glial cells and affects the nerve regeneration for instance in the spinal cord after injury, where regeneration beyond the glial scar is problematic. Another field for regenerative studies is regeneration induced by neural grafts in neurodegenerative disorders. Much effort has been dedicated to study reinnervation of the striatum induced by transplanted dopaminergic cells in animal models of Parkinson's disease in vitro (DIV), without the presence of glial cell bodies, and retracts usually after some weeks. The later appearing nerve fiber formation is found in the presence of astroglia and is persistent over time. Based on studies using a mitotic inhibitor, a strong relationship between the two growth patterns has been suggested In ventral mesencephalic organotypic cultures, two morphologically different types of nerve fiber growth patterns have been observed Extracellular matrix molecules, integrins or integrin-associated protein, also known as CD47, are factors that are expressed by the astrocytes. It is known that the extracellular matrix proteins, such as the proteoglycans, widely affect nerve fiber outgrowth, however, little is known about the effects of CD47 −/− Balb/c mice has previously been described ad libitum. Mating was made over one night, and fetuses collected at embryonic day 14. These experiments had been approved by the local ethics committee, Umeå Ethics Committee for Animal Studies (permit number: A23-08).The generation of CD47−/−, SIRPα mutants, and their wildtype (WT) mice were deeply anesthetized using isoflurane . The ventral mesencephalic (VM) area of the fetal brain was dissected under a microscope in sterile conditions. For the dissection, Dulbecco’s modified Eagle’s medium  was used. The dissected tissue pieces were sliced into 300 µm coronal sections using a tissue chopper and transferred to DMEM. The tissue slices was cut in the midline into two pieces and for each culture, one such piece was used. These tissue slices were attached to sterile, poly-d-lysine coated  coverslips (12×24 mm) in two drops of chicken plasma (Sigma Aldrich) and one drop of thrombin . The tissue slice/plasma/thrombin mixture was dried for 15–20 min before placed in 15 ml Falcon tubes containing 0.9 ml of medium. The tubes were inserted in a “roller-drum” placed in an incubator at 37°C in 5% CO2. The cultures were continuously rotating at a speed of 0.5 turns per minute.Pregnant CD47The medium contained 55% DMEM (Gibco), 32.5% Hanks’ balanced salt solution (Gibco), 10% fetal bovine serum (Gibco), 1.5% glucose (Gibco), and 1% Hepes (Gibco). All ingredients were mixed and filtered through a sterile filter . Antibiotics were added to the medium in a final concentration of 1%  and used at plating. At the first medium change and thereafter, antibiotics were excluded. The medium was changed twice a week.+/+ n = 20; CD47−/− n = 9) or 14  days. The n’s were based from several experiments: 7 DIV CD47+/+ n = 4; CD47−/− n = 5 and 14 DIV: CD47+/+ n = 7; CD47−/− n = 6; SIRPα mutants n = 4; SIRPα WT n = 2. Antibodies against thombospondin-1  were added to the medium of cultures derived from CD47+/+ tissue from plating and throughout the experiment. Antibodies were purified from hybridoma supernatants by ammonium sulfate precipitation and affinity chromatography using Protein G High Trap columns . Control cultures to the antibody treatment were treated with vehicle added to the medium of CD47+/+ tissue.The tissue slices were kept in culture for 7 (CD47in vitro (DIV), the ventral mesencephalic cultures were fixed in 2% paraformaldehyde in 0.1 M phosphate buffer (pH = 7.4) for 1 h and rinsed 3 times for 10 min in phosphate buffered saline  at room temperature. Afterwards, the cultures were incubated in antibodies raised against tyrosine hydroxylase , and in antibodies against vimentin to visualize astrocytes  for 48–72 h at 4°C. After rinsing in PBS, cultures were incubated in secondary antibodies Alexa 594  and Alexa 488  for 1 h at room temperature. For staining of cell nuclei, the cultures were incubated with DAPI  for 10 min at room temperature. All antibodies and DAPI were diluted in 1% Triton X-100 in PBS. Incubations were performed in a humidified atmosphere. All cultures were triple labeled, which was performed in sequence with one antibody at the time. Controls for unspecific immunohistochemistry were performed where the primary antibodies were