Nowadays, chaos generators are an attractive field for research and the challenge is their realization for the development of engineering applications. From more than three decades ago, chaotic oscillators have been designed using discrete electronic devices, very few with integrated circuit technology, and in this work we propose the use of field-programmable gate arrays (FPGAs) for fast prototyping. FPGA-based applications require that one be expert on programming with very-high-speed integrated circuits hardware description language (VHDL). In this manner, we detail the VHDL descriptions of chaos generators for fast prototyping from high-level programming using Python. The cases of study are three kinds of chaos generators based on piecewise-linear (PWL) functions that can be systematically augmented to generate even and odd number of scrolls. We introduce new algorithms for the VHDL description of PWL functions like saturated functions series, negative slopes and sawtooth. The generated VHDL-code is portable, reusable and open source to be synthesized in an FPGA. Finally, we show experimental results for observing 2, 10 and 30-scroll attractors. Chaos theory deals with nonlinear and complex dynamic behavior that is associated to unpredictable phenomena. The main characteristic is that small changes in the initial conditions lead to drastic changes in the results. It is deterministic because one knows its model parameters and it is unpredictable because one does not know the evolution of the trajectories, and then one cannot predict its behavior.Three decades ago, the author in  introducn-double scrolls . It was a generalization of Chua’s circuit where the 1-double scroll corresponds to the classical double scroll one. Generating more than 2-scrolls was the challenge after Chua’s circuit. In 2004 , and one must decide how many places to use for the integer and fractional parts  is updated to x1 + h∗y1, where x1 and y1 are initial conditions and h is the step sizeState variable yn[j] is updated to y1 + h∗z1, where y1 and z1 are initial conditions and h is the step sizeState variable x1 is between the break points values B[q] and B[q+1]. If it is satisfied thenVerifies if variable d1PWL is equated to d1∗k[p], where d1 corresponds to coefficient d1 in k[p] to a saturation level in Eq (2)Variable n − 1) times, where n is the number of scrolls being generatedLoop to iterate 2(x1 is between the break points B[q+1] and B[q+2]. If it is satisfied thenVerifies if d1PWL is equated to d1∗(((k[p+1]-k[p])/(B[q+2]-B[q+1]))∗(x1-((B[q+1]+B[q+2])/2))+((k[p+1]+k[p])/2))If step 9 is not satisfied thend1PWL is equated to d1∗k[p+1]q by 2Increases p by 1Increases zn[j] is updated to z1+h(-a∗x1 − b∗y1 − c∗z1 + d1PWL), where a, b, c are coefficients in Eq (1)State variable x1 is updated to the next iterationState variable y1 is updated to the next iterationState variable x1 is updated to the next iterationState variable j from step 1 is incremented to the next iteration, until st is accomplished.Index To generate even and odd number of scrolls from Algorithm 1. Generating 2-scrolls using saturated function serieswhile j<st:1   p = 02    q = 03    xn.insert4    yn.insert5    if x1 > B[q] and x1 < B[q+1]:6    d1PWL = d1∗k[p]7     while q < 2∗(n-1):8    if x1 >= B[q+1] and x1 <= B[q+2]:9     d1PWL = d1∗(((k[p+1]-k[p])/(B[q+2]-B[q+1]))∗(x1-((B[q+1]+B[q+2])/2))+((k[p+1]+k[p])/2))10      elif x1 > B[q+2] and x1 < B[q+3]:11     d1PWL = d1∗k[p+1]12      q = q+213     p = p+114     zn.insert)15    x1 = xn[j]16    y1 = yn[j]17    z1 = zn[j]18    j = j+119    f(x) must be described as shown in Eqs  is lower than the break point B[0]. If it is satisfied thenVerifies if PWL is equated to m∗(x-B[0])+k[0]), where m is the slope, B[0] the break point and k[0] the amplitudeVariable n − 3) times, where n is the number of scrolls being generatedLoop to iterate (2x in f(x) is between the break points B[j] and B[j+1]. If it is satisfied thenVerifies if PWL is equated to m[j]∗(x-B[j+1-j1])+k[j+1-j1]j is equal to n-3. If it is satisfied thenVerifies if j is equated to j+2Index j1 is equated to 1Index j fails the above thenVerifies if j is equated to j+1Index x in f(x) is between the break points B[n-2] and B[n-1]. If it is satisfied thenVerifies if PWL is equated to m[n-2]∗x, where m[n-2] correponds to slope m1 or m2 in Eqs (2 in Eqs  and 6)PWL is eqx is higher than B[2∗n-3]. If it is satisfied thenVerifies if PWL is equated to m∗(x-B[2∗n-3])+k[2∗n-3]x[n] is updated to x+h∗), where alpha is coefficient α in x and y are initial conditions, and h is the step sizeState variable y[n] is updated to y+h∗gamma∗(x-y+z), where gamma is coefficient γ in z is the initial conditionState variable z[n] is updated to z+h∗(-beta∗y), where beta is coefficient β in Eq (4)State variable x is updated to the next iterationState variable y is updated to the next iterationState variable z is updated to the next iterationState variable i from step 1 is incremented to the next iteration, until st is accomplished.Index For Chua’s circuit based on negative slopes, to generate even and odd number of scrolls from n in Eqs  and 6),,f(x) musAlgorithm 2. Generating 2-scrolls using negative slopeswhile i < st:1   j = 02    j1 = 03    if x < B[0]:4    PWL = m∗(x-B[0])+k[0])5     while j < 2∗n-3:6    if x >= B[j] and x < B[j+1]:7     PWL = m[j]∗(x-B[j+1-j1])+k[j+1-j1]8      if j == n-3:9      j = j + 210       j1 = 111       else:12      j = j + 113       if x >= B[n-2] and x < B[n-1]:14     PWL = m[n-2]∗x15      if x >= B[2∗n-3]:16     PWL = m∗(x-B[2∗n-3])+k[2∗n-3]17      xn.insert))18    yn.insert)19    zn.insert)20    x = xn[i]21    y = yn[i]22    z = zn[i]23    i = i + 124    f(x) must be described as shown in Eqs (st timesLoop to iterate x in f(x) is lower than the break point B[0]. If it is satisfied thenVerifies if PWL is equated to m(x − B[0]) + k[0], where m is the slope, B[0] the break point and k[0] the amplitudeVariable x in f(x) is higher than B[n-2], where n is the number of scrolls being generated. If it is satisfied thenVerifies if PWL is equated to m∗(x-B[n-2])+k[2∗n-3], where m is the slope, B[n-2] the break point and k[2∗n-3] the amplituden-2 timesLoop to iterate x in f(x) is between the break points B[j] and B[j+1]. If it is satisfied thenVerifies if PWL is equated to m[j]∗(x-(B[j+1]+B[j])/2)x[n] is updated to x+h∗alpha(y-PWL)), where alpha is coefficient α in x and y are initial conditions, and h is the step sizeState variable y[n] is updated to y+h∗gamma(x-y+z), where gamma is coefficient γ in z the initial conditionState variable z[n] is updated to z+h(-beta∗y), where beta is coefficient β in Eq (7)State variable x is updated to the next iterationState variable y is updated to the next iterationState variable z is updated to the next iterationState variable i from step 1 is incremented to the next iteration, until st is accomplished.Index For Chua’s circuit based on sawtooth function, to generate even and odd number of scrolls from n in Eqs  and 9),,f(x) musAlgorithm 3. Generating 2-scrolls using sawtooth functionwhile i < st:1   if x <= B[0]:2    PWL = m∗(x-B[0])+k[0]3     elif x > B[n-2]:4    PWL = m∗(x-B[n-2])+k[2∗n-3]5     for j in range(n-2):6    if x > B[j] and x <= B[j+1]:7     PWL = m[j]∗(x-(B[j+1]+B[j])/2)8      