The “Z-shaped” flexure hinges and the parallel-six-connecting-rods structure are utilized to obtain the lowest working stress while compared with other types of flexure hinges. In order to achieve the optimized structure, matrix-based compliance modeling (MCM) method and finite element method (FEM) are used to evaluate both the static and dynamic performances of the proposed 2-DOF piezoelectric-driven platform. Experimental results indicate that the maximum motion displacements for x-stage and y-stage are lx = 17.65 μm and ly = 15.45 μm, respectively. The step response time for x-stage and y-stage are tx = 1.7 ms and ty = 1.6 ms, respectively. A compact 2-DOF (two degrees of freedom) piezoelectric-driven platform for 3D cellular bio-assembly systems has been proposed based on “Z-shaped” flexure hinges. Multiple linear motions with high resolution both in  The 3D cellular bio-assembly system has become a hot research topic during recent years. In order to manipulate cells and biological tissues in 3D space, micro/nano-positioning systems with high resolution and multi-DOF (multiple degrees of freedom) should be utilized in 3D cellular bio-assembly systems . PI, Attocube and SMartAct, have already developed the commercial parallel-type piezoelectric-driven multi-DOF platforms [Piezoelectric-driven multi-DOF platforms are widely utilized in 3D cellular bio-assembly systems for their own advantages: high speed, simple control, high resolution, and multiple degrees of freedom ,3. Up tolatforms ,9,10, thlatforms ,12,13, ilatforms ,15,16,17latforms ,19,20. Alatforms ,21,22,23latforms ,25,26. NIn 3D cellular bio-assembly systems, the size of tissue cells is almost around 10 μm, which means that the positioning platform with submicron accuracy and multiple degrees of freedom is enough for the cell manipulation. The working frequency needs not to be too high. Additionally, high reliability is also required, since the 3D cellular bio-assembly process always takes a long time for thousands of cells or more. Hence, the nested form serial-type piezoelectric-driven platform is quite suitable for 3D cellular bio-assembly systems to achieve the submicron accuracy, simple structure and control system, and low cost. In order to achieve high reliability and long service lifetime, the used flexure hinges for piezoelectric platforms should have low working stress. However, the most utilized right-angle and right-circle flexure hinges always have high working stress, which greatly influences the working lifetime and the reliability. Therefore, it is of great significance to develop the piezoelectric-driven platforms with lower stress on the flexure hinges.In this study, in order to reduce the working stress of flexure hinges and obtain multiple degrees of freedom motions for 3D cellular bio-assembly systems with high reliability, a compact serial-type piezoelectric-driven platform based on “Z-shaped’’ flexure hinges is proposed and manufactured. The “Z-shaped’’ flexure hinge is compared with right-angle and right-circle flexure hinges by matrix-based compliance modeling (MCM) method and finite element method (FEM) to confirm it has the lowest working stress under the same condition. The parallel-six-connecting-rods structure is optimized by FEM method to enhance platform stiffness and make the platform stable. The experimental system is established to test the performance of the proposed two degrees of freedom (2-DOF) piezoelectric-driven platform. Additionally, the open-loop control method with respect to the coupling error is developed to achieve high positioning resolution.Flexure hinges are widely exploited in piezoelectric-driven platforms, due to their unique superiorities: high resolution, compact size, no friction and simple structure. Several kinds of flexure hinges have been developed for different applications. Among them, the right-angle flexure hinge and right-circle flexure hinge are the most used ones. However, during the application of right-angle and right-circle flexure hinges, the working stress is always high, which greatly influences the working lifetime of the whole piezoelectric-driven platform. Compared with other kinds of flexure hinges, the “Z-shaped” flexure hinge achieves larger output displacement with smallest working stress, which is ideally qualified for the piezoelectric-driven platform with long working lifetime.OaOa1; the structure is L = 16 mm, w = 6 mm and t = 1 mm. OrOr1 (basic right-angle), part Or1Or2 (basic right-circle) and part Or2Or3 (basic right-angle); the structure is l1 = l2 = 5 mm, R = 3 mm. The “Z-shaped” flexure hinge is shown in OzOz1, part Oz1Oz2 and part Oz2Oz3; the size is l3 = l5 = 8 mm, l4 = 6 mm. According to the references [Cra for the right-angle flexure hinge is gotten by the following:E is the elastic module of the applied material; G is the shear module; L, w, t are the structural parameters of the basic right-angle flexure hinge.In order to compare the working stress of right-angle, right-circle and “Z-shaped” flexure hinges under the same condition, matrix-based compliance modeling (MCM) method is applied for the theoretical calculation. ferences ,28, the Crc is shown in the following [For the basic right-circle flexure hinge, the compliance matrix ollowing :(2)Crc=[g = R/t:ux, uy and uz are the linear deformation along the x, y and z directions; θx, θy and θz are the rotational angles around the x, y and z directions; Fx, Fy, Fz, Mx, My and Mz are the applied forces and torques in the x, y and z directions.Define Since the right-angle flexure hinge shown in Ori with respect to point Or, which are achieved by:Rori(αori) and Sori are the rotation matrix and position matrix of point Ori with respect to point Or, respectively; αori is the rotation angle of point Ori relative to point Or; rxri, ryri and rzri are the relative positions between point Ori and point Or. The right-circle flexure hinge is divided into three parts b, and itOzi with respect to point Oz; Rozi(αozi) and Sozi are the rotation matrix and position matrix of point Ozi to point Oz, respectively. The developed “Z-shaped” consists of three basic right-angle flexure hinges c, and thTherefore, the total compliance matrixes of the right-angle, right-circle and “Z-shaped” flexure hinge are achieved by the Equations (11), (12) and (16), respectively.E = 206 GPa, and the Poisson ratio is μ = 0.288 during the FEM simulation. The structural parameters are the same with those shown in F = 20 N along y direction is applied at the right end surface of each flexure hinge. The FEM mesh size is 0.1 mm for these three flexure hinges.In order to confirm the theoretical calculation by MCM method, FEM has been applied to simulate the deformation of these three flexure hinges. Three kinds of flexure hinge  have been calculated by FEM, and the material of them is steel. The elastic module is set as y direction is Dz = 323.5 μm by the MCM method, while that from the FEM method is Dz’ = 340.1 μm; the error between the MCM and FEM methods is ez = 4.9% based on Equation (20), which confirms the application feasibility of the MCM method. The maximum deformation for the right-angle flexure hinge is Dra = 265.8 μm with an error of era = 1.7%. The error of the right-circle flexure hinge is erc = 2.2%, considering the influence of the torque induced by input force F in Equation (10). The comparison of FEM and MCM methods under other forces can be obtained in Dy along y direction is applied on the right end surface of each flexure hinge. The left end of each flexure hinge is fixed to the ground without freedom. The FEM mesh size is 0.1 mm for these three flexure hinges. Since the working stroke of the utilized piezo stack is around 10 μm while the driving voltage is 100 V, the applied displacement is chosen as Dy = 10 μm. It is shown that the maximum working stress of these three flexure hinges are σra = 10.9 MPa, σrc = 46.5 MPa and σz = 8.8 MPa. The working stress of the “Z-shaped” flexure hinge is the smallest among them, which is of great significance to extend the working lifetime of piezoelectric-driven platforms. Hence, the “Z-shaped” flexure hinge is utilized for the design of the proposed 2-DOF nano-positioning platform.The deformation results both from the MCM and FEM methods are shown in x-stage and y-stage instead of the generally used parallel-four-connecting-rods, and it is helpful to enhance the output stiffness of the whole platform. Additionally, it makes the platform more stable and the coupling error smaller, since the stiffness of x- and y-stages is enhanced to reduce coupling deformation of reaction forces. FEM method is also used to calculate the performance of the proposed 2-DOF piezoelectric-driven platform. During the FEM calculation, the four holes on the base corners are fixed to the ground without motion degrees of freedom; an input force F = 50 N was applied on x-stage in the surface to nest the piezoelectric stack; meanwhile, the reaction force Fr = 50 N was also applied on the surface to nest the piezoelectric stack on y-stage. y-stage are “Z-shaped” flexure hinges, the deformation of area A in y-stage is about 3 μm, as shown in x direction is nested inside y-stage, the reaction force makes the y-stage generate this unwanted deformation, which should be avoided. Meanwhile, in the case that two right-angle flexure hinges are utilized to replace “Z-shaped” flexure hinges for y-stage, the unwanted deformation of area A is rather small since the stiffness of right-angle flexure hinge along x direction is higher than the “Z-shaped” flexure hinge. Additionally, the output force of the used piezoelectric stacks is around Fstack = 800 N, which is large enough to reduce the influence of the increased stress for y-stage motion stroke. Therefore, both “Z-shaped” flexure hinges and right-angle flexure hinges are utilized in the 2-DOF piezoelectric-driven platform. Based on “Z-shaped” flexure hinges, a 2-DOF piezoelectric-driven platform has been proposed and manufactured. The parallel-six-connecting-rods structure is applied both in x-piezo stack and y-piezo stack), x-stage, y-stage, screws, “Z-shaped” flexure hinges, right-angle flexure hinges and the base. The x-piezo stack is used to drive x-stage; y-piezo stack is applied for y-stage. These two piezo stacks  are from NEC COMPANY; the maximum output force of them is around Fstack = 800 N, and the motion stroke is about Lstack = 17 ± 2 μm in the case of input voltage V = 150 V; they are working on the d33 model and the value is d33 = 720 × 10−12 m/V. The x-stage is nested inside of y-stage, which means that the proposed nano-positioning platform is in serial type. This serial structure can make the