omitted. After additional rinsing, the cultures were mounted in 90% glycerin in PBS.At 7 or 14 days Image analysis was used to measure the distances that astrocytes had migrated and the length of TH-positive nerve fiber outgrowth. The distances were measured from the periphery of the tissue slice to the distal end that nerve fibers or the migrating astrocytes had reached using a microscale mounted in one of the oculars. Estimations of overall TH-positive outgrowth were based on 3–4 measurements in each culture over areas that displayed nerve fiber outgrowth, and estimations of vimentin-positive astrocytes were performed on measurements made in 4 perpendicular directions from the tissue slice. Nerve fiber density measurements were performed using NIH image analysis program in images captured using a 20X lens and a CCD camera . The density measurements were performed over areas distal to the migration of astrocytes, near the astrocytic borderline. DAPI staining confirmed that no other cell types than vimentin-positive had migrated from the tissue slice. All measurements were performed on blind-coded slides. Images for publication were captured using a Retiga-4000RV CCD camera  and processed with Openlab software .Statistical analysis were performed on means per slice culture and type of measurement using two-factor analysis of variance (ANOVA) for genotype and DIV interactions, and the analysis was followed by post hoc analysis using Student’s t-test. All results were expressed as means ± SEM, and significant level was set at p<0.05.+/+ and CD47−/− fetuses were cultured for 7 and 14 DIV and examined for TH- and vimentin-immunohistochemistry. The results revealed that the distance for TH-positive nerve fiber outgrowth was affected by time without any significance in interaction between time and genotypes . No difference in the length of outgrowth was found between CD47+/+ and CD47−/− at 7 DIV, while at 2 weeks the TH-positive nerve fibers had reached significantly longer distances in cultures derived from CD47−/− than in CD47+/+ cultures . The total distance reached by the nerve fiber outgrowth was around 1 mm at 7 DIV and around 3.5 mm at 14 DIV for CD47−/− cultures, which was longer than the astrocytic migration had reached at the respective time points  and within each genotype significantly enhanced length was reached at 14 DIV , while no difference was found between genotypes  and vimentin-  –immunoreactivity in the tissue slices at 7 DIV, these images are here shown separately to complement +/+  and CD47−/−  cultures. Scale bar = 100 µm.(TIF)Click here for additional data file.


The TK2−/− mouse model displays symptoms similar to humans harboring TK2 deficient infantile fatal encephalomyopathy. Here, we have studied the TK2−/− mouse model to clarify the pathological role of progressive mtDNA depletion in liver for the severe outcome of TK2 deficiency. We observed that a gradual depletion of mtDNA in the liver of the TK2−/− mice was accompanied by increasingly hypertrophic mitochondria and accumulation of fat vesicles in the liver cells. The levels of cholesterol and nonesterified fatty acids were elevated and there was accumulation of long chain acylcarnitines in plasma of the TK2−/− mice. In mice with hepatic mtDNA levels below 20%, the blood sugar and the ketone levels dropped. These mice also exhibited reduced mitochondrial β-oxidation due to decreased transport of long chain acylcarnitines into the mitochondria. The gradual loss of mtDNA in the liver of the TK2−/− mice causes impaired mitochondrial function that leads to defect β-oxidation and, as a result, insufficient production of ketone bodies and glucose. This study provides insight into the mechanism of encephalomyopathy caused by TK2 deficiency-induced mtDNA depletion that may be used to explore novel therapeutic strategies.Thymidine kinase 2 (TK2) deficiency in humans causes mitochondrial DNA (mtDNA) depletion syndrome. To study the molecular mechanisms underlying the disease and search for treatment options, we previously generated and described a TK2 deficient mouse strain (TK2 The TK2−/− mice were born with mtDNA depletion in adipose tissue, brain and skeletal muscle. Although the mice behaved and grew normally the first 7 days they demonstrated progressive loss of mtDNA content in several organs and at postnatal day 12 the mtDNA content was also decreased in liver. Although the liver