xn.insert))9    yn.insert)10    zn.insert)11    x = xn[i]12    y = yn[i]13    z = zn[i]14    i = i+115    From the pseudocodes listed above, one can infer the kind of digital hardware for generating chaotic behavior. For instance, to generate more than 2-scrolls or odd scrolls, one just needs to extend the PWL descriptions from Eqs , 5), ,  and 9),,9), and k1 = -1, k2 = 1, B1 = -0.0165, B2 = 0.0165, a = 0.7, b = 0.7, c = 0.7, and d1 = 0.7. The double-scroll attractor is shown in For the chaotic oscillator based on saturated function series: m1 = -0.276, m2 = -3.3036, B1 = -0.1, B2 = 0.1, α = 10, β = 15, γ = 1, k1 = 0.3036, k2 = -0.3036. The double-scroll attractor is shown in For Chua’s circuit using negative slopes: m = 0.25, B1 = 0, α = 10, β = 16, γ = 1 and k = 0.25. The double-scroll attractor is shown in For Chua’s circuit using sawtooth function: To compute the fixed-point format being used for the VHDL descriptions, one needs to simulate the chaotic oscillator to know parameters like coefficient values and PWL characteristics, namely: break points, amplitudes, and slopes. Those parameter values for generating 2-scroll attractors for the three chaotic oscillators are the following:Other simulation results for generating 30-scrolls are shown in Figs From the simulation results, one can identify the ranges of the state variables and parameters that will serve to define the computer arithmetic to translate simulation parameters to VHDL-code. The binary digits will have integer and fractional parts and our algorithm converts real numbers to their 2’s complement. Basically, the integer part is converted through successive divisions and the fractional part with multiplications. From the 2’s complement numbers, one can generate VHDL code by interconnecting the required blocks that solve a discretized system of equations, e.g. Eqs , 11) an an11) anFor example: Using 32 bits, the ranges of the state variables for generating 2-scrolls in Again, by using 32 bits, the ranges of the state variables for generating 30-scrolls in As one can infer, an important issue when implementing a dynamical system using fixed-point arithmetic is related to the necessary number of bits for the fractional part. The required number of bits can be estimated by trial and error techniques until observing the number of scrolls in the phase-space portrait, as already shown in , ch. 8. e1, e2, e3] = I, and I is the identity matrix of size 3 × 3. Thus, ei, for i = 1, 2, 3, are each unitary column vector of the identity matrix I.To measure the Lyapunov exponents, the initial state of the chaotic oscillator is set tot > 0, then the states in the three new observational systems will be y = T. The observational system is integrated by several steps until an orthonormalization period TO is reached. After this, the state of the variational system is orthonormalized by using the standard Gram-Schmidt method. The next integration is carried out by using the new orthonormalized vectors as initial conditions T is the state after the matrix  is orthonormalized, then the Lyapunov exponent λi, for i = 1, 2, 3 is evaluated byThe Lyapunov exponents measure the long time sensitivity of the flow in TO was set to 5 seconds and the period T to 2000 seconds, respectively. The number of bits for the integer part was established by looking at the phase-space portraits of the expanded system, so that 8-bits are good enough for the integer part. For the fractional part, we test the behavior by using from 10 to 30 bits, therefore, In this work the simulation of the expanded system was carried on with fixed integer arithmetic, using Forward Euler method with a time-step of 0.001. The computation of the Lyapunov exponents using st listed in the pseudocodes in Sect. 3. It also processes the initial conditions from the first iteration loop.As detailed in , 21, thex and y are easy to implement as shown in z needs more hardware, as shown in From the discretized equations of the oscillator based on saturated functions in x and their corresponding PWL functions for negative slopes and sawtooth. As state variables y and z have the same description from Eqs x and theAgain, for both Chua’s chaotic oscillators based on negative slopes and sawtooth function, the number of building blocks required in the Chaotic Oscillator Unit from Very High-Speed Integrated Circuit Hardware Description Language, it was developed around 1980 at the request of the U.S. Department of Defense. At the beginning, the main goal of VHDL was the electric circuit simulation; however, tools for synthesis and implementation in hardware based on VHDL behavior or structure description files were developed later. With the increasing use of VHDL, the need for standardized was generated. In 1986, the Institute of Electrical and Electronics Engineers (IEEE) standardized the first hardware description language through the 1076 and 1164 standards. Nowadays, VHDL is technology/vendor independent, then VHDL codes are portable and reusable.VHDL is the acronym of In Subsection 4.1 one can see the main parameters of a 2-scroll chaotic oscillator, like: coefficient values, break points, amplitudes, and slopes. They are used to perform a high-level simulation to identify the ranges of the state variables and iteration parameters that serve to define the computer arithmetic to translate simulation parameters to VHDL-code. Our approach can automatically generate VHDL code by interconnecting the required blocks that solve a discretized system of equations, e.g. Eqs –12), as, as12), Step 1: Select the number of scrolls being generated, coefficient values and characteristics of the PWL function of the desired multi-scroll chaotic oscillator. The break points are directly related to the number of scrolls and must be provided from the left (the most negative) to the right (the most positive), according to Figs Step 2: The high-level simulation is performed and our approach verifies the number of desired scrolls. From simulation data, the fixed-point format is established, as discussed in Subsection 4.1, where the ranges in Figs Step 3: Our approach creates a file containing the libraries and code for the blocks implementing The VHDL-codes for all the required blocks are shown in the following Algorithms. As one sees, they are described using 28 bits as the word length. This is not an issue, since our approach can generate the codes automatically. The signals are also described for each case.CLK: This is the master clock of the FPGA. All the operations are performed based on the clock frequency.RST: Reset of the system, puts the outputs to zero. Restarts the system.N1 and N2: Input data.OUTS: Output data.sena1: Internal signal with the double of bits from the word length to perform the multiplication.The blocks multiplier, adder and subtractor have the following signal descriptions:Algorithm 4. Multiplier in VDHLentity multiplier is1   port;5    N2: in std_logic_vector(27 downto 0);6    OUTS: out std_logic_vector(27 downto 0) := (others=>‘0’));7    end multiplier;8   architecture code of multiplier is9   signal sena1: signed(55 downto 0) := (others=>‘0’);10   begin11   process12   begin13   if RST = ‘1’ then14    OUTS <= (others => ‘0’);15     elsif rising_edge(CLK) then16    