proposed platform in a compact size and with small coupling errors. Two kinds of flexure hinges (“Z-shaped” and right-angle) are exploited to form a parallel-six-connecting-rods structure for y-stage; the flexure hinges in x-stage are all “Z-shaped” flexure hinges. The material of the proposed 2-DOF piezoelectric-driven platform is structural steel. Wire electrical discharge machining (WEDM) was utilized to manufacture the platform, so as to make the whole platform in an integrated structure. The size of the proposed 2-DOF piezoelectric-driven platform is l1 × l2 × l3 = 130 mm × 150 mm × 8mm.Ek and potential energy Et keeps the same during the motion. In the case that force Fa is applied on y-stage, the platform moves a distance s. Kinetic energy Ek of the proposed platform is divided into two parts: kinetic energy Ek1 induced by the linear motion, kinetic energy Ek2 induced by the rotation motion of flexure hinges. Kinetic energy Ek is achieved by the following equation [m1 is the mass of the “Z-shaped” flexure hinge; m2 is the mass of the right-angle flexure hinge; m3 is the mass of y-stage; m4 is the mass of x-stage; s is the motion displacement of y-stage; θ = arctan (s/L) ≈ s/L is the rotation angular displacement of flexure hinge; Jz is the inertia moment of the “Z-shaped” flexure hinge; Jr is the inertia moment of the right-angle flexure hinge. equation :(21)EK1=Et is mainly induced by the six flexure hinges for y-stage, and it is obtained by the following equation:kθz is the rotational stiffness of the “Z-shaped” flexure hinge; kθr is the rotational stiffness of the right-angle flexure hinge. Potential energy E0 keeps the same and it can be achieved by the following equation:Therefore, the total energy f1 = 678 Hz, f2 = 965 Hz, f3 = 1297 Hz, f4 = 1553 Hz, f5 = 1832 Hz and f6 = 2002 Hz. It can be seen that the undesired transverse motion (z and θx) is introduced in high frequency since the use of “Z-shaped” flexure hinges, which is not good for the 2-DOF nano-positioning platform. Generally, the proposed 2-DOF nano-positioning platform works under several dozens of Hertz; hence, the resonance phenomenon could be avoided. We are still working on this platform, and hope to overcome this disadvantage in future work.However, it is difficult to get the accurate solution of Equation (25). Hence, FEM method is applied to analyze the dynamic performance of the proposed 2-DOF piezoelectric-driven platform. FEM dynamic analysis results are obtained in x, y) and one industrial personal computer (IPC). The signal producer  generates the needed voltage signal for the piezo-stacks, and the voltage signal is amplified by the voltage amplifier  to a high value to drive the piezo-stacks in x- and y-stages. The maximum output voltage of this HPV is UHPV = ±150 V with a voltage resolution of eHPV = 1%; the maximum output current is IHPV = 0.5 A. The laser reflectors are attached to the x- and y- stages to reflect the laser signal. Laser sensor LK-G10 (KEYENCE COMPANY) is utilized to generate the laser signal to record the motion of the laser reflector. The laser head is placed in line with the motion direction of x-/y- stage which has the attached laser reflector. The measurement range of LK-G10 is LLK = ±1 mm with a resolution of eLK = 10 nm which is high enough for the proposed piezoelectric-driven platform. All data is processed and changed by the AD/DA card which is integrated inside the laser sensor LK-G10, and then saved by the IPC. The commercial software from KEYENCE COMPANY is utilized to control the laser sensor and save the data with a sampling time of tsa = 0.1 ms. The software from BOSHI COMPANY is exploited to control the signal producer. All this equipment is placed on the air-floating vibration isolation platform to isolate the vibration out of the experimental system.x, y and z directions (80 mm × 80 mm × 120 mm). However, the 3-DOF motor-driven platform is a traditional one, not the originality of our paper, and the resolution is not high enough for cell manipulation; hence, it is not mentioned in our paper. Here, the shown experiments are for the proposed 2-DOF piezoelectric-driven platform.In the real application, the proposed piezoelectric-driven platform is fixed to a motor-driven 3-DOF platform to obtain enough motion strokes. This motor-driven platform is composed of three motors, three ball screws and linear guiders to obtain large working stroke in x direction(a) Motion performance in x-stage under different input voltage Vx is illustrated in Vx goes up from Vx = 0 V to Vx = 150 V, and then falls down to Vx = 0 V. The motion displacement lx of x-stage increases to lx = 17.65 μm in the case of input voltage Vx = 150 V. This experiment is repeated three times, and the repeatability is found excellent. The maximum displacement error between the loading (up) and unloading (down) curves is ex = 0.36 μm, which is thought to be induced by the hysteresis and system errors. Compared with the maximum motion displacement, the maximum relative displacement error is exr = 2.03%. Many researchers have paid attention to reduce the hysteresis of piezoelectric actuators by closed-loop control systems and compensation algorithms. However, these methods make the whole system complicated and very costly. The requirement of the resolution in 3D cellular bio-assembly systems is not so high, since the size of cells is almost around 10 μm. Therefore, the influence of hysteresis is not discussed deeply here.The motion performance of y direction(b) Motion performance in y-stage under different input voltage Vy. The input voltage Vy also increases from Vy = 0 V to Vy = 150 V, and then decreases to Vy = 0 V. The maximum motion displacement of y-stage is ly = 15.45 μm in the case that the input voltage Vy = 150 V. This experiment is also repeated three times, and the maximum displacement error between the loading (up) and unloading (down) curves is ey = 0.56 μm, which is also thought to be induced by the hysteresis and system errors. Hence, the maximum relative displacement error is about eyr = 3.62%, which is a little larger than that in x-stage. The difference between the repeated three groups is thought to be caused by the assembly errors of the piezoelectric stacks and the system errors. (c) Displacement coupling errorecy and ecx are achieved from Vx is applied to the x piezo stack, the laser sensor is utilized to measure the displacement of y-stage in y direction. Hence, the displacement coupling error ecy is shown in ecy increases with input voltage Vx. The maximum displacement coupling error for y-stage is ecy = 0.76 μm, in the case that the input voltage Vx = 138 V. The relationship between the input voltage Vx and the displacement coupling error ecy is simplified by the following equation with a correlation coefficient of R2 = 0.95: The displacement coupling errors ecx for x-stage is obtained in the case that input voltage Vy is applied for y-piezo stack  Output forcex-stage during the measurement. The motion displacement ly of y-stage keeps almost the same, in the case that stand weight increases from m = 0 g to m = 1000 g, which indicates that the proposed 2-DOF piezoelectric-driven platform works stably with high output force.The performance of the proposed 2-DOF piezoelectric-driven platform under different output force has been measured, as shown in (e) Step responsex-piezo stack and y-piezo stack. The sampling time of the laser sensor is about 0.1 ms. The step response time for x-stage is tx = 1.7 ms; the stable displacement is about lx = 12.84 μm; therefore, the response velocity is vx = 7.84 × 10−3 m/s. The step response time for y-stage is around ty = 1.6 ms; the stable displacement is about ly = 10 μm; therefore, the response velocity is vy = 6.25 × 10−3 m/s. x and y-stages. Generally, if the closed-loop control method is utilized, it is easy to avoid these coupling errors ecx and ecy. However, displacement sensors are needed for the closed-loop control system, which makes the whole system complicated and expensive. In order to get high positioning accuracy with open-loop control system, we have to analyze the motion displacement and the coupling error. For example, the real motion displacement Ly of y-stage contains two parts: one is the displacement ly caused by input voltage Vy, the other part is the coupling error ecy induced by input voltage Vx. Therefore, the real motion displacement Ly is achieved by the following:ly and Vy is obtained from Since the designed platform is a 2-DOF platform, this means that there are coupling errors in both Ly: firstly, the input voltage Vx should be known to calculate the error ecy; next, wanted ly is got by using wanted Ly subtracts the error ecy; finally, the need Vy is obtained by Equation (28). Vy under different input voltage Vx. It is illustrated that, in order to get high accuracy, different Vy should be given under different Vx. The largest voltage error of the applied Vy is evy = 5.39 V, in the case that Vx = 0 V and 100 V. By applying the needed Vy from the Equation (19), the wanted real motion displacement Ly is obtained by this open-loop method.In this open-loop control method, three steps are needed to achieve the real motion displacement x-stage is lx = 17.65 μm in the case of input voltage Vx = 150 V; the maximum motion displacement of y-stage is ly = 15.45 μm in the case of input voltage Vy = 150 V. The step response time for x-stage is about tx = 1.7 ms, while that for y-stage is ty = 1.6 ms. All the experimental data indicate that “Z-shaped” flexure hinges are quite suitable for the design of piezoelectric-driven platforms to obtain submicron accuracy and microsecond response time. This study is meaningful for the application of piezoelectric-driven platform in 3D cellular bio-assembly system. A compact 2-DOF piezoelectric-driven platform based on “Z-shaped” flexure hinges has been proposed; the experimental system was established to test its performance. Different types of flexure hinges are calculated and compared; and “Z-shaped” flexure hinge was confirmed to have the smallest working stress under the same deformation. MCM and FEM methods were utilized for the optimization of the proposed 2-DOF piezoelectric-driven platform. The maximum motion displacement for


Nature Communications 10.1038/s41467-018-05874-2, published online 17 August 2018.Correction to: The original version of this Article contained an error in Fig. 1g in which a top x-axis was shown with incorrect labels for the wavelength. The correct version of Fig. 1g, which removes this labelling, is:which replaces the previous incorrect version:This has been corrected in both the PDF and HTML versions of the Article.