structure was similar in the TK2+/+ and TK2−/− mice, the latter showed an increased number of intracellular lipid vesicles. The TK2−/− mice were also progressively hypothermic and displayed abnormal brown adipose tissue and an almost complete loss of hypodermal fat at 14 days of age when this fat layer is normally well developed. In addition, the adipocytes in the 14 days old TK2−/− mice showed heterogeneity in size and accumulation of lipid vesicles in contrast to a homogeneous pattern seen in the TK2+/+ mice −/− mice. Here we analysed the TK2−/− mouse model to clarify the importance of mtDNA for liver function by studies of the lipid phenotype and functionality of the hepatic mitochondria as well as alterations in plasma. The obtained data show that the TK2-deficiency induced gradual loss of mtDNA in the liver leads to defect β-oxidation and, as a result, insufficient production of ketone bodies and glucose.Currently there is no cure available for MDS and to increase the understanding of the mechanisms behind the syndrome we previously generated and described a mouse strain deficient in TK2 . TK2Electron images of mitochondria were obtained as previously described 2 lattices according to Weibel +/+ and TK2−/−). To determine the numbers of mitochondria per cell, estimations were done based on the two-dimensional structures in the images and the assumption that the size of mitochondria in the tissue correlates with the size of the area in the image.A stereological method was used to determine the volume density of the mitochondria in the cytoplasm of the hepatocytes. The areas of mitochondria profiles on printed images and the total area of the cytoplasm was measured by point counting using 2 cmhypoxanthine-guanine phosphoribosyltransferase (Hprt) was selected as endogenous control. Data was collected on individual cDNA samples.Total RNA from 7 and 12 days old mice (four of each genotype and age) was extracted by TRIzol® (Life Technologies) and quantified with the NanoDrop™ 2000 (Thermo Scientific). RNA quality assessment was performed using the Experion™ RNA StdSens analysis kit (Bio-Rad Laboratories). Oligo-dT-primed cDNA synthesis was performed with 1 µg total RNA using the High Capacity Reverse Transcriptase Kit (Applied Biosystems). The cDNA was diluted 1 to 10 before real-time PCR was performed on the Applied Biosystems PRISM 7000 using the SYBRgreen reagent (Applied Biosystems). Reactions were performed in triplicates and −/− mice, a PCR screening array for 84 genes in the mouse fatty acid oxidation pathway  was conducted with liver tissue from 14 days old mice (three of each genotype). Total RNA was extracted using the RNA easy mini kit (Qiagen). The RNA integrity was assessed spectrophotometrically, and all samples had 260/280 ratios between 1.8–2.0 and 260/230 ratios above 1.7. Reverse transcription was done on 0.5 µg total RNA per sample using the RT2 First strand kit (Qiagen) according to the manufacturer’s protocol. SYBR green RT-PCR analysis was performed on an ABI 7500 fast cycler. The array evaluated the expression of 84 genes involved in fatty acid oxidation. The acquired data were analyzed for statistical differences using Qiagen’s software RT2 Profiler PCR Array Data Analysis version 3.5. Briefly the PCR array data were calculated by the comparative cycle threshold method, normalized against housekeeping genes, and expressed as mean fold change in TK2−/− samples relative to TK2+/+ control samples. Genes of TK2−/− mice that differed by >2-fold compared with TK2+/+ controls are included and highlighted in bold when p≤0.05.To identify additional genes involved in fatty acid oxidation that might be altered in TK2Liver tissues from 7 and 12 days old mice (three of each genotype and age) were ground in liquid nitrogen, dissolved in 400 µl RIPA buffer, incubated on ice for two hours, centrifuged at 16000 g for 20 min at 4°C and the supernatants retrieved. Western blot analyses to determine adipophilin, cytochrome c oxidase subunit II (COX II) and voltage-dependent anion channel (VDAC) protein levels in liver protein extracts were performed as previously described +/+ and nine TK2−/−), 10 days old (two of each genotype), and 12 days old (five of each genotype) mice were also used for quantification of hepatic triglyceride content using the TG GPO-PAP reagent (Roche). Samples were run in duplicates and the measurement was repeated