sena1 <= (signed(N1)∗signed(N2));17     OUTS <= std_logic_vector(sena1(51 downto 24));18     end if;19    end process;20   end code;21   Algorithm 5. Adder in VHDLentity adder is1   port;5    N2: in std_logic_vector(27 downto 0);6    OUTS: out std_logic_vector(27 downto 0) := (others => ‘0’));7    end adder;8   architecture code of adder is9   begin10   process11   begin12   if RST = ‘1’ then13    OUTS <= (others => ‘0’);14     elsif rising_edge(CLK) then15    OUTS <= std_logic_vector(signed(N1) + signed(N2));16     end if;17    end process;18   end code;19   Algorithm 6. Subtractor in VHDLentity subtractor is1   port;5    N2: in std_logic_vector(27 downto 0);6    OUTS: out std_logic_vector(27 downto 0) := (others => ‘0’));7    end subctractor;8   architecture code of subtractor is9   begin10   process11   begin12   if RST = ‘1’ then13    OUTS <= (others => ‘0’);14     elsif rising_edge(CLK) then15    OUTS <= std_logic_vector(signed(N1) − signed(N2));16     end if;17    end process;18   end code;19   Algorithm 7. Iterations control block in VHDLentity IterCtrl is1   port;5    Yini: in std_logic_vector(27 downto 0);6    Zini: in std_logic_vector(27 downto 0);7    Xout: out std_logic_vector(27 downto 0) := (others => ‘0’);8    Yout: out std_logic_vector(27 downto 0) := (others => ‘0’);9    Zout: out std_logic_vector(27 downto 0) := (others => ‘0’));10    end IterCtrl;11   architecture code of IterCtrl is12   signal dz: std_logic_vector(27 downto 0) := “0000000000000000000000000000”;13   signal dy: std_logic_vector(27 downto 0) := “0000000000000000000000000000”;14   signal dx: std_logic_vector(27 downto 0) := “0000000101100110011001100110”;15   begin16   process17   variable count: integer := 0;18   begin19   if RST = ‘1’ then20    dx <= (others => ‘0’);21     dy <= (others => ‘0’);22     dz <= (others => ‘0’);23     elsif rising_edge(CLK) then24    if count = 8 then25     count := 0;26      dx <= Xini;27      dy <= Yini;28      dz <= Zini;29      else30     count := count +1;31      end if;32     end if;33    end process;34   Xout <= dx;35   Yout <= dy;36   Zout <= dz;37   end code;38   CLK and RST are the same for all blocks.Xini, Yini and Zini: Feeds input signals to the Chaotic Oscillator Unit, until accomplishing st iterations as sketched in Sect. 3.st iterations.Xout, Yout and Zout: Output data that will be processed by the Iterations control block to perform dx, dy and dz: Initial conditions for each iteration. These registers save the output data of the state variables and then feeds them as input signals Xini, Yini and Zini.count: Signal to update the initial values to the Chaotic Oscillator Unit at each iteration. This signal is enabled after 8 clock cycles (CLKs), which is the time required by the maximum number of series-connected blocks.Iterations control block (Algorithm 7) has the signals:Algorithm 8. Comparator for implementing saturated functions in VHDLentity comparator is1   port;5    dato_sat1: in std_logic_vector(27 downto 0);6    dato_pen0: in std_logic_vector(27 downto 0);7    dato_X: in std_logic_vector(27 downto 0);8    dato_S: out std_logic_vector(27 downto 0) := (others => ‘0’));9    end comparator;10   architecture code of comparator is11   constant B0: std_logic_vector(27 downto 0) := “1000000000000000000010101000”;12   constant B1: std_logic_vector(27 downto 0) := “1111111110111100011010101000”;13   constant B2: std_logic_vector(27 downto 0) := “0000000001000011100101011000”;14   constant B3: std_logic_vector(27 downto 0) := “0111111111111111111111101111”;15   begin16   process17   begin18   if RST = ’1’ then19    dato_S<= (others => ‘0’);20     elsif rising_edge(CLK) then21    if dato_X > B0 AND dato_X < B1 then22     dato_S <= dato_sat0;23      elsif dato_X > B2 AND dato_X < B3 then24     dato_S <= dato_sat1;25      else26     dato_S <= dato_pen0;27      end if;28     end if;29    end process;30   end code;31   The PWL functions are implemented using comparators. If the PWL segments increase, as detailed in Sect. 3, then also the number of comparisons do. The comparator blocks have CLK and RST signals as the previous blocks.dato_sat0: Input data for the first level of saturation.dato_sat1: Input data for the second level of saturation.dato_pen0: Input data for the slope.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_sat0, dato_sat1, or dato_pen0.B0, B1, B2 and B3: Constants to represent the break points from the left to the right of the PWL functions.The PWL function for the oscillator based on saturated functions is implemented by Algorithm 8, where:dato_pen0: Input data for the first slope located on the left of the PWL function.dato_pen1: Input data for the last slope, which is the same as the first.dato_penm: Input data for the central slope.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_pen0, dato_pen,1 or dato_penm.B0 and B1: Constants to represent the break points from the left to the right of the PWL functions.The PWL function for the oscillator based on negative slopes is implemented by Algorithm 9, where:Algorithm 9. Comparator for implementing negative slopes in VHDLentity comparador is1   port;5    dato_pen1: in std_logic_vector(27 downto 0);6    dato_X: in std_logic_vector(27 downto 0);7    dato_S: out std_logic_vector(27 downto 0) := (others => ‘0’));8    end comparador;9   architecture complicado of comparador is10   constant B0: std_logic_vector(27 downto 0) := (others => ‘0’)11   begin12   process13   begin14   if RST = ’1’ then15    dato_S<= (others => ‘0’);16     elsif rising_edge(CLK) then17    if dato_X < B0 then18     dato_S <= dato_pen0;19      else20     dato_S <= dato_pen1;21      end if;22     end if;23    end process;24   end complicado;25   Algorithm 10. Comparator for implementing sawtooth function in VHDLentity comparator is1   port;5    dato_pen1: in std_logic_vector(27 downto 0);6    dato_penm: in std_logic_vector(27 downto 0);7    dato_X: in std_logic_vector(27 downto 0);8    dato_S: out std_logic_vector(27 downto 0) := (others => ‘0’));9    end comparator;10   architecture complicado of comparador is11   constant B0: std_logic_vector(27 downto 0) := “1111111100110011001100110100”;12   constant B1: std_logic_vector(27 downto 0) := “0000000011001100110011001100”;13   begin14   process15   begin16   if RST = ’1’ then17    dato_S<= (others => ‘0’);18     elsif rising_edge(CLK) then19    if dato_X < B0 then20     dato_S <= dato_pen0;21      elsif dato_X >= B1 then22     dato_S <= dato_pen1;23      else24     dato_S <= dato_penm;25      end if;26     end if;27    end process;28   end complicado;29   dato_pen0: Input data for the first slope.dato_pen1: Input data for the last slope, which is the same as the first.x to perform comparisons.dato_X: Input data of the state variable dato_S: Output data that takes the value from one input like dato_pen0, dato_pen1.B0: Constant to represent the break point.The PWL function for the oscillator based on sawtooth function is implemented by Algorithm 10, and it is similar as the last two comparators, where:The generated VHDL-code is ready to be synthesized into an FPGA, so that the following Section shows experimental results.x is shown on the top left-side, y on the bottom left-side, and the phase-space portrait x − y on the right-side. One can