Increasing evidence shows that real networks interact with each other, forming a network of networks (NONs). Synchronization, a ubiquitous process in natural and engineering systems, has fascinatingly gained rising attentions in the context of NONs. Despite efforts to study the synchronization of NONs, it is still a challenge to understand how do the network sizes affect the synchronization and its phase diagram of NONs coupled with nonlinear dynamics. Here, we model such NONs as star-like motifs to analytically derive the critical values of both the internal and the external coupling strengths, at which a phase transition from synchronization to incoherence occurs. Our results show that the critical values strongly depend on the network sizes. Reducing the difference between network sizes will enhance the synchronization of the whole system, which indicates the irrationality of previous studies that assume the network sizes to be the same. The optimal connection strategy also changes as the network sizes change, a discovery contradicting to the previous conclusion that connecting the high-degree nodes of each network is always the most effective strategy to achieve synchronization unchangeably. This finding emphasizes the crucial role of network sizes which has been neglected in the previous studies and could contribute to the design of a global synchronized system. For these real-world systems, one component is often an individual network within a much larger complex multilayer network10, forming a part of a network of networks (NON). Comparing to a single network, the connection between networks brings more complex and complicated properties of NONs, such as robustness13, epidemic eruption14, and the self-organization in evolutionary game15. Among these properties, synchronization16 is an important behavior in biological17, engineering18, climatological19, and ecological systems20. The synchronization of a single network has been intensively investigated in the past few decades. However, the synchronization of NONs, which is more universal in real world, has not gained enough understanding21 yet. There has been a limited number of studies on the synchronization of NONs. Martin et al.22 showed the existence of a critical number of diagonal interlinks, beyond which synchronization could not be enhanced, while Francesco23 introduced a unique Master Stability Function that determines the stability of synchronization. The study on the synchronization of NONs has certain practical significance. For instance, the smart-grid, which can be regarded as NONs24, is a combination of microgrids and the traditional power networks. More and more microgrids are integrated into the power system, consequently decreasing the couplings of the power system and thus driving this NON into the boundary of its synchronization18.The present high attention to network science roots in its capability of explaining the behavior of complex real systems, including Interneti.e. the linkage between different networks; the dynamics, i.e. the universally nonlinear behavior in the real world; and the structure, i.e. the connections between a certain amount of nodes. While research on these three factors have led to some important findings, some scientific gaps remain. Some studies found that the optimal connection strategy for the case of a single connection is always to link the high-degree nodes from different networks26. No model, to our best knowledge, are available to systematically study and theoretically explain the optimal connection strategies for more than one connection. Previously, the most commonly applied predictive tools to study the dynamics of NONs, such as Master Stability Function27 and the eigenvalues of the adjacency matrix28, are based on the linearization . We present a mechanical analog of this model in Fig. λ. The nodes are synchronized if they rotate around a circle with angular velocity ω and avoid collisions. In the single network Kuramoto model, the system undergoes a first order38 or a hysteresis transition39 when the coupling strength crosses a critical value. Comparing to a single network, NONs behaves much more complex since it has two kinds of coupling strengths: the internal coupling strength λ and the external coupling strength β , the hub connects to the leaf (H-L) and the leaf connects to the leaf (L-L). Here, we not only consider the connection strategy with one link but also with two links (see Method Connection Strategy).One of the most widely studied models of synchronization is the Kuramoto model37 is r(t) is the order parameter at time step t, N is the network size, θ(t) is the phase at time t, and i denotes an imaginary number. The order parameters for network I and network II are set as rI(t) and rII(t). The upper indices (I) and (II) stand for ‘Network I’ and ‘Network II’ respectively. Furthermore, we define the order parameter for interconnected networks R(t) as:H-H strategy. Then numerically mimic the order parameter r(t) by solving the ordinary differential equations of the coupled Kuramoto model (See Method Coupled Kuramoto model) for the given internal and external coupling strengths, where β, i.e. βs in the legend) both order parameters rI(t) and R(t) show a periodic function as time t and we denote the period as ΔT (Case I), indicating that the interconnected networks are not globally synchronized. When i.e. βl in the legend), however, both rI(t) and R(t) converge to their steady states (Case II), demonstrating that the interconnected networks are in the globally synchronized state, showing in Fig. λ, determines the synchronization of interconnected networks, we decrease λ from 3 to 0 gradually with a constant interval λ at the final state asr is similar to equation . Meanwhile, for the large value βl, there exists the critical value of λ that separates the synchronized phase and the unsynchronized phase, indicating the existence of a minimum value of β, below which the system can never synchronize , only when Next, we derive the analytical solutions for the case of two-links connection. The analytical critical internal coupling strength can be mathematically described asβc but also has the upper limit βmax.It means that in this case the system not only has the lower limit H-L equals to L-H. Figure NI, NII) presents the critical coupling strength with certain networks sizes NI and NII. Figure H-H, H-H L-L, and H-L L-H. This λc increases with 43 and is analogous to the discovery in etiological systems44. Figure λc for H-L, L-L, L-L L-L and H-L L-L respectively. We observe that Fig. H-L) is the combination of Fig. H-H) and Fig. L-L), along the diagonal line H-L L-L, i.e., L-L L-L. Such property depends on whether the hub with larger network size acts as the connector  and and\docunge (Eqs ). If \doT), while the y-label denotes the simulation results (using superscript S). When the derived results are as the same as the simulation results, the corresponding dots are on the diagonal line H-L L-L etc. In Fig. λc, βc and βmax respectively. The comparisons show that our theory aligns with the simulation.The derived critical coupling strengths are validated in Fig. The above analysis points out the significant role of network sizes for the synchronization of NONs. In real systems, the network sizes are not constant and may change over time. Thus, it’s important to develop an approach to keep the system synchronized in a fast and efficient way, facing the time-variable network sizes.45. The “maximum allowable network sizes” refers to the maximum network sizes that the system can tolerate. If the network sizes are larger than this value, the system fails to remain synchronized. For H-H, the maximum allowable network sizes can be obtained from equation  is much easier to be synchronized than that with H-L L-L (yellow zone), although the system with H-L L-L has a high-degree node as the connector. Besides, all the connection strategies except H-L L-L share similar operating space when i.e. near the diagonal line in Fig. H-H L-L connects the high-degree nodes while H-L L-H connects the high-degree node to the low-degree node. However, the former  and the external coupling strength (β) on operating space; the former one sizes the operating space while the latter, shapes. This finding helps to decide which coupling strength should be utilized and how much it should be enhanced, guiding the time-demanding control to maintain synchronization.Facing the fast-changed network sizes, our results contribute to the time-demanding control. With certain coupling strengths  and SF networks (SASF), the ratios of different connection strategies are almost along a universal line , λ. The other is the coupling strength between networks , β.The coupled Kuramoto model for Network I and Network II isλ drops gradually and everytime λ changes, the initial phases are the end states of the last λ. In all these plots, the natural frequency of all oscillators were 1 and β remained unchanged. λ was a fixed value in the simulation shown in Figs λ was kept unchanged for 10000 iterations and passed to another value for another 100000 iterations with The coupled Kuramoto model was applied to the simulations shown in Figs 46. The initial network had one node. A new node was added to a existed node i with the probability k is the degree of one node and the mean degree is 1. η is the fitness randomly chosen between Scale-Free network was generated using BB modelH-H , H-L (short for hub-leaf), L-L, H-H L-L, H-L L-H, L-L L-L, and H-L L-L. Note that our results can be extended to multiple couplings between two networks. Connection Strategy indicates the adjacency matrix between networks M. When the nodes i and j act as the connector linking two networks, then Connection strategy refers to which node acts as the connector that linking different networks. All the connections between two networks discussed in this study are λ were randomly chosen from λ for all nodes started from 1.20 that the system could be synchronized, then λ for all nodes deceased gradually until 0.98 to check whether the system could be synchronized or not.The derived results were validated on the heterogeneous networks, as shown in Fig. Supplementary Information


Firstly, two wave-breaking criteria are derived by employing the transport equation theory and the localization analysis method. Secondly, a sufficient condition of the blow-up solutions is established by using the classic method. The results obtained in this paper are new and different from those in previous works.The two-component  Moreover,(III)X into X, is bounded on bounded sets in Y, and satisfiesWe make the following assumptions, where Assume that conditions (I), (II), and (III) hold. Giventhere is a maximaldepending only onand a unique solutionvto system . Then  and 40)\document). Using , we getequality , we deduSolving , we obtaμ-version Camassa–Holm equation under μ-version Camassa–Holm equation, the reader is referred to [If we let mentclass2pt{minimμ-version Camassa–Holm equation essentially. So the blow-up phenomena in [μ-version Camassa–Holm equation.It is worthwhile to mention that comparing with the results in , our blomentclass2pt{minim


Correction to: npj Digital Medicine 10.1038/s41746-018-0066-9, Article Published online 16 Oct 2018The original version of the published Article contained an error in the spelling of the third Author’s name. “John Halamaka” has been changed to “John Halamka”. This has been corrected in the HTML and PDF version of the Article.


O-glycosylation has been made. Thus, predicting the likelihood of O-glycosylation with sequence and structural information using classical regression analysis is quite difficult. In particular, if a binary response is used to distinguish between O-glycosylated and non-O-glycosylated sequences, an appropriate set of non-O-glycosylatable sequences is hard to find.To-date, no claim regarding finding a consensus sequon for S/T-site are analyzed: N-glycosylation, O-mucin type  glycosylation, and phosphorylation. Results found include: 1) The consensus composite sequon for O-glycosylation is: ~(W–S/T–W), where “~” denotes the “not” operator. 2) The consensus sequon for phosphorylation is ~(W–S/T/Y/H–W); although W–S/T/Y/H–W is not an absolute inhibitor of phosphorylation. 3) For linear probability model (LPM) estimation, N-glycosylated sequences are good approximations to non-O-glycosylatable sequences; although N – ~P – S/T is not an absolute inhibitor of O-glycosylation. 4) The selective positioning of an amino acid along the sequence, differentiates the PTMs of proteins. 5) Some N-glycosylated sequences are also phosphorylated at the S/T-site in the N – ~P – S/T sequon. 6) ASA values for N-glycosylated sequences are stochastically larger than those for O-GlcNAc glycosylated sequences. 7) Structural attributes  are significant LPM predictors of O-GlcNAc glycosylation. The LPM with sequence and structural data as explanatory variables yields a Kolmogorov-Smirnov (KS) statistic of 99%. 