once with new extracts. Absorption was measured at 492 nm  and analysed using the Magellan™ Data Analysis Software (Tecan).Liver tissues were extracted for lipids as described by Folch Size exclusion chromatography of plasma lipoproteins in samples from 7 and 12 days old mice (four of each genotype and age) was performed as described earlier Blood glucose levels were determined from whole blood by using the Accu-Chek (Roche). Blood ketone levels were determined in whole blood by using the Precision Exceed (Abbott). Six mice of each genotype were analyzed to determine the average ketone and glucose levels on postnatal day 14.The determination of acylcarnitines was performed in 12 days old mice (three of each genotype) by a modification of previously described methods Sample preparation: 50 µl plasma was diluted with a corresponding amount of isotope labelled internal standard mix  and deproteinised with 500 µl acetonitril containing 0.3% v/v formic acid. The sample was vortexed vigorously and centrifuged. The supernatant was transferred to a clean vial and evaporated, and derivatised with 100 µl 3 M HCl in n-butanol . The sample was evaporated again and reconstituted with 100 µl methanol and 50 µl 10 mM formic acid.LC-MS/MS procedure: 5 µl of the prepared sample was injected onto an Acquity UPLC™ BEH C18 2.1×50 mm, 1.7 µm chromatographic column (Waters). A gradient elution with water (mobile phase A) and methanol (mobile phase B), both containing 0.1% v/v formic acid, was set up to separate the analytes. The acylcarnitines in the resulting eluate were ionized in positive ion mode electrospray and determined with a tandem mass spectrometer  in MRM mode. Quantification of each compound was performed by internal standard methodology using the TargetLynx software (Waters). Plasma from 12 days old mice (three of each genotype) were analyzed in separate experiments.14C] palmitic acid (Sigma) into acid-soluble metabolites was studied in triplicate samples in the presence or absence of 50 µM antimycin A. The peroxisomal β-oxidation (antimycin A insensitive) was subtracted from the total β-oxidation. Livers of 14 days old mice (three of each genotype) were analyzed in separate experiments. Comparisons of mitochondrial β-oxidation were made within each run.Mitochondrial β-oxidation was determined in liver homogenates according to a previously described method Mitochondria were isolated from liver tissue samples of 14 days old mice (three of each genotype). Mitochondrial ATP production rates (MAPR) and citrate synthase (CS) activity were determined as previously described Total CPT activity was measured in isolated liver mitochondria from 14 days old mice (four of each genotype) according to a previously described method +/+ and TK2−/− mice of the same age. Significance was set at p≤0.05.Two tailed, unpaired Student’s t-test was used to test for statistical significance of difference between TK2+/+ and the TK2−/− mice. Sections shown are representative of the samples  and correlated to the total areas of the cytoplasm giving a mean number of mitochondria/µm2 cytoplasm of 0.52±0.02 for TK2+/+ and 0.54±0.08 for TK2−/−. However, the mean area/mitochondria was 0.540±0.010 µm2 for TK2+/+ and 0.866±0.046 µm2 for TK2−/−, all together showing that the mitochondria in the 12 days old TK2−/− mice were hypertrophic.Transmission electron microscopy was performed on liver samples from the TK2 samples . The vol+/+ mice . To dete−/− mice constitutes approximately 70% compared to the TK2+/+−/− mice affected the protein expression in the hepatic mitochondria the expression levels of the mitochondrial protein cytochrome c oxidase subunit 2 (COXII) were measured in 7 and 12 days old mice. For both time points the COXII expression in the TK2−/− mice was similar to the expression in the TK2+/+ mice (It was previously shown that the mtDNA content in livers of 12 days old TK2+/+ mice .−/− mice (150%) compared to TK2+/+ mice  (3-hydroxy-3-methyl-glutaryl-coenzyme A reductase (Hmgcr), the rate limiting enzyme in cholesterol synthesis, was similar in both 7 and 12 days old TK2−/− and TK2+/+ mice, but 3-hydroxy-3-methyl-glutaryl-coenzyme A synthase (Hmgcs) -1 displayed a trend towards decreasing expression levels with increasing age in the TK2−/− mice , a key enzyme in fatty acid synthesis, was decreased in the 12 days old TK2−/− mice compared to TK2+/+ mice of the same age and the hepatic mRNA expression of the sterol regulatory element binding protein 1c (Srebp1c), which is responsible for the regulation of genes required for de novo lipogenesis, such as Fas, also displayed a similar expression pattern  . The hep−/− mice . Hepaticlder age . The hep pattern .