count the number of scrolls from the phase-space portraits, and from Lyapunov exponents evaluation, one can infer that the more scrolls are generated the more complex behavior. In this manner, and from the experimental results, it is clear that engineering applications like in [The generated VHDL-codes for the synthesis of multi-scroll chaotic oscillators based on PWL functions were implemented in the Altera’s FPGA EP4CGX150DF31C7 Cyclone IV GX. The used resources are listed in We have introduced an approach programmed in Python for the generation of VHDL-code associated to multi-scroll chaotic oscillators that are based on PWL functions. The pseudocodes for simulating three kinds of chaotic oscillators were listed to infer the FPGA implementation of the PWL functions based on saturated functions series, negative slopes and sawtooth one. From the high-level simulation, our algorithm determines the fixed-point format to interconnect digital blocks associated to the discretized equations of the chaotic oscillators. The PWL functions are then implemented by using comparator blocks that can increase the number of comparisons according to the number of scrolls being generated. It was highlighted that these tasks are performed from the description of the dynamical equations and PWL functions, to the interconnection of the digital blocks and generation of the VHDL-code, which is portable, reusable and open source to be synthesized in an FPGA of any vendor.It is worthy mentioning that the FPGA resources depend on the discretization approach. For instance, Subsection 4.2 showed that Forward Euler and the fourth-order Runge-Kutta methods computed similar values of the Lyapunov exponents, and also when using floating point and fixed integer arithmetic. However, the number of bits for the fractional part matters. As shown in Finally, it can be concluded that our approach for generating VHDL descriptions can estimate the number of required blocks from the equations listed in


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Genome rearrangement describes gross changes of chromosomal regions, plays an important role in evolutionary biology and has profound impacts on phenotype in organisms ranging from microbes to humans. With more and more complete genomes accomplished, lots of genomic comparisons have been conducted in order to find genome rearrangements and the mechanisms which underlie the rearrangement events. In our opinion, genomic comparison of different individuals/strains within the same species (pan-genome) is more helpful to reveal the mechanisms for genome rearrangements since genomes of the same species are much closer to each other.Pseudomonas aeruginosa and Escherichia coli, and investigate the inversion events among different strains of the same species. We find an interesting phenomenon that long  inversion regions are flanked by a pair of Inverted Repeats (IRs). This mechanism can also explain why the breakpoint reuses for inversion events happen. We study the prevalence of the phenomenon and find that it is a major mechanism for inversions. The other observation is that for different rearrangement events such as transposition and inverted block interchange, the two ends of the swapped regions are also associated with repeats so that after the rearrangement operations the two ends of the swapped regions remain unchanged. To our knowledge, this is the first time such a phenomenon is reported for transposition event.We study the mechanism for inversion events via core-genome scaffold comparison of different strains within the same species. We focus on two kinds of bacteria, Pseudomonas aeruginosa and Escherichia coli strains, IRs were found at the two ends of long sequence inversions. The two ends of the inversion remained unchanged before and after the inversion event. The existence of IRs can explain the breakpoint reuse phenomenon. We also observed that other rearrangement operations such as transposition, inverted transposition, and inverted block interchange, had repeats (not necessarily inverted) at the ends of each segment, where the ends remained unchanged before and after the rearrangement operations. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events.In both The online version of this article (doi:10.1186/s12864-017-3655-0) contains supplementary material, which is available to authorized users. Comparative genomics studies show that genome rearrangement events often occur between two genomes. Genome rearrangement events play important role in speciation. The rearrangement operations include deletions, insertions, inversion, transposition, block interchange, translocation, fission and fusion, etc. Here we study the mechanism for inversion events via core-genome scaffold comparison of different strains within the same species.Drosophila melanogaster and two closely related species,D. simulans and D. yakuba, and reconstructed the molecular events that underlie their origin [P-value <10−3) [P<10−4).By comparing two genomes, we can find candidate rearrangement operations. However, the set of rearrangement operations to transform one genome into the other is not unique in many cases. Computing the rearrangement operations between two genomes under different assumptions is an active area, where intensive research have been conducted . It is rr origin . Rajarame <10−3) . Darmon e <10−3) . The asse <10−3) , 9–12. Ae <10−3) . They alDrosophila genus [Saccharomyces pastorianus [Pevzner and Tesler found extensive breakpoint reuse for inversion events in mammalian evolution when comparing human and mouse genomic sequences –15. Statla genus , 19 as wtorianus .It is well known that recombination (crossing-over) of homologous or non-homologous DNAs can lead to various genetic variations including inversions, transpositions, insertions/deletions, and will leave some direct or inverted repeats on both ends, and existing repeats can further promote more variations.Pseudomonas aeruginosa has been done by Ozer et al. [To study the rearrangement operations, comparison of different individuals/strains within the same species (pan-genomes) can be more helpful since strains within the same species are conserved. A pan-genome, or supra-genome, describes the full complement of genes in a clade , which can have large variation in gene content among closely related strains. Pan-genomes were first studied by Tettelin more than a decade ago . Severalr et al. . For panPseudomonas aeruginosa and Escherichia coli, and investigate the inversion events among different strains of the same species. We find an interesting phenomenon that long  inversion regions are flanked by pairs of Inverted Repeats (IRs) which are often Insertion Sequences (ISs). This mechanism also explains why the breakpoint reuses for inversion events happen. We study the prevalence of the phenomenon and find that it is a major mechanism for inversions. The other observation is that for different rearrangement events such as transposition and inverted block interchange, the two ends of the swapped regions are also associated with repeats so that after the rearrangement operations the two ends of the swapped regions remain unchanged. To our knowledge, this is the first time such a phenomenon is reported