8) With only sequence data, the KS statistic erodes to 80%, and 21% of out-of-sample O-GlcNAc glycosylated sequences are mispredicted as not being glycosylated. The 95% confidence interval around this mispredictions rate is 16% to 26%.Three sequences from similar post-translational modifications (PTMs) of proteins occurring at, or very near, the O-glycosylation; and underscores the germaneness of structural information for predicting the likelihood of O-glycosylation.The data indicates the existence of a consensus sequon for The online version of this article (10.1186/s12860-019-0200-9) contains supplementary material, which is available to authorized users. N-linked and O-linked )–81)76–81ldberger  suggesteficients . And theficients . Yes, “p0≤ Xb ≤1  = Xb and ||v||p denotes the “p-norm” of a vector v. In contrast, RR uses the constraint (||b||2)2≤ t. That is, the sum of squared coefficients, instead of absolute coefficients, is constrained. In Lagrangian form [The LPLM is estimated by minimizing the negative log-likelihood function, ian form , 96, the–L(·) + λ × sum(|b|). This minimization is done using Nesterov’s method [ρi, where 0 < ρ < 1 and i is the ith step in Lasso selection, is used for λ in the ith step. Variable subset selection, via the Lasso, is done, using up to 200 steps, so as to minimize the Schwarz Bayesian Criterion (SBC) [ρ and the resultant number of variables (excluding the intercept) selected, the KS statistic values, the Brier Scores, and twice the negative log-likelihood values are shown in Table ρ = 0.815; and this is also nearly “optimal” in terms of Brier Score and KS statistic value. To assess LPLM performance out-of-sample as ρ varies, 5-fold cross validation (CV) is done. LPLMs are fit on a set of 4 “training” folds and the Brier Score is calculated on the holdout  fold. Thus, for a particular choice of ρ, five out-of-sample Brier Scores are available. The signal-to-noise ratio (SNR), which is μ/σ, of these 5 Brier Scores is calculated. Per the Rose criterion [ρ of 0.82, the SNR is a maximum and satisfies the Rose criterion. Thus, overall, it is reasonable to take ρ = 0.82 as “optimal”.s method . Here, aon (SBC) . In-sampriterion , SNRs abO-glycosylation in Nature, would they, post-Lasso, continue to be statistically significant if they were put into a classical logit model to reestimate Y ?The resultant LPLM is shown in Table Y with the independent variables in Table Y separates a subset of the predictors to a large enough degree. To resolve this problem, we used Firth’s penalized maximum likelihood method [Y is reestimated using the LPM with the predictors in Table t-values, at the 10% level of significance are noted in Table Using the classical logit model, we reestimated d method , 101. SeY. However, a philosophical question arises. Suppose, Nature operates along a logit surface to reveal the probability of O-glycosylation, given sequence and structural information in the neighborhood of the S/T site; and the Lasso has identified the relevant predictors. It appears to us that the selected predictors should, for the most part, be statistically significant, in the classical sense, when they are put into a classical logit model. The fact that 23 of the 51 LPLM predictors are not significant is troubling.Of course, given a set of predictors, the estimated coefficients produced by the classical logit model, the LPM, and the LPLM are not comparable, because different optimization procedures are used to estimate Y as 1 when it is empirically 0  in the estimation dataset, and 0 when it is empirically 1 , is shown in Table The rates at which the estimated models mispredict As can be seen in Table psi angle. It is not a predictor in Table phi and psi angles go together, an important question is whether the LPM in Table t-values [p-values less than 10% – i.e., variable subset selection is done, manually, using White t-values instead of classical t-values. The selected model is in Table Consider the case of the t-values , are useR2 of about 76%. However, its standardized (studentized) residuals are not normally distributed as indicated by the Q-Q plot in Fig. psi is a “circular” measure and has heavy tails as can be seen by its tendency to cluster (with phi) in certain (and remain sparse in other) regions of the X-Y plane, which can be seen in the Ramachandran plot in Fig. psi. However, as fitting a model to psi is not the focus of this paper, we did not pursue that goal at this time (our focus is on answering the question raised in the last paragraph). Nonetheless, the t-values can be viewed as approximations  for model selection by viewing linear regression, in this case, as purely an optimization method. In particular, we held the 99 variables in Table t-values), the new CVPRESS tends to exceed the minimum CV PRESS of about 4,253,609 – the reduction of 106 variables to 99 variables is fairly small and likely explains this movement. This variability, with its associated summary statistics, is shown in Fig. The model in Table ρ varies in Table ρ takes 13 values in Table ρ, an optimal LPLM is selected. In all, there are 13 optimal LPLMs. All possible variables are shown in a rectangular format in Table S/T in Table G; and this cell would represent the dummy variable p3G – that is, the dummy variable representing amino acid G in position +3. The variables (predictors) in the LPM of Table m3G as a predictor. Similarly, the 13 in cell  means that p8N was selected as a predictor by all of the 13 LPLMs; and, additionally, because this cell is shaded green, p8N is also a predictor in the selected LPM of Table BH (Beta Hairpin) was only selected in 3 of the 13 LPLMs, but is in the LPM of Table ρ varies, is captured in Table The number of times predictors are selected by the LPLM as The KS statistic is an alternative measure to the well-known Gini coefficient –107. TheThe LPM in Table ASA measures the extent to which a sequence is exposed. As mentioned previously, there are two sequences in our data that have ASA values of 0, but are deemed to be O-glycosylated. In our model, ASA_zero is a statistically significant variable. Because of its special status as an “observation specific” dummy variable, it is not estimable within the classical logit modeling framework [O-glycosylated. Although, we do not know the answer to that, we think that until empirical evidence to the contrary comes up, it does not hurt keeping ASA_zero in the LPM. Furthermore, the good news is that dropping it from the model does not change any of our major conclusions. That is, in our case, dropping ASA_zero as an explanatory variable, simply allowed its effect to be distributed among the other coefficients without disrupting anything. However, we did not drop it, because it is of interest to consider (until evidence to the contrary) its marginal contribution toward encouraging O-glycosylation.ramework . Its staASA_zero in the LPM, but not ASA, raises another important question: is ASA_zero a spurious dummy variable in the LPM, because the fundamental variable underlying it, ASA, is not in the LPM? The answer is “no”. And we discuss the reason for this next.The presence of O-glycosylation. This means a significant amount of the information carried by ASA should be embedded in the amino acids and the positions they occupy in the sequences. To test this assumption, we searched for a linear regression model with the natural logarithm (log) of ASA as the dependent variable and sequence information as independent variables. Our search followed the same principles that were used to model psi. We used the log transformation to get ASA “closer” to normality, and, as a consequence, the two observations with ASA values of 0 also get eliminated from the estimation dataset (because log(0) is undefined). ASA, like psi, has “heavy” tails. The selected model is shown in Table R2 of about 80%. Its in-sample average squared error (ASE) is about 0.0425. The studentized residual distribution in Fig. ASA values of 0  can discriminate between O- and non-O- glycosylated sequences . The KS statistic would be a measure of this. In the estimation sample, if Y is ranked by estimated log(ASA), the resultant KS statistic is about 32.0%, which is a strong enough value to not reject the assumption that the distribution of amino acids along the sequence does meaningfully drive ASA. Like in the case of psi, we do 5-fold CV (holding the 103 predictors “constant”), 20 times, to study the variability in the resultant 100 realizations of the out of sample ASEs and KS statistics. These variabilities, which are not unreasonable relative to the baseline ASE and KS statistic, are shown in Fig. The information content in N-glycosylation occurs in the rough ER and the process ends in the Golgi apparatus; phosphorylation occurs in the cytoplasm/nucleus; and O-GlcNAc proteins are also cytosolic/nuclear. So, the question is whether our “control” dataset  is the correct “negative control”? The answer is “yes”. By virtue of their differences in their mechanisms and cellular compartmentalization, it makes logical sense to use N-glycosylated sequences as the negative control for O-glycosylation. For example, in the absence of a consensus sequon for O-glycosylated sequences, using sequences with the S/T motif within the O-glycosylated set of sequences makes little sense, because there is no guarantee that such sequences will not get O-glycosylated.The PTMs occur in different parts of the cell by a very complex process involving many different enzymes. For example, the initial processing of the The LPM models a binary outcome. For example, when tossing a “fair” coin, if one wants to model  whether the coin will come down as “head”, then one needs to start with two distinct data subsets: a set of heads and a set of tails . One canO-glycosylated. Now the way we do this, is by starting with “partial” or “incomplete” information. That is, we start with explanatory variables in the ±10 positions around the S/T site. The distribution of amino acids in this neighborhood is considered. And the structural information at S/T is considered. For example, we have not considered the distribution of amino acids further away from S/T, which must also carry key information. So, for example, the LPM is “unaware” of the existence of the signal peptide in N-glycosylated proteins that sends it to a specific part of the cell.Similarly, our use of the complex concept called “probability” , in thisS/T site is sufficient for predicting the likelihood of O-glycosylation. To validate this assumption, we started with Table O-glycosylated proteins having the N – ~ P – S/T sequon is tiny . Given this empirical observation, we assumed that the “signature” N – ~ P – S/T discourages O-glycosylation – i.e., it appears to be an approximate inhibitor of O-glycosylation. Now, a reason that only a tiny number of sequences with the N – ~ P – S/T sequon are O-glycosylated may be because the remaining signature around the O-glycosylated S/T site is quite similar to the corresponding one around the N-glycosylated S/T site. And, there may be some “signal” outside of our chosen ±10 neighborhood that may occasionally allow a protein with the N – ~ P – S/T sequon to get O-glycosylated. However, the LPM is not trained on data outside of the ±10 neighborhood and, thus, such a signal, if it were to exist, is unobservable from the LPM’s frame of reference. So, our selecting the N-glycosylated sequences as the negative control set is independent of cell “compartments”. We are simply training the LPM on the signatures around the ±10 neighborhood of the N- and O- glycosylated sequences. Furthermore, if our assumption regarding the signature is unreasonable, we would have seen material deterioration in prediction accuracy under CV, but did not.So, we assumed that sequence and structural information in the neighborhood of the composite sequon for O-glycosylation is ~ (W–S/T–W), where “~” denotes the “not” operator. This means ~W – S/T – W, W – S/T – ~W, or ~W – S/T – ~W are necessary for O-glycosylation. The RR estimated LPM approach was instrumental in making this finding. The consensus sequon for phosphorylation is ~ (W–S/T/Y/H–W); although W–S/T/Y/H–W is not an absolute inhibitor of phosphorylation. Structural attributes  are significant predictors of O-GlcNAc glycosylation.The consensus N-glycosylated sequences are good approximations to non-O-glycosylatable sequences; although N – ~P – S/T is not an absolute inhibitor of O-glycosylation. The selective positioning of an amino acid along the sequence, differentiates the PTMs of proteins. Some N-glycosylated sequences are also phosphorylated at the S/T-site in the N – ~P – S/T sequon. ASA values for N-glycosylated sequences are stochastically larger than those for O-GlcNAc glycosylated sequences.For LPM estimation it is found that and structural data as explanatory variables yields a KS statistic value of 99%. With only sequence data, the KS statistic erodes to 80%. With 50% as the cutoff probability for predicting O-GlcNAc glycosylation, this LPM mispredicts 21% of out-of-sample O-GlcNAc glycosylated sequences as not being glycosylated. The 95% confidence interval around this mispredictions rate is 16% to 26%. This study underscores the germaneness of structural information for predicting the likelihood of O-glycosylation.The LPM with sequence O-glycosylation. These studies have employed “machine learning” techniques, like random forests and factor analysis, to predict O-glycosylation [explanatory process driving O-glycosylation. Thus, classical hypothesis testing is germane to the present work, and algorithmic modeling with its pervasive focus on prediction is not, in this paper, an end in itself. In this sense, Cox’s crucial comment to Breiman is echoed, which is that the “starting point” in this paper is not the data, but the underlying process generating it [O-glycosylation prediction process.Several other studies are available for the predicting the likelihood of sylation –114. Howating it . Furtherglycos_public.xlsx. This data, and key operations performed on it, are described in Table Data collected on the glycosylated sequences used for analysis herein is stored in an Excel spreadsheet: glycos_public.xlsx in the column labeled PDBID. In dbOGAP, the total number of UniProt sequences is 376; and, of these, 39 have structural information. Several unique structures exist for each of the 39 sequences. This results in the 39 sequences having a total of 998 structures with unique PDB-IDs. Each structure is unique in terms of different ligands/drugs bound to them. Although there is redundancy in this data from a sequence standpoint, these 39 sequences have different structural attributes as defined by phi-psi angles and ASA, which are important structural variables. In order to make sure that these are real structural differences, similar structures were superposed to calculate a RMSD (root-mean-squared-deviation). It is confirmed that the resolution of the structures is high enough to believe in the subtle conformational changes captured in the data. The 10 residues to the left of the S/T-site, and the 10 residues to the right of the S/T-site, are superposed to see if the S/T-site displays slightly different conformations. An example is presented in Fig. S-site, in this example, is flexible enough to accommodate its role in a multitude of cellular processes. Residues that have maximum deviation (0.5-1.0Å) are close to the S-site. Residues farther out superimpose with less than 0.1Å RMSD.Structural data is obtained from the Protein Data Bank (PDB) , and thewww.uniprot.org) and shown in columns labeled Uniprot_Accession_No and Protein_Organism in glycos_public.xlsx [Ligand_Sequence in glycos_public.xlsx) is obtained from the PDBsum database [The sequence and taxonomy information for the data used in the analysis are extracted from the UniProtKB database (lic.xlsx . The ligdatabase ASA in glycos_public.xlsx) are obtained using the ASAView tool and database [R, K, H), negatively charged residues , polar uncharged residues , Cysteine (C), and hydrophobic residues .Accessible Surface Area (ASA) values


Nature Communications 10.1038/s41467-019-08536-z, published online: 05 February 2019Correction to: The original version of this Article contained an error in the title, which was previously incorrectly given as “Negative differential resistance as a critical indicator for the discharge capacity of lithium-oxygene batteries”. The correct version states “lithium-oxygen” in place of “lithium-oxygene”. This has been corrected in both the PDF and HTML versions of the Article.