−/− mice containing <20% mtDNA, were analysed with PCR screening arrays and compared to TK2+/+. The expression levels of several genes were altered in the TK2−/− mice  was significantly reduced and the levels of methylmalonylcarnitine (C4-DC) and buturylcarnitine (C4) were also decreased, although not significantly, in the TK2−/− mice.In mitochondrial fatty acid oxidation defects, metabolic blocks exist in the catabolic pathways of mitochondrial coenzyme A esters. Screening for disorders of fatty acid β-oxidation is commonly performed by determination of plasma acylcarnitine levels −/− mice demonstrated considerably reduced mitochondrial palmitate oxidation rate within each measurement, exhibiting only 9–40% of the oxidation rate observed in the TK2+/+ mice.The accumulated data pointed to impaired β-oxidation and for that reason the fatty acid oxidation was determined in liver homogenates from 14 days old mice . The TK2−/−mice the mtDNA levels had drastically dropped below 20% compared to TK2+/+, however, the mtDNA depletion did not affect the MAPR at this point , which is responsible for the formation of long-chain acylcarnitines from activated fatty acids and free carnitine, was decreased . A key regulatory site controlling the flux of long-chain fatty acids through β-oxidation is the mitochondrial carnitine system that catalyzes the transport of long-chain fatty acids into the mitochondrial matrix. Since the gene expression level of ecreased , the CPTecreased . The met−/− mice were used to clarify how hepatic mitochondria are involved in the development of the encephalopathic phenotype of the TK2−/− mice. The hepatic cells of TK2−/− mice progressively lose the mtDNA and contain approximately 90% at postnatal day 7, 70% at postnatal day 12 and less than 20% at postnatal day 14, compared to the TK2+/+ mice. The TK2−/− mice retained normal expression of the mtDNA encoded protein COXII in the hepatic cells both at day 7 and 12, showing that they had enough mtDNA to maintain normal protein levels. It has been shown before that partially mtDNA depleted cells can retain COXII protein levels In this study liver and plasma from both symptom-free (7 days old) and symptomatic (12 and 14 days old) TK2−/− mice at day 12. Mitochondrial volume homeostasis is a housekeeping function that is essential to maintain the structural integrity of the mitochondrion. Increased matrix volume has also been suggested to activate the respiratory chain and thereby increase ATP production −/− mice. In addition, hepatic adipophilin protein expression, a cellular indicator of stored lipids, was significantly increased in 12 days old TK2−/− mice. Macrovesicular steatosis has also been seen periportally in liver of human patients harboring the multi-organ involvement form of TK2 mutations in which mtDNA depletion has been detected in brain and liver as well as muscle −/− and TK2+/+, suggesting an alteration in the components of lipid vesicles in these mice which needs further investigation.The loss of mtDNA content in hepatocytes was accompanied by an increase in mitochondrial volume, which was almost doubled in the TK2−/− mice. Consistent with an altered lipid metabolism in the TK2−/− mice abnormal adipose tissue development together with a moderate impairment of mitochondrial respiratory function was also found in TK2 knock-in mice −/− mice were filled with milk and the ketone-bodies were decreased in the TK2−/− mice, not reflecting a typical profile for starvation. It is possible that hormonal imbalances e.g. caused by loss of pancreatic functions are also contributing to the deteriorating condition. However, our own preliminary investigations of insulin levels in plasma have not yet indicated any difference between the TK2−/− and TK2+/+ mice, although it should be mentioned that these measurements were performed on non-fasted mice (data not shown).Lipids, such as cholesterol and NEFAs were elevated in 12 days old TK2−/− mice. The increased blood lipids may stem from a number of factors, such as intake from food and hepatic and peripheral clearance from and output to the bloodstream. However, high