for transposition event.We study two types of bacteria, We develop a pipeline to generate the core-genome blocks, dispensable blocks and strain-specific blocks based on the multiple sequence alignment produced by Mugsy .Pseudomonas aeruginosa, before merging, there are 185 blocks in the core genome of the 25 strains. After merging, the scaffolds contain 69 blocks.We then develop a computer program to generate the scaffolds of the strains from the core-genome blocks by repeatedly merging two consecutive blocks appearing in all the strains of the same species. In this way, the number of distinct blocks in the core-genome scaffold is reduced dramatically. For example, for After that, we compute the inversion distance between two scaffolds. Computing the inversion distance between two scaffolds is a very hard and complicated combinatorial problem. Several algorithms have been developed. Due to the difficulty of algorithm design, most of the algorithms only consider inversion events. However, a transposition/block-interchange event can be represented as 3 inversion events, and an inverted transposition/block-interchange event can be represented as 2 inversion events. Therefore, some of the computed inversion events may not be real. There are algorithms dealing with inversion and other rearrangement events such as block interchanges simultaneously. However, the weights for different events are different . Thus, those algorithms still suffer from the problem of outputting inversions that are not real.Our strategy here is to eliminate some obvious (independent) transposition, inverted transposition, block interchange, and inverted block interchange events before computing the inversion distance between two scaffolds.G1=+1+2…+n is the first input scaffold and G2=π1π2…πn is a sign permutation of the n blocks over the set N={1,2,…,n} of n distinct blocks, where each integer i∈N appear once in G2 in the form of either +i or −i. All the rearrangement operations are on G2.For simplicity, we always assume that transposition swaps the order of two consecutive blocks/regions without changing their signs. A transposition  on regions πi,…,πj−1 and πj…πk−1 transforms the sign permutation π1…πi−1πi…πj−1πj…πk−1πk…πn into π1…πi−1πj…πk−1πi…πj−1πk…πn.A independent if it transforms the sign permutation π1…πi−2πi−1πi+1πiπi+2πi+3…πn into π1…πi−2πi−1πiπi+1πi+2πi+3…πn, where πi−1πiπi+1πi+2 is either +(q−1)+q+(q+1)+(q+2) or −(q+2)−(q+1)−q−(q−1) for {q−1,q,q+1,q+2}⊆N={1,2,…,n}. Though an independent transposition swaps two consecutive blocks πi+1 and πi instead of two regions πi,…,πj−1 and πj…πk−1 as in the definition of a general transposition, a pre-process allows us to merge two consecutive blocks if they are consecutive in both input genomes. Thus, we can still handle some cases for swapping two consecutive regions. For example, the genome +1+2+6+7+3+4+5+8 becomes +1+2+4+3+5 after merging +6+7 (represented as +4)and +3+4+5 (represented as +3) and re-number +8 as +5 in the new representation. An independent transposition can change +1+2+4+3+5 into +1+2+3+4+5. In terms of breakpoint graph, the two blocks πi+1πi in an independent transposition is involved in a 6-edge cycle and after the transformation the 6-edge cycle becomes three 2-edge cycles. In other words, the three breakpoints involved in the 6-edge cycle disappear after the transformation. See Fig. A transposition is inverted transposition swaps the order of two consecutive blocks/regions with one of the block’s sign changed. An inverted transposition  on regions πi,…,πj−1 and πj…πk−1 transforms the sign permutation π1…πi−1πi…πj−1πj…πk−1πk…πn into π1…πi−1−πk−1…−πjπi…πj−1πk…πn or π1…πi−1πj…πk−1−πj−1…−πiπk…πn.An independent if it transforms the sign permutation π1…πi−2πi−1−πi+1πiπi+2πi+3…πn or π1…πi−2πi−1πi+1−πiπi+2πi+3…πn into π1…πi−2πi−1πiπi+1πi+2πi+3…πn, where πi−1πiπi+1πi+2 is either +(q−1)+q+(q+1)+(q+2) or −(q+2)−(q+1)−q−(q−1) for {q1,q,q+1,q+2}⊆N={1,2,…,n}.An inverted transposition is block interchange swaps the locations of two separated blocks without changing their signs. A block interchange  on regions πi…πj and πk…πl transforms π1…πi−1πk…πlπj+1…πk−1πi…πjπl+1 … πn into π1…πi−1πi…πjπj+1…πk−1πk…πlπl+1…πn.A independent if it transforms the sign permutation π1…πi−1πkπi+1…πk−1πiπk+1…πn into π1…πi−1πiπi+1…πk−1πkπk+1…πn, where πi−1πiπi+1 is either +q+(q+1)+(q+2) or −(q+2)−(q+1)−q and πk−1πkπk+1 is either +p+(p+1)+(p+2) or −(p+2)−(p+1)−p for {q,q+1,q+2}⊆N and {p,p+1,p+2}⊆N. Similarly, the two blocks πk and πi are involved in two (interleaving) 4-edge cycles in the breakpoint graph and after the transformation, they become four 2-edge cycles. In other words, there are four breakpoints at the two ends of the two blocks, after the transformation, the four breakpoints disappear. See Fig. A block interchange is +(q+1)+q+ or − on regions πi…πj and πk…πl transforms π1…πi−1−πl…−πkπj+1…πk−1−πj…−πiπl+1…πn into π1…πi−1πi…πjπj+1…πk−1πk…πlπl+1…πn.An independent if it transforms the sign permutation π1…πi−1−πkπi+1…πk−1−πiπk+1…πn into π1…πi−1πiπi+1…πk−1πkπk+1…πn, where πi−1πiπi+1 is either +q+(q+1)+(q+2) or −(q+2)−(q+1)−q and πk−1πkπk+1 is either +p+(p+1)+(p+2) or −(p+2)−(p+1)−p for {q,q+1,q+2}⊆N and {p,p+1,p+2}⊆N. Again, there are four breakpoints at the two ends of the two blocks −πi and −πk, after the transformation, the four breakpoints disappear.An inverted block interchange is After eliminating independent transposition, inverted transposition, block interchange and inverted block interchange events, we use GRIMM-Synteny , 28 to cFinally, we developed a pipeline to compare sequences at the two ends of each inversion region to see whether a pair of inverted repeats exists. Once the inverted repeats are found, the pipeline can also search all the strains and mark down its positions in different strains.Pseudomonas aeruginosa strains PACS2, F22031,NCGM1900, LES431, NCGM2.S1, Carb01_63, SCV20265, UCBPP-PA14, VRFPA04, DSM_50071, 19BR, 213BR, B136-33, PA7, PA1, YL84, LESB58, M18, RP73, DK2, MTB1, PAO1, PA1R, NCGM1984,and FRD1 were downloaded from NCBI GenBank. The details of these 25 Pseudomonas aeruginosa strains are listed in Additional file Pseudomonas aeruginosa strain is in Additional file Complete genome sequences of 25 Pseudomonas aeruginosa strains NCGM1984, B136-33, YL84, M18, LESB58, SCV20265, LES431, UCBPP-PA14, DK2, MTB-1, DSM_50071, Carb01_63, and F22031. Group 2 contains 6 strains, which are strains RP73, 213BR, PA1, PA1R, 19BR, and PAO1. Groups 3-8 contain 1 strain each and the respective strains are PACS2, FRD1, NCGM2.S1, VRFPA04, NCGM1900, and PA7.Group 1 contains 13 strains, which are We computed the pairwise inversion distance between scaffolds after eliminating other kinds of independent rearrangement events such as transpositions, inverted-transpositions, block-interchanges, and inverted-block-interchanges. For each of the 8 scaffolds, we chose a scaffold with the minimum inversion distance (after eliminating other independent rearrangement events) to compare. The purpose was to compare two scaffolds with a small number of inversions so that we can observed real inversions between them. From Table B that appears four times in Both Scaffold 1 and Scaffold 5, where B appear as −B once and as +B three times in Scaffold 1. The