Scientific Reports 10.1038/s41598-019-52159-9, published online 30 October 2019Correction to: The original version of this Article contained an error in the title of the paper, where the word “Striatum” was incorrectly given as “Stratum”. This has now been corrected in the PDF and HTML versions of the Article.


While previous word-frequency approaches calculated rough measures of sequence similarity or dissimilarity, some new alignment-free methods are able to accurately estimate phylogenetic distances between genomic sequences. One of these approaches is Filtered Spaced Word Matches. Here, we extend this approach to estimate evolutionary distances between complete or incomplete proteomes; our implementation of this approach is called Prot-SpaM. We compare the performance of Prot-SpaM to other alignment-free methods on simulated sequences and on various groups of eukaryotic and prokaryotic taxa. Prot-SpaM can be used to calculate high-quality phylogenetic trees for dozens of whole-proteome sequences in a matter of seconds or minutes and often outperforms other alignment-free approaches. The source code of our software is available through Github:  Here, groups of orthologous sequences have to be identified first, for which then multiple alignments are to be calculated. There are generally two different strategies of resolving phylogenies based on multiple alignments. In the so-called supermatrix approach, multiple sequence alignments of single genes or proteins are concatenated. A phylogenetic tree is inferred from the resulting matrix, kelihood  or Bayeskelihood . Alternakelihood  or superkelihood  approachAll of these steps are time consuming, and manual intervention is often required. Therefore, word-based or alignment-free alternatives have been proposed recently, which are much faster and require much less data preparation. Most alignment-free methods compare the word composition of sequences , 9, 10, Accurate alignment-free tools are urgently needed because of the huge volume of data generated by new sequencing techniques. Another advantage of alignment-free methods, compared to alignment-based approaches, is the fact that they can be applied to incomplete data, e.g., to unassembled sequencing reads or to partially sequenced genomes . Note thad hoc measures of sequence similarity or dissimilarity. They are not based on stochastic models of molecular evolution, and they do not try to estimate distances between sequences in a statistically rigorous way. More recently, some alignment-free approaches have been proposed that are based on explicit models of DNA evolution. These methods are able to estimate the number of substitutions per site that have happened since two nucleic-acid sequences have evolved from their last common ancestor [The above-mentioned approaches to alignment-free sequence comparison calculate ancestor , 34, 35.A main application of alignment-free approaches is comparison of whole genomes. Consequently, most alignment-free methods have been designed to work on DNA sequences. If distantly related species are studied though, phylogenetic trees are usually inferred from protein sequences rather than from DNA sequences. The reason for this is that protein sequences are more conserved than DNA sequences, as synonymous substitutions are not visible in proteins. Thus, for distal species, it may be hard to detect similarities between genes at the DNA sequence level, while homologies may still be detectable among protein sequences. It is therefore highly desirable to have accurate alignment-free software tools that work on protein sequences, in addition to the available tools for DNA sequence comparison. Generic word-frequency methods can be applied to both DNA and protein sequences; the program Feature Frequency Profile (FFP), e.g., has been used for whole-proteome comparison . As mentHere, we propose an alignment-free method that estimates the phylogenetic distance between two taxa based on the average number of amino-acid substitutions in the whole proteoms since they evolved from their last common ancestor. Our approach is based on filtered spaced word matches (FSWM), a concept we introduced recently for whole-genome sequence comparison ; see 3131, 35 foTo evaluate our approach, we used simulated protein sequences and real-world whole proteomes. Test runs on the simulated sequences show that our distance estimates are very close to the true distances for distance values of up to around 2.0 substitutions per sequence position. On the real-world sequences, we evaluated our approach indirectly by phylogenetic analysis, as is common practice in the field. We used Prot-SpaM to estimate pairwise distances for various sets of taxa, and we applied the neighbor-joining algorithm  to calcuP of length ℓ – , i.e., for a length-ℓ word P over {0, 1} –, a spaced-word with respect to P is a length-ℓ word W over the alphabet W(i) = * if and only if P(i) = 0. An index i ∈ {1, …, ℓ} is called a match position of P or W, respectively, if P(i) = 1, and a don’t care position otherwise. The number of match positions in a pattern or spaced-word is called its weight w. We say that a spaced word W with respect to P occurs in a sequence S at some position i if one has W(k) = S(i + k − 1) for all k ∈ {1, …, ℓ} with P(k) = 1 – ,i.e., for all match positions of P.We consider sequences over an alphabet P between two sequences S1 and S2 at  if the same spaced word w.r.t P occurs at position i1 in S1 and at position i2 at S2. In other words, there is a spaced-word match between S1 and S2 at  if and only if one has S1(i1 + k − 1) = S2(i2 + k − 1) for all match positions k of P. Below is an example for a spaced-word match between two sequences S1 and S2 at  with respect to the pattern P = 1100101; the spaced word TN**D*P occurs at position 2 in S1 and at position 3 in S2:Moreover, we say that there is a spaced-word match w.r.t. Similar to our original FSWM approach, we estimate distances between protein sequences based on selected spaced-word matches between them, with respect to one or several pre-defined patterns. Distance values are obtained by comparing the amino-acid residues that are aligned to each other at the don’t-care positions of the selected spaced-word matches. This is similar to estimating distances in standard alignment-based approaches; the only difference from those standard approaches is that we are using don’t-care positions of spaced-word matches instead of full sequence alignments.T, then a spaced-word match is considered a random match and is not further considered. As a default, we use a threshold value of T = 0. To see that this threshold accurately separates homologous from background spaced-word matches, one can plot the number of spaced-word matches with a score s against s, see Fig. -match histogram, or spamogram for short. In these plots, two peaks are typically visible, a peak on the right-hand side representing homologous spaced-word matches and a peak on the left-hand side representing background matches. By default, we are using patterns with a weight of w = 6 and with 40 don’t-care positions, i.e., with a length of ℓ = 46.To estimate distances in this way, one has to make sure that only those spaced-word matches are selected that represent homologies, i.e., that the involved spaced-word occurrences go back to the same origin in the last common ancestor of the two proteins that are compared. To distinguish such "homologous" spaced-word matches from random background matches, we calculate a score for each spaced-word match using the BLOSUM62 substitution matrix . SimilarS1 and S2 are compared and a spaced word W occurs n time in S1 and n′ times in S2, then this gives rise to a total of n × n′ spaced-word matches. Taking all of these spaced-word matches into account for phylogeny reconstruction would overemphasize repeated regions where the same spaced words occur multiple times. Instead of using all possible spaced-word matches, we therefore use a one-to-one mapping of spaced-word occurrences in the compared sequences. That is, we ensure that each spaced word occurrence is involved in at least one of the selected spaced-word matches. Formally, if there are two spaced word matches at  and at , respectively, then we can include both of them simultaneously in our list of selected spaced-word matches only if W, we calculate the scores of all spaced-word matches involving W. We then select them one-by-one in descending order of their scores, always ensuring that each occurrence of W is used in at least one of the selected spaced-word matches.Moreover, we use a one-to-one mapping of spaced-word occurrences. Note that if sequences eviously . For a gFinally, in order to estimate pairwise distances between two input sequences, we consider the pairs of amino acids aligned to each other at the don’t-care positions of the selected spaced-word matches. Here, we are using the Kimura model  that appP. More precisely, we are using a set m binary patterns such that all patterns in w but have their match and don’t-care positions arranged differently. We then use spaced-word matches with respect to all patterns m = 5 patterns. To find suitable pattern sets, we integrated the tool rasbhari [The accuracy and statistical stability of the described approach depends on the number of selected spaced-word matches; the more matches we obtain, the more accurate and stable the results of our method will be. To increase the number of spaced-word matches, the default version of our program uses multiple patterns instead of one single pattern rasbhari  into ourrasbhari  of patteT = 0 to discard background spaced-word matches, and sets of m = 5 patterns. We compared our program to the following four other alignment-free methods that can be run on protein sequences: Average Common Substring Approach (ACS) [k = 0. The competing tools were used with their default parameters. In addition to evaluating these tools on protein sequences, we ran filtered spaced word matches on the complete genome sequences of the same taxa. All test runs were done on a 10 x Intel(R) Xeon(R) central processing using E7-4850 with 2.00 GHz with four cores each, equaling 40 cores and 1,000 GB of random access memory.To assess the quality of our new approach and to compare it to other alignment-free methods, we used artificially generated as well as real-world protein sequences. For the test runs, we used the default parameters of our program, namely, 6 match positions and 40 don’t care positions, i.e., a total pattern length of 46 –, a threshold of ch (ACS) , FFP [36ch (ACS) , kmacs [ch (ACS) , and CVTch (ACS) . Here, wm = 5 patterns , we did a second series of test runs with m = 1, i.e., with single