levels of circulating lipids may be an indicator of fat being released from the adipose tissue, contributing to the accumulation of lipid droplets in the liver cells of the TK2−/− mice plasma of 12 days old mice was screened for fatty acid oxidation disorders. Long-chain acylcarnitines and 3-OH acylcarnitines were found to accumulate in TK2−/− mice suggesting an impaired β-oxidation. Therefore the palmitate oxidation rate was analyzed in liver mitochondria and found to be decreased by 60–90% in the TK2−/− mice. However, due to limited sample size and inability to fast the mice the absolute palmitate oxidation rates could not be determined.Since the accumulated data suggested a defect in the fatty acid oxidation in the TK2−/− mice occurred rapidly between postnatal day 12 and 14, and it is likely that residual activities together with activities of newly synthesized mtDNA encoded proteins, were enough to maintain the respiratory chain activity at this point. The increased mitochondrial volume in the TK2−/− liver cells may also contribute to further activate the respiratory chain. Other compensatory mechanisms such as increased mitochondrial transcript or protein half-lives, which have been suggested to occur in muscle tissue harboring severe mtDNA depletion due to TK2 deficiency CPT1A, which is responsible for conjugation of carnitine to long chain acyl-CoAs, was reduced. To test if the CPT1A activity was altered in 14 days old TK2−/− mice a method determining total CPT activity was used and the activity was found to be reduced to 65% compared to TK2+/+. Since CPT1A is the overall rate-limiting step in the β-oxidation of long chain fatty acids it is likely that the reduced CPT1A activity is participating in the altered β-oxidation in the TK2−/− mice. CPT1A is regulated both by malony-CoA, which inhibits its activity, and by the curvature and lipid characteristics of the outer mitochondrial membrane −/− mice but the lipid profile of the TK2−/− mice was altered and it is possible that it contributes to a decrease in CPT activity.The severe drop in mtDNA levels in the TK2−/− mice the fatty acid oxidation was impaired but the observed ketone and glucose levels were normal. However, in 14 days old TK2−/− mice the β-oxidation was reduced in the liver and the observed ketone and glucose levels were dramatically decreased. Since the brain cannot utilize fatty acids directly it is dependent on the liver to produce ketone bodies through fatty acid β-oxidation. These mechanisms are of vital importance for normal brain function when the organism is on a carbohydrate restricted diet. The brain has high ATP requirement and low regenerative capacity and is highly sensitive to mtDNA depletion. We have previously shown that lack of TK2 activity causes a decline in mtDNA levels in neurons in vivo−/− mouse model. Besides the major reduction of mtDNA in muscle, liver and brain they also show an early disease onset, accumulation of fat in the liver in form of macrovesicular steatosis, and seizures as a late manifestation similar to the mice −/− mouse completely lacks TK2 activity and displays more severe neurological features compared to patients with TK2 mutations −/− mice resembles symptoms seen in patients with progressive liver failure due to DGUOK deficiency that commonly present with hypoketotic hypoglycemia in combination with lactic acidosis and mild neurological involvement In 12 days old TK2−/− mice was lower already at birth, and was somewhat further decreased at 14 days of age. However, the TK2−/− mice were born with normal levels of mtDNA in the liver and developed severe mtDNA depletion by postnatal day 14 . The severe mtDNA depletion coincided with severely reduced mitochondrial β-oxidation in the liver most likely due to loss of CPT1A activity. We propose that the gradually increasing failure of the liver to produce ketone bodies and glucose for the brain diminishes the remaining brain function and leads to loss of consciousness and ultimately death of the TK2−/− mice. A better understanding of the primary and secondary factors that influence TK2 progression, and liver damage due to mtDNA depletion, may contribute to better prognosis and adequate treatments. In the absence of a cure for MDS the achieved results provide additional insight in the pathogenesis that can be used to explore novel therapeutic strategies.The mtDNA content in the brain of the TK2