four occurrences of B form a pair of IRs at the two ends of each of the 3 inversion regions , we can see that for the first inversion (-B-6+B to -B6+B), the real cutting points (breakpoints) are at the left end of -B and the right end of +B, while for the other two inversions  1 to 31) with complete sequences from 17 genome families at NCBI’s GenBank. These 31 strains are SE15, IAI39, EC4115, CFT073, CE10, O103:H2 str. 12009, C227-11, 536, K-12 substr. MG1655, ST2747, NA114, 042, O111:H- str. 11128, O145:H28str.RM13514, O104:H4 str. 2011C-3493, SE11, SS52, APEC O78, SMS-3-5, DH1Ec095, 1303, O157:H7 str. Sakai, 55989, B str. REL606, O83:H1 str. NRG 857C, UMN026, PCN033, 789, O127:H6 str. E2348/69, P12b, and ED1a. The detailed information of these 31 strains is listed in Additional file Escherichia coli strain is given in Additional file We selected 31 Escherichia coli strains EC4115, CE10, C227-11, K-12 substr. MG1655, ST2747, 042 O104:H4 str. 2011C-3493, SE11, SS52, APEC O78, DH1Ec095, 1303, O157:H7 str. Sakai, 55989, B str. REL606, O83:H1 str. NRG 857C, UMN026, PCN033, 789, O127:H6 str. E2348/69, and ED1a. Group 2 contains 3 strains, SE15, CFT073 and 536. Groups 3-9 contain 1 strain each and the respective strains are O145:H28 str. RM13514, SMS-3-5, P12b, IAI39, O103:H2 str. 12009, NA114, and O111:H- str. 11128.Group 1 contains 21 strains which are After computing pairwise inversion distance among the 9 scaffolds, we selected a scaffold with minimum inversion distance for each of the 9 scaffolds as shown in Table In total, there are 17 inversion events among the 8 distinct pairs in Table For all the inversions listed in Table We find a total of 12 different types of pairs of inverted repeats and use letters from +D/-D to +M/-M, +S/-S and +Q/-Q to label and differentiate these 12 pairs of IRs. The locations of these IRs in the scaffolds are shown in Fig. The three inversion steps from Scaffolds 1 to 7 are illustrated in Fig. It is worth pointing out that the two +Ms in Scaffold 1 form a pair of directed repeats (DRs). After inversion , the pair of directed repeats (DRs) of M becomes a pair of inverted repeats. This means that a pair of DRs has the potential to mediate inversions.We find an inverted block interchange between Scaffold 6 and 1 and we use Fig. The other explanation is that an inverted block interchange can be replaced by two inversions. Figure Figure mentclass2pt{minimThe other explanation is that the inverted transposition can be replaced by two inversions: the first inversion is from Blocks 16 to -15 and the second inversion is from Blocks -19 to -16 see Fig. . Both ofPseudomonas aeruginosa and Escherichia coli strains, IRs were found at the two ends of long sequence inversions. The two ends of the inversion remained unchanged before and after the inversion event. We also observed that other rearrangement operations such as transposition, inverted transposition, and inverted block interchange, had repeats (not necessarily inverted) at the ends of each segment, where the ends remained unchanged before and after the rearrangement operations. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events. Past studies reveal that insertions and deletions (indels) can be mediated by directed repeats (DRs) [For both ts (DRs) , 30. Seqts (DRs) , 30. Howts (DRs) .The mechanism for breakpoint reuse is also interesting. The fact that long inversions are flanked by a pair of inverted repetitive elements can clearly explain why breakpoint reuse happens for inversions. Our observations show that the breakpoint reuse is actually the repeated segment reuse. The breakpoints at the nucleotides level for the reused repeat differ depending on the repeat is at the left or right end of the inversion.Pseudomonas aeruginosa and Escherichia coli strains. We have found that repeats were at the ends of different kinds of rearrangement events including inversion, transposition, inverted transposition, and inverted block interchange. In many cases, these repeats keep the ends of rearrangement events unchanged. This suggests that the conservation of ends could possibly be a popular phenomenon in many types of chromosome rearrangement events.In this paper, we have studied the rearrangement events for both Additional file 1Table S3. Information on Repeats found in Pseudomonas aeruginosa strains. Table S4. Scaffolds of each of the 25 Pseudomonas aeruginosa strains. Table S5. Physical positions of repeats and breakpoints of rearrangements in Pseudomonas aeruginosa strains. Table S6. Scaffolds of each of the 31 Escherichia coli Strains. Table S7. Physical positions of repeats and breakpoints of rearrangements in Escherichia coli strains. Table S8. Information on repeats found in the Escherichia coli Strains. (XLSX 103 kb)Additional file 2Cases where multiple transposition or block interchange events happen between two groups. (DOCX 20 kb)


We propose a comprehensive delayed HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We prove the positivity and boundedness of solutions and analyze the global stability of two boundary equilibria and then study the local asymptotic stability and Hopf bifurcation of the positive (infection) equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model by numerical simulation. Hepatitis B virus (HBV) has become one of the serious infectious diseases threatening global human health, which can cause chronic liver infection and further result in liver inflammation, fibrosis, cirrhosis, or even cancer . Each yeMathematical modeling and analysis of the dynamics of such infectious viruses as HBV play important roles in understanding the factors that govern the infectious disease progression and offering insights into developing treatment strategies and guiding antiviral drug therapies . So far,Among these works, the development of virus models with immune responses is gaining much attention , 11, 12.In this paper, we will propose a more comprehensive model than those existing ones, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. We first discuss the existence of two boundary equilibria and one positive (infection) equilibrium. We then analyze the global stability of the two boundary equilibria, the local asymptotic stability and Hopf bifurcation of the positive equilibrium and also the stability of the bifurcating periodic solutions. Moreover, we perform numerical simulations to illustrate some of the theoretical results we obtain and also illustrate how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.The paper is structured as follows. In  Wang et al.  proposedBased on this model, we propose a new and comprehensive HBV model, which not only considers the immune response to both infected cells and viruses and a time delay for the immune system to clear viruses but also incorporates an exposed state and the proliferation of hepatocytes. To better understand our model, we illustrate its mechanism in x,  e,  y,  v,  z, and w denote the number of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and alanine aminotransferases (ALT), respectively. The parameter λ represents the natural production rate of uninfected cells. rx is a new term which is introduced to represent the proliferation of hepatocytes, where r is the proliferation