patterns. Figure To evaluate the distances estimated by our program, we simulated sequences with the tool pyvolve . Pyvolveentclass1pt{minimaNext, we applied the above alignment-free methods to calculate phylogenetic trees from real-world protein sequences. For four different groups of species, we downloaded all available protein sequences from GenBank , in addin taxa is 2 · n − 6 [As mentioned above, Prot-SpaM uses a probabilistic algorithm to generate pattern sets, so the results of different program runs on a sequence set can differ slightly, even if the same parameter values are used. We therefore performed 100 program runs on each dataset. Table  · n − 6 . Program · n − 6 . Neighbo · n − 6 .Escherichia coli and Shigella. For each strain, we were able to download about 4,000-5,000 protein sequences; the total size of this dataset is around 41 MB. Figure Our first dataset consists of 29 strains of Wolbachia strains, a group of Alphaproteobacteria that are intracellular endosymbionts of arthropods and nematodes [Wolbachia, 16 distinct genetic lineages (“supergroups”) are currently distinguished (named by letters A-F and H-Q) that may differ in host specificity and type of symbiosis [Wolbachia I). A tree generated with Prot-SpaM from this dataset is shown Fig. As a second test case for benchmarking, we analyzed the phylogeny of ematodes . Within ymbiosis . We re-aymbiosis , therebyWolbachia benchmarking dataset, we analyzed relationships between supergroups based on available (draft) genomes see below (Wolbachia II). For within supergroup relationships (Wolbachia I), a program run of Prot-SpaM on the whole proteome recovered a tree that is largely congruent in topology and branch lengths in comparison to a phylogenomic supermatrix analysis of 252 single-copy orthologs that excluded genes that showed signs of recombination. A comparison based on RF distances showed that our new method outcompetes other available alignment-free programs  proteomes for all available Wolbachia draft and fully assembled genomes  of the binary patterns and spaced words, their length ℓ, the number m of different binary patterns used by the program, and the cutoff value T to separate homologous from background spaced-word matches. To see how these parameters influence the results of our software and to find suitable default values, we ran Prot-SpaM with varying values of these four parameters. Here, we modified one parameter at a time, using the respective default values of the remaining three parameters. The results are summarized in Tables w, ℓ, and T that work best for all datasets, but our default values seem to be a reasonable compromise. Using sets of m = 5 binary patterns does not improve the quality of the produced trees in terms of their RF distances to the reference trees compared to program runs with single patterns. Table E. coli/Shigella dataset where most taxa are closely related to each other and another one in the Metazoan dataset that contains taxa with very large evolutionary distances. In the pairwise comparison of E. coliO157:H7 strain EDL933 with E. coliO157:H7 Sakai (EHEC), Prot-SpaM selected more than 6,000,000 spaced-word matches. These two proteomes have less than 1,600,000 amino acids each, so in this case >3.75 spaced-word matches per sequence position were selected. By contrast, less than 13,000 spaced-word matches were selected in the comparison of Brugia malayi and Homo sapiens. The latter proteome has a length of more than 75,000,000 amino acids, so here less than 0.00017 spaced-word matches per sequence position were selected.The number of spaced-word matches in a pairwise sequence comparison depends on how similar the two sequences are to each other; see  for detak-mer frequencies or on the length of common substrings. These approaches have been applied not only to DNA but also to protein sequences. A drawback of these methods is that they can only calculate rough measures of sequence similarity or dissimilarity; they do not estimate phylogenetic distances in a rigorous way. More recently, word-based methods have been developed that can accurately estimate phylogenetic distances between genomic sequences based on stochastic models of DNA evolution. One of these approaches is FSWM.A number of so-called alignment-free approaches have been proposed in recent years to rapidly calculate phylogenetic distances between genomic sequences. Earlier approaches are based on In this study, we introduced Prot-SpaM, a new implementation of FSWM to compare complete or incomplete proteome sequences to each other. To our knowledge, Prot-SpaM is the first tool that can accurately estimate phylogenetic distances between protein sequences without the need to calculate full sequence alignments. Our benchmark results show that distance estimates obtained with our approach are accurate for a large range of phylogenetic distances. Distances calculated with CVTree,ACS,FFP, and kmacs, by comparison, are monotonously increasing with the number of substitutions between the compared sequences. The obtained distance values are far from proportional to the real distances, and they flatten out somewhere between 0.5 and 1.5 substitutions per position see Fig. . By contk-mer length of k = 4, so we used this value in our study. It has been reported, however, that FFP may perform better on protein sequences if larger values of k are used [In our program evaluation, we used all competing software tools with their respective default parameters, if such default values were recommended by their developers. It should be mentioned, however, that some of the evaluated programs might produce better results with different parameter settings. The program FFP, e.g., uses a default are used . A comprE.coli/Shigella and plant datasets, as shown in Table Wolbachia datasets, it still performed reasonably well and was again superior to competing approaches on the whole-proteome sequences, but it was outperformed by word-frequency methods on the 252 selected orthologous proteins. A possible explanation of this result is discussed below. On the large prokaryote dataset and for the metazoan set, by contrast, none of the compared programs could reproduce the reference trees that we used in our evaluation. These are difficult datasets since they span very large evolutionary distances. Also, it should be mentioned that there are no absolutely reliable reference trees available for these datasets. For the metazoan dataset, e.g., the position of the ctenophores is still a matter of debate [Prot-SpaM produced high-quality trees and was superior to other alignment-free methods for the f debate , 74, 75.An interesting result is the performance of Prot-SpaM compared to our previous approach FSWM that takes genomic sequences as input. For most groups of taxa in our study, the results of Prot-SpaM and FSWM were of similar quality in the sense that the RF distances to the reference trees were comparable for both approaches. However, for the set of 813 prokaryote taxa, our new spaced-words approach performed better on whole proteomes than our previous approach on whole genomes, as shown in Fig. Wolbachia I, we actually obtained better RF distances with Prot-SpaM and FSWM when we applied these programs to the whole-protein or whole-genome sequences, respectively, than when we applied them to the 252 selected orthologous proteins or to the genes coding for those proteins : linuxProgramming language: C++Other requirements: noneLicense: GNU GPLAny restrictions to use by non-academics: nonehttp://projects.gobics.de/data/protspam/paperData.tgz. Additional supporting data and snapshots of the code are available in the GigaScience repository, GigaDB [The sequence datasets and trees used for the program evaluation can be downloaded from , GigaDB ACS: Average Common Substring Approach; BLOSUM: Blocks Substitution Matrix; CVTree: Composition Vector Tree; FFP: Feature Frequency Profile; FSWM: filtered spaced word matches; indel: insertions and deletions ; kmacs: k-Mismatch Average Common Substring Approach; PHYLIP: PHYLogeny Inference Package; Prot-SpaM: proteome-based spaced-word matches; rasbhari: Rapid Approach for Seed optimization Based on a Hill-climbing Algorithm that is Repeated Iteratively; RF: Robinson-Foulds; spamogram: Spaced-Word-Match Histogram.The authors declare that they have no competing interestsThe project was partially funded by the Volkswagen Foundation, project VWZN3157. We acknowledge support by the Open Access Publication Funds of the Göttingen University.Wolbachia and wrote parts of the manuscript. B.M. co-supervised the project and wrote most of the manuscript.C.A.L. designed the study, co-supervised the project, and drafted parts of the manuscript. J.S. implemented the software and did most of the program evaluation. S.D. contributed to the program evaluation. M.G. and C.B. analyzed and discussed the results on GIGA-D-18-00169.pdfClick here for additional data file.GIGA-D-18-00169_R1.pdfClick here for additional data file.GIGA-D-18-00169_R2.pdfClick here for additional data file.Response_to_Reviewer_Comments_Original_Submission.pdfClick here for additional data file.Response_to_Reviewer_Comments_Revision_1.pdfClick here for additional data file.Response_to_Reviewer_Comments_Revision_2.pdfClick here for additional data file.Reviewer_1_Report_ -- Sharma Thankachan5/31/2018 ReviewedClick here for additional data file.Reviewer_3_Report_ -- Se-Ran Jun6/18/2018 ReviewedClick here for additional data file.Reviewer_3_Report_revision_1 -- Se-Ran Jun9/17/2018 ReviewedClick here for additional data file.Reviewer_3_Report_Revision_2 -- Se-Ran Jun9/29/2018 ReviewedClick here for additional data file.Reviewer_4_Report_ -- Sriram Chockalingam6/20/2018 ReviewedClick here for additional data file.Reviewer_4_Report_Revision_1 -- Sriram Chockalingam9/18/2018 ReviewedClick here for additional data file.Reviewer_5_Report_ -- Alexey Kozlov6/21/2018 ReviewedClick here for additional data file.Reviewer_5_Report_Revision_1 -- Alexey Kozlov9/18/2018 ReviewedClick here for additional data file.Supplemental FileClick here for additional data file.


Scientific Reports 10.1038/s41598-018-34386-8, published online 29 October 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “eGFR” was incorrectly given as “Egfr”This has now been corrected in the PDF and HTML versions of the Article.


Communications Biology 10.1038/s42003-019-0360-3, Published online 26 March 2019Correction to: In the original version of the paper, Fig. 4 contained an error in the labels for “Intermediate” and “Insignificant” in the figure key, making the symbols partially illegible. This error has been corrected in the HTML and PDF versions of the paper.