rate. Parameters d, (and the following) a1,  ε,  k4, and k7 represent the natural death rate of uninfected cells, exposed cells, infected cells, free viruses, CTLs, and ALT, respectively. β represents the infection rate from uninfected cells to exposed cells and a2 the transfer rate from exposed cells to infected cells. The production rate of free viruses from infected cells is denoted by k, and the production rate of CTLs by k3. k5 represents the production rate of ALT from the extrahepatic tissue and k1k6 the production rate of ALT when the infected hepatocytes are killed by CTL. The immunity-induced clearance for infected cells is modeled by a term k1yz, where k1 represents the clearance rate of infected cells. Similarly, the immunity-induced clearance for free viruses is modeled by k2vz, where k2 represents the clearance rate of free viruses. τ is time delay. All the parameters in this paper are positive and d > r. For convenience, we define new parameter ρ = d − r.The model is then given as follows:In this subsection, we prove the positivity and the boundedness of solutions of system .x(0) ≥ 0,  e(0) ≥ 0,  y(0) ≥ 0,  w(0) ≥ 0,  v(t) ≥ 0,  z(t) ≥ 0,  t ∈ . From the first equation of system (x(t) = e0t(ρ+βv(s))ds−∫x(0) + λ∫0test[ρ+βv(ξ)]dξ−∫ds; therefore, x(t) ≥ 0 for ∀t > 0 if x(0) ≥ 0. Next, we consider the second, third, and fourth equation in system (e(t), y(t), v(t):We denote f system , we haven system  as a none(t) ≥ 0,  y(t) ≥ 0,  v(t) ≥ 0 if e(0) ≥ 0,  y(0) ≥ 0,  z(0) ≥ 0.Based on Theorem 2.1 in , we havez(t) = ek4t−y(0) + ∫0tk3v(t − γ)z(t − γ)ek4(t−γ)−dγ, we have z(t) ≥ 0,  ∀t > 0 if z(t) ≥ 0,  t ∈ .w(t) = ek7t−w(0) + ek7t−∫0t[(k5 + k1k6y(s)z(s)]ek7t−ds, because y(t), z(t) ≥ 0, so we have w(t) ≥ 0,  ∀t > 0 if w(0) ≥ 0.G(t) as a linear combination of x,  e,  y,  v,  z:Hence, the nonnegative is proved. In what follows, we will study the boundedness of solutions. We define t→∞G(t) ≤ λ/δ, namely, x(t) + e(t) + y(t) + (a1/2k)v(t) + (a1k2/2kk3)z(t + τ) ≤ λ/δ. So we have 0 ≤ x(t), e(t), y(t), v(t), z(t) ≤ λ/δ Because the boundedness of x(t), e(t), y(t), v(t), z(t),  limt→∞w(t) ≤ (k5 + k1k6)λ2/k7δ2.Therefore, we obtain limThe boundedness is proved.In this subsection, we study the equilibria of system . The metE00 in which x ≠ 0,  w ≠ 0,  e = y = v = z = 0 and an equilibrium without immune response E11 in which x ≠ 0,  e ≠ 0,  y ≠ 0,  v ≠ 0,  w ≠ 0,  z = 0) and a positive (infection) equilibrium E22 in which x ≠ 0,  E ≠ 0,  y ≠ 0,  v ≠ 0,  z ≠ 0,  w ≠ 0.The system  has two E00 = , where x0 = λ/ρ, and w0 = k5/k7, and the basic reproductive number is obtained by the following method.The infection-free equilibrium is R0 = ρ(FV−1), whereR0 = a2kx0β/a1ε(a1 + a2).Based on integral operator spectral radius, the basic reproductive number is E11 = , where x1 = a1ε(a1 + a2)/a2kβ,  e1 = (a1ε/a2k)v1,  y1 = (ε/k)v1,  v1 = a2kλ/a1ε(a1 + a2) − ρ/β, and w1 = k5/k7. Similarly, we have the basic reproductive number is R1 = k3v1/k4 + kk3y1k1/a1k2k4 at E11 = .The equilibrium without immune response is E22 = , whereThe infected positive equilibrium is In this section, we will employ the direct Lyapunov method and LaSalle's invariance principle to establish the global asymptotic stability of the two boundary equilibria.E00 is globally asymptotically stable if and only if R0 < 1.The infection-free equilibrium See E11 is globally asymptotically stable if and only if R1 < 1.The equilibrium without immune response See E22.In this section, we will discuss the local asymptotic stability and Hopf bifurcation of the positive equilibrium E22 is as follows:The characteristic equation of system  at E22 DefineH above becomesτ = 0,  lie to the left of the imaginary axis when τ = 0. However, with τ increasing from zero, some of its roots may cross the imaginary axis to the right. In this case, there are some roots having positive real parts, and therefore the equilibrium E22 becomes unstable. Next, we will discuss the stability of system (E22 when τ > 0.From f system  at E22 We first divide  into twoλ = iω  (ω > 0). Substituting λ = iω into  = 0 has a positive real root ω2.Suppose  has a pu iω into  yields = 0 has no positive real roots, then the positive equilibrium E22 is locally asymptotically stable for any τ > 0.If G(x) = 0 has no positive real roots, then obviously  = 0 has j = 0,1, 2,….Substituting  iω into , we obtaH = 0 has a pair of purely imaginary roots j and λnj)((τ) = αnj)((τ) + iωnj)((τ) as the root of (τnj)((τnj) = 0 and Therefore, the characteristic equation  root of  near τnG(x) = 0 has some positive real roots, then E22 is locally asymptotically stable for τ ∈ . Therefore, system  = F1δ(θ)dθ + F2δ(θ + τ)dθ.By the Riesz representation theorem , there eϕ ∈ C, we further defineφ ∈ C, defineϕ, φη(θ) = η and ϕ ∈ C. Then A(0) and A∗(0) are adjoint operators.For n system  can be wh(θ) and h∗(s) be the eigenvectors of A(0) and A∗(0) corresponding to the eigenvalues iω0 and −iω0, respectively. We choose h(θ) and h∗(s) as h∗(s), h(θ)〉 = 1 is satisfied. We give the detailed computation of (Let ∗(s) as hθ=1,h2,hg20,  g11,  g02, and g21, using the method given in  = sign[(dG/dx)|x=xn].Then the following values can be computed:ectively . From (Cu2∗ = −Re(c1(0))/G′(ω02). We obtain the following theorem.Let Assume the hypotheses (1), (2), and (3) at the beginning of u2∗ > 0  (u2∗ < 0), then the bifurcating periodic solutions exist for τ > τ0  (τ < τ0) in a τ0-neighborhood.(1) If β2 < 0  (β2 > 0), the bifurcating periodic solutions are orbitally asymptotically stable as t → +∞  (t → −∞).(2) If τ0 = τn0(0), where c1(0)).(1) If β2 < 0, namely, Re(c1(0)) < 0, then there exist stable periodic solutions for τ > τn0(0) in a τ0-neighborhood. So the bifurcating periodic solutions are orbitally asymptotically stable as t → +∞.(2) If In this section, we will numerically illustrate the theoretical results obtained above and also discuss how the factors such as the immune response to infected cells and viruses and the proliferation of hepatocytes affect the dynamics of the model under time delay.For the following simulations, we choose the parameter values for system  as follox(t) = 1,  e(t) = 1,  y(t) = 1,  v(t) = 1,  z(t) = 1, and w(t) = 1, where t ∈ .We set the initial values to E22 =  and the critical time value τn0(0) = 0.041.With the parameter values given in , we haveτ > 0.041, we obtain stable bifurcating periodic solutions. For example, when τ = 0.05, the simulation result is shown Figures When τ < 0.041, the bifurcating periodic solutions are unstable. For example, when τ = 0.03, the simulation result is shown in Figures E22 is asymptotically stable and the system will converge to E22.When τ), the radius of limit cycle will increase. The simulation result is shown With the increasing of time delay  equilibrium and also the stability of the bifurcating periodic solutions. We also numerically illustrate the Hopf bifurcation and the stability of the bifurcating periodic solutions.Moreover, we numerically illustrate how the factors such as the time delay, the immune response to infected cells and viruses, and the proliferation of hepatocytes affect the dynamics of the model, which shows that the former two factors have a big effect on the model dynamics, while the latter one does not have a big effect.