Multi-trait single step genetic evaluation is increasingly facing the situation of having more individuals with genotypes than markers within each genotype. This creates a situation where the genomic relationship matrix ( Within the last decade, genotyping thousands of individuals with single nucleotide polymorphism (SNP) chips has become common practice in breeding programs of many species of economic relevance. However, due to cost effectiveness these individuals are being genotyped with low- to medium-density SNP chips, with usually not more than 50,000 markers.To date, genetic evaluation systems accommodate SNP genotypes via the so-called single step model, in which most often markers are used to pre-calculate a relationship matrix, which subsequently augments the usual pedigree derived relationship matrix into a so-called mentclass2pt{minimHowever, the situation described above of having more genotyped individuals than markers has led to a situation where  such as  offer hu such as  or deman such as . However such as , 6. SS-TSNP-BLUP , 5, 7.In this paper, we will examine the computational advantage of SS-T-BLUP for a large-scale multi-trait BREEDPLAN single step genetic evaluation of Australian Angus beef cattle. We will compare the results to those obtained by using an ordinary SS-H-BLUP approach.In the following, three equivalent representations of the inverse of the mentclass2pt{minimlsewhere , 6, 8–11The  markers .Replacing mentclass2pt{minimAssuming that The differences between the three approaches regarding computational time spent on preparing necessary data and solving the MME iteratively can be reduced to a set of very specific operations unique to the respective representation of the inverse of Assuming that mentclasspt{minimaion once .z, with a dense matrix, which is ]. It requires the multiplication of a vector with the MME coefficient matrix once per iteration. Therefore, this method is affected by the way the inverse of Differences in peak memory requirement directly result from the size of the arrays, which must be kept in RAM simultaneously during preparation and iteration. Furthermore, for SS-H-BLUP and SS-The SS-H-BLUP, SS-aluation . The dataluation . To incrDummy genotypes for 91,295  individuals, sampled from the pool of non-genotyped individuals, were generated by The BREEDPLAN multi-trait model included pre-corrected phenotypes , a singlecessary . The totFor all three models matrix, The system of equations was solved with AGBU’s current large-scale linear mixed model library solver, which uses the PCG algorithm for iteratively solving linear mixed models and integrates Intel(R) MKL(R), version 2017 update 8. For research and commercial purposes, the solver is available on request. Block-diagonal and diagonal pre-conditioners were used for random and fixed factors, respectively.Xb = y, where X is the coefficient matrix, b is the solution vector and y is the right hand side vector, convergence was achieved when the L2 norm of vector (y − Xb) scaled by the L2 norm of vector y was Denoting the MME as Results for the different parts of the setup and solving steps are in Table mentclass2pt{minimThe preparation time for model specific parts for SS-T-BLUPy, with matrices y =  requirements in gibabyte due to matrices and operations that are unique to the approaches are in the last row in Table The last column in Table SS-T-BLUP has been proposed as a single step model which can be helpful for datasets for which the number of genotyped individuals exceeds the number of markers and the i may then be i in the different traits. This expansion is not possible for the models applied here. However, SS-SNP-BLUP usually comes at the cost of much higher model dimensionality and slow convergence rates when solved iteratively [In terms of modelling capacity SS-T-BLUP, SS-H-BLUP, and SS-ratively . The latratively . To our Since all models were equivalent, it was expected that the number of iterations needed for convergence was the same. However, surprisingly there was no difference in the number of iterations for convergence when using only the 58,705 real genotypes (results not shown), 150k genotypes or 400k genotypes. A possible explanation is the way the dummy genotypes were generated. Thus, it is very likely that a dataset with 400k real genotypes may require more iterations but that the time needed for preparation and per iteration will be similar to that observed in this study.As shown by the results, SS-T-BLUP clearly outperforms SS-H-BLUP in terms of total processing time mainly due to the huge computational cost of setting-up These results support the conclusion that SS-T-BLUP provides a feasible algorithm to calculate exact solutions for estimated breeding values when the number of genotyped individuals exceeds the number of markers. A limitation to the number of genotyped individuals is only set by the available RAM. Therefore, SS-T-BLUP allows solving single step equation systems iteratively without generating


However, SBM still has great potential in complex applications due to the following two points; Bregman Iteration (BI), employed in SBM, may not make good use of the phase information for complex variables. In addition, the converting technique may consume more time. To address that, this paper presents the complex-valued Split Bregman method (CV-SBM), which theoretically generalizes the original SBM into the complex domain. The complex-valued Bregman distance (CV-BD) is first defined by replacing the corresponding regularization in the inverse problem. Then, we propose the complex-valued Bregman Iteration (CV-BI) to solve this new problem. How well-defined and the convergence of CV-BI are analyzed in detail according to the complex-valued calculation rules and optimization theory. These properties prove that CV-BI is able to solve inverse problems if the regularization is convex. Nevertheless, CV-BI needs the help of other algorithms for various kinds of regularization. To avoid the dependence on extra algorithms and simplify the iteration process simultaneously, we adopt the variable separation technique and propose CV-SBM for resolving convex inverse problems. Simulation results on complex-valued l1-norm problems illustrate the effectiveness of the proposed CV-SBM. CV-SBM exhibits remarkable superiority compared with SBM in the complex-to-real transforming technique. Specifically, in the case of large signal scale n = 512, CV-SBM yields 18.2%, 17.6%, and 26.7% lower mean square error (MSE) as well as takes 28.8%, 25.6%, and 23.6% less time cost than the original SBM in 10 dB, 15 dB, and 20 dB SNR situations, respectively.The Split Bregman method (SBM), a popular and universal CS reconstruction algorithm for inverse problems with both  Thus, a more general algorithm for CS recovery in complex variables is needed.In recent years, there have been many papers focusing on this problem, such as Complex Approximate Message Passing (CAMP)  and M-lal1-norm and TV-norm regularization problems. By the idea of decomposing the original problem into several subproblems worked out by Bregman Iteration (BI) [A to 2m × 2n, which consumes more memory and time within the iteration process.The Split Bregman method (SBM) proposed in  is a uniion (BI) ,30, SBM ion (BI) ,32, e.g.ion (BI) , SAR imaion (BI) , forwardion (BI) , SAR imaion (BI) , and masion (BI) . Howeverl1-norm problems reveal the effectiveness of CV-SBM compared with existing methods. Particularly, the proposed CV-SBM exhibits 18.2%, 17.6%, and 26.7% lower mean square error (MSE) and takes 28.8%, 25.6%, and 23.6% less time than SBM through the complex-to-real transforming technique in 10 dB, 15 dB and 20 dB SNR cases with large signal scale n = 512.To tackle the aforementioned problems, this paper theoretically generalizes the original SBM into the complex domain, named the complex-valued Split Bregman method (CV-SBM). We first define the complex-valued Bregman distance (CV-BD) and replace the associated regularization term with the CV-BD in the inverse problem. Then, complex-valued Bregman Iteration (CV-BI) is proposed to solve this new problem. In addition, according to the calculation rules, Wirtinger’s Calculus, and optimization theory for complex variables, how well-defined the CV-BI is and its convergence are analyzed in detail. The proof of the above two properties reveals that CV-BI can settle inverse problems if the regularization term is convex. Since CV-BI requires the help of additional algorithms to find the solution to the specific regularization, as BI does, its solution is still complicated and computationally expensive. Inspired by SBM, we adopt the variable separation technique to avoid the requirement of other optimization algorithms and then present CV-SBM to settle the convex inverse problems with the simplified solution. Simulation results on the complex-valued The rest of this paper is organized as follows. In l1-norm and TV-norm regularization [SBM, whose main idea is to decompose the original unconstrained problem into several equivalent subproblems solved by BI ,30, has rization . For theJ(x) is a real convex function but not necessarily differentiable, and H is a real convex and differentiable function. By replacing J(x) with corresponding Bregman distance (BD) J(x) at the point v, and J(x) at v, BI focuses on the optimization problemDJp ,38, BI tDefinition 1.T at z0 if T at z0 is called the subdifferential of T at z0 and is denoted by Let found in .J(x) so that (5) is still a convex problem. Furthermore, it can be used as the measurement of the closeness of two points in J(x) [A key property of the BD is that it has the same convex characteristic as  in J(x) ,30.l1-norm and TV-norm problems, Equation (6) is easily calculated, whereas dealing with (5) is more complicated and needs the help of another algorithm [As for the lgorithm , such aslgorithm  or FPC [lgorithm . To avoid and considers an equivalent constrained problem to Equation (7)SBM aims to find a solution to the unconstrained problemE = ||d||1 + H(x) and μ is a positive constant to balance the two terms in Equation (9) within the iterations. Since there are two unknown variables completely fulfilling the demand of BI for optimization problems, BI can be performed for each of them:The corresponding unconstrained version of (8) can be formulated as follows:Let kx by fixing kd−1 and then clearing up kd by fixing kxFor (13), one can decompose it into resolving two subproblems alternately, i.e., working out Evidently, the combination of BI and SBM can be adopted to settle plenty of convex optimization problems in a real system ,41. HoweT(c) = RT(c) + IjT(c) in complex variables c = Rc + Ijc, the complex differentiability equals to the satisfied Cauchy–Riemann conditions:As is well-known, convex optimization theory requires the differentiability of the objective function. For l1-norm regularization problem:F(x) does not obey (17) so that calculating the complex gradient directly is unavailable. For a complex-valued T at Rc and Ic, which can be obtained by the traditional ways. According to Equations (19) and (20), one can calculate the W-gradient of c* and the CW-gradient of c:To overcome such a problem, an alternative tool for computing the complex gradient was brought into light recently called Wirtinger’s calculus . It rela*c and the CW-gradient for c are equal to zero, in Wirtinger’s calculus we can treat c and c* as two irrelevant or independent variables, which is the main approach allowing us to utilize the elegance of Wirtinger’s calculus. Here is an example: if T(c) = c(c*)2, then we have Considering that both the W-gradient for found in .l1-norm in F(x) at zero. In this case, the conjugate Wirtinger’s subgradients (CW-subgradients) [T at c if T at Rc and Ic. The set of all CW-subgradients of T at c is called Wirtinger’s differential of T at c and is represented by T, the Wirtinger’s differential only contains one element, i.e., its CW-gradient. Wirtinger’s differential of modulus |ix| and i is the index for the element of vector x and s is some complex number verifying In general, for the convex function in complex variables, the optimization condition is the CW-gradient equal to zeros vector. Nevertheless, in practice, some functions may not be differentiable everywhere, e.g., adients)  can be a follows .(25)∂x*∈∂x*F(x) . By the To prevent the BD becoming complex-valued, we first generalize the BD into the complex domain and introduce the CV Bregman distance (CV-BD) theoretically.Definition 2.For Lemma 1 and Lemma 2 prove that the CV-BD is as the same convex as J(x) and can measure the closeness at two points in J.To ensure that the CV-BD can be utilized as the objective function as the BD, in the following Lemma 1.Letbe a CV-BD associated with real convex or strictly convex function J, whereThenis as the same convex property as J for variable u in each v.Proof. J is a real convex function and let Assume J is a real convex function, J satisfiesu as J. Considering that J is a real strictly convex function, we assume J satisfiesu as J. Then, it can be concluded that J for variable u in each v. □For Lemma 2. Letbe a CV-BD associated with real strictly convex function J and assume a pointis on the line segment connecting u and v, where,. Thenand equality holds if and only if u = v.Proof. Lemma 1Lemma 2. Then, the CV-BD at two points in convex function J would decrease when they get closer, and may become zero if and only if the two points coincide. This property makes the CV-BD the measurement of closeness at two points.Assume J(x) as the objective function:Thus, inspired by the original BI, we use the CV-BD between the variables to be solved and the current solution to replace real convex function H(x) and the CV-BD. However, the CV-BD kx-1, which inevitably interferes with the solution of kx. As we shall prove below, this issue is not vital, since CV-BI introduced in the following subsection automatically chooses a suitable CW-subgradient when dealing with Equation (36).Obviously, CV-BI for Equation (36) is proposed directly and the definition and the convergence are proved in the following.x0 = 0, p0 = 0, for k = 1,2,Algorithm 1. Let kx as a minimizer of the convex function kQ(x)1. compute 2. compute x0 and p0 whatever satisfy x0≠0, its CW-subgradient requires optional calculation, which is not desired in practical.Generally, we can initialize kx exists in kQ(x) and the iteration can find an appropriate CW-subgradient kp automatically.In this subsection, we reveal that the iterative procedure in Algorithm 1 is well defined. Specifically, a minimizer Proposition 1. Assume that0=0,J(x) is convex and bounded, and let x. Then, for eachk in Qk(x), and there exists an appropriate CW-subgradient, there exists a minimizer xandsuch thatk is unique.Moreover, if A has no null space, a minimizer xProof. k = 1, Q1(x) becomes the original function F(x), of which the existence of minimizers and the optimality condition p1 + q1 = p0 = 0 is well known [p1 = HrA1.We prove the result by induction. In the case of ll known . In addik−1 to k and assume kQ(x). Recalling the l2-norm greater than or equal to zero, kQ(x) can be estimated asThen, we proceed from J(x) is not constant, the boundedness of kQ(x) implies the boundedness of J(x). This shows that the level sets of kQ are weak-* compact [kQ exists due to the optimization theory. Besides, if A has no null space and H(x) as well as J(x) is strictly convex, kQ(x) is also strictly convex, and therefore the minimizer is unique. This completes the proof of the existence of minimizers for all k > 1.Since only  compact . Hence, kp and kq exist for all k > 1. According to the optimality conditions for kQ(x)kp−1 exists, one can get that The following proves p0 = 0, we obtain thatRecalling Equation (38) and The definition of CV-BI has been proved as mentioned above. The whole CV-BI can be summarized as follows:Algorithm 1: CV-BIx0 = 0 p0 = 0 k = 1 λ, Initialization:   While “stopping criterion is not met” do     K = k + 1;End whileJ(x) in Equation (18) if J(x) is a real convex function in complex variables. Furthermore, since each step of Algorithm 1 obeys the optimization rules in the complex domain instead of converting the objective function kQ(x) and variable x into the real domain, one can summarize that CV-BI preserve phase information for complex variables.Review the entire process of proof and one can find that CV-BI possesses the ability to solve any kind of regularization term In this subsection, the convergence property of CV-BI is analyzed. To be specific, two monotonicity properties are proved with the help of the CV-BD.Proposition 2.k) obtained from the CV-BI is monotonically nonincreasing, we getUnder the above assumption, the sequence of H < ∞, then we even haveProof. kx is the minimizer of the convex function kQ(x), we obtainRecall the nonnegative property of the CV-BD and that We can derive a formula motivated by the identity of the original BD :(48)DJpkProposition 3. Under the same assumption as Proposition 2, letbe a minimizer of H(x) with, then we haveProof. H(x), we get an inequalityRecall the nonnegative property of the CV-BD and that According to Equations (49), (51) can be derived asfound in . □For various kinds of regularization terms corresponding to Equation (43), CV-BI still has to employ other algorithms as BI does, which makes the solution process complicated and computationally expensive. Inspired by SBM, we separate the original variable and present CV-SBM to settle the convex inverse problems with the simplified solutions.A constrained optimization problem in complex variablesF is convex in x and d. Thus, by applying CV-BI to Equation (54) in each variable, we can derive thatEvidently, C1 is a constant. Substituting Equation (58) in Equations (55)–(57) yieldsx and dTo simplify the above iteration step Equation (55), we assume J(x) in any convex optimization task.The above two subproblems can be worked out easily. Considering the property of CV-BI, it can be inferred that CV-SBM is also universal for convex The overall CV-SBM is shown as Algorithm 2Algorithm 2: CV-SBMx0 = 0, d0 = 0, p0 = 0, λ, μ, k = 1Initialization:   While “stopping criterion is not met” do    k =k+1;End whileAssuming Φ = I, then we can work out Equation (18) through CV-SBM by three steps :x subprobleml2-norm is differentiable, Equation (63) can be tackled by taking the CW-gradient of x equal to zero, and yieldStep1: Clear up the d subproblemStep2: Find a solution to the bStep3: Update l1-norm problem can be presented as Algorithm 3CV-SBM for Algorithm 3: CV-SBM for l1-norm problemx0=0, d0=0, p0=0, λ, μ, k=1Initialization:   While “stopping criterion is not met” do            k = k+1;        End whilel1-norm problems in the complex domain. We apply the proposed method to recover a complex-valued random sparse signal x from the noisy measurements y generated by y = Ax + ε, where x consists of L nonzero elements and the amplitudes of both x’s and A’s real part and imaginary part obey Gaussian Distribution N . The noise vector ε is assumed to be i.i.d zero-mean complex Gaussian noise. The contrastive means for the proposed scheme in the following subsections are as follows: classical OMP [This section presents the performance of the proposed CV-SBM by conducting a wide range of experiments solving ical OMP , CAMP, Mical OMP . Noted ttol = 2e−4 denotes the tolerance and kmax = 2000 is the maximum iteration times. All the experiments are carried out in MATLAB 2016b on the PC with Intel I7 7700K @4.2 GHz with 32 GB memory.The stopping criterion for all algorithms is given as follown = 256 and m = 128, respectively. Moreover, the sparsity level of x is fixed L = 32 and the Signal to Noise Ratio (SNR) is set to 15 dB.In this subsection, an illustrative example is devised to demonstrate the effectiveness of the proposed method in comparison with OMP, CAMP, M-lasso, and RV-SBM. We consider that the signal and measurement dimension is x remain hidden to emphasize the nonzero ones in In this subsection, CV-SBM is applied in ISAR imaging with real data of the Yak-42 plane to demonstrate its superiority, comparing with RV-SBM, the range-Doppler (RD) algorithm, and the CS recovery method . DetaileIn this subsection, we test the robustness of the proposed technique against the measurement noise. The experimental parameters are set as follows: SNR varies from 5 dB to 20 dB and other parameters are fixed the same as in the previous subsection. For each SNR, we average the MSE of 100 independent trials as the experimental result, as shown in As the SNR increases, the MSE of the proposed scheme declines, which implies CV-SBM is robust to the noise. Before the SNR reaches 7 dB, the MSEs of CAMP and CV-SBM are almost the same, but when SNR goes beyond 7 dB, the MSE of CV-SBM surpasses CAMP’s and becomes the lowest among all the algorithms. Both OMP’s and RV-SBM’s MSE numerically exceed CV-SBM’s. In addition, CV-SBM behaves better than M-lasso, except at the point when the SNR is equal to 7 dB, at which they are approximately the same. This demonstrates that the proposed algorithm has better robustness against the measurement noise among the methods.n = 256, SNR = 15 dB, L = 32, and m varies from 29 to 128. As in the previous subsection, we measure the average MSE over 100 independent trials, as shown in In this subsection, how the dimension of measurements influences the recovery result is presented. We set In n from 128 to 1024 and fix SNR = 20 dB, m = 0.5n, L = 0.125n. For each n, we conduct 20 independent trials and average the CPU time cost as the result, as shown in In this subsection, the computational cost of the proposed method is measured with increasing dimension of the signal. To this end, we vary A to 2m × 2n, which leads to an inverse matrix with 2n × 2n elements and takes more memory and time within the resolving process, while CV-SBM only needs to calculate a complex inverse matrix with n × n elements. Nevertheless, CV-SBM takes a little more time than CAMP in large signal scale situations thanks to CAMP’s specific design for l1-norm problems, whereas CV-SBM contains an inverse operator. However, CV-SBM still has great potential to exceed CAMP considering that the gap between CAMP and CV-SBM is not large.The result shows that CV-SBM takes less CPU time than OMP and M-lasso in all test dimensions. Before the dimension reaches 512, RV-SBM requires the least time. However, when the dimension is more than 512, CV-SBM requires less CPU time than RV-SBM. This is because the complex-to-real transformation utilized in RV-SBM expands the dimension of the sensing matrix λ = 0.005 and μ = 120 and the stopping criterion (the tolerance tol and kmax) varies. Furthermore, other experimental parameters are as follow: L = 0.125n, m = 0.5n and SNR varies from 10 dB, 15 dB, and 20 dB. For each stopping criterion and SNR, 20 independent trials were carried out and the average MSE, CPU time cost, and iteration time are selected as the result. The average MSE and CPU time cost of the proposed method are also presented if CV-SBM presents better performance.The tests mentioned above have shown that the proposed CV-SBM presents remarkable performance compared with RV-SBM in the same experimental environment. Thus, in this subsection, we focus on the convergence, time cost, and performance of CV-SBM and RV-SBM by implementing experiments with various parameters. In the following experiments, two main parameters for CV-SBM and RV-SBM are set to n = 256 and fix tol = 2e−4, kmax = 2000, as shown in In the first test, we examine CV-SBM and RV-SBM in small scale tol = 2e−5 and keep the other parameters consistent with the previous experiment (To inspect the performance regardless of the convergence condition, we reduce the tolerance to periment . For anyn = 512 is taken into consideration (tol and the maximum iteration times kmax are set the same as in At last, large scale deration . The toll1-norm problem to recover a complex-valued sparse signal. Experimental results demonstrate the superiority of CV-SBM over other existing CS reconstruction methods, especially RV-SBM, in both recovery accuracy and time cost for large-scale cases.In this paper, a new CS recovery algorithm named CV-SBM is presented, which generalizes the widely employed SBM into the complex domain. CV-SBM induces a theoretical support to directly reconstruct the sparse signal in complex-valued variables, instead of converting them into real ones. We apply the proposed CV-SBM to a A significant goal for future work lies in applying CV-SBM to more complicated regularization problems, since CV-BI and CV-SBM are theoretically able to deal with any convex optimization problem.