There is an incorrect figure referenced in the third paragraph of the “3.1 Grouping Taxis by Incomes” section of the Results. In the section titled “Working time”, “Fig 5” should be listed as “Fig 6”.There is an incorrect figure referenced in the fourth paragraph of the “3.1 Grouping Taxis by Incomes” section of the Results. In the section titled “Passenger load”, “Fig 6” should be listed as “Fig 5”.


Scholars of Luke–Acts have struggled to define the apostles’ proclamations of judgment on those who threatened the early Christian community. Ananias and Sapphira (Acts 4.32–5.11), Simon magus (8.4-25) and Bar-Jesus (13.4-12) all fall victim to the apostles’ words of power, yet scholars have typically shied away from categorizing their speeches as curses. Close analysis of the structure, style, phonaesthetic and dramatic aspects of the Greek texts suggests, however, that Luke indeed intends the apostles’ speeches to be heard as curses whilst simultaneously presenting them as legitimate acts of power. A comparison with Greek and Coptic ‘magical’ texts helps to place the curses of Acts in the context of cursing traditions in the wider ancient Mediterranean world. They have generally regarded Paul’s words to Bar-Jesus as a form of a curse,7 An analysis of each episode will, in the first place, attend to the thematic, grammatical and phonaesthetic properties of the apostle’s speech to highlight its dramatic power in a performance setting, both within the narrative and in the narrative’s recitation amongst Christian communities. Secondly, in each case I will employ a comparative perspective to evaluate the curses of Acts in relation to a selection of Greek and Coptic ‘magical’ texts. This approach aims to demonstrate how an ancient audience of Luke–Acts would have heard the apostles engaging with curse traditions typical of ancient Mediterranean culture while also viewing them as divinely authorized acts of power superior to acts of ‘magic’.This article, in contrast, will employ literary analysis to demonstrate key links between these three episodes in Acts, arguing that these conflict narratives should be read in continuity with each other.8 In fact, curses play an important role in the Jewish and Christian scriptures.9 In this article I will follow the more anthropologically informed definition of a curse employed by David Frankfurter in his analysis of blessings and curses in Roman Imperial and Late Antique Egypt (or the subversive power that plagues one following such a performance (as in “her curse still rests on me”)’. However, I would also add the qualification that this subversive power can be operated in a way deemed to be socially legitimate or illegitimate. Thus I use ‘curse’ as a neutral term that is open to construal as either ‘miracle’ or ‘magic’ according to its performer’s perceived authority and status. Following the studies of Poupon, Reimer, Marguerat and others,10 this article will be sensitive to the strategies employed by the author of Acts to navigate a context in which the apostles’ words of power are susceptible to accusations of magic. It will go further than previous studies by placing the strategies themselves within cursing traditions of Late Antiquity, showing greater overlap between them than previously acknowledged and the potential tensions in the claim to distinction found in Acts. Given that scholars widely accept Paul’s words to Bar-Jesus to be a form of a curse, we will begin with that episode, identifying the key features of the curse itself before progressing to discuss the earlier episodes in the narrative.The hesitancy of scholars to categorize the apostles’ judgments as curses may be linked to the continued association of the label ‘curse’ with uncritical conceptions of ‘magic’ that define magic as some impure or primitive form of religion.ue Egypt : 158. Fr11 in a similar manner to the OT prophets . Bar-Jesus is described as a ψευδοπροϕήτης (13.6), the prophets’ natural antagonist.12 Paul is ‘filled with the holy spirit’ (13.9) whereas Bar-Jesus is said to be ‘full of all deceit and villainy’ (13.10). Such rivalry is reminiscent of the prophetic clashes attested throughout the Jewish scriptures, none more famous than Elijah’s challenge to the prophets of Baal (1 Kgs 18.1-40). The text also describes Bar-Jesus as μάγος, which is reminiscent of Moses and Aaron’s encounter with the magicians of Pharaoh (Exod. 7–9). These intertextual frames prepare the audience to expect a power conflict between Paul and Bar-Jesus.The beginning of Acts 13.4-12 sets the scene in Gentile Salamis and introduces some of the key themes that are to follow in the narrative. It employs intertextual allusions in order to parallel Paul with Bar-Jesus and to posit the pair as natural opponents. Paul is ‘sent out by the Holy Spirit’ (13.4),Paul’s speech can be divided into the following parts and contains three themes that are also found in Peter’s pronouncements in Acts 4.32–5.11 and 8.9-25.Paul begins by re-labelling his opponent. He then asks a rhetorical question which provides an effective contrast to his previous statements. His speech comes to a climax as he vividly curses his opponent by ‘the hand of the Lord’.Paul’s pronouncement is notable for its direct style, created by its repeated re-labelling of the adversary. In addressing Bar-Jesus (‘son of Jesus’) as υἱὲ διαβόλου (13.10), Paul employs irony to accuse him of being an impostor : 617. InThe first of these three themes is the presence of the demonic, introduced through the re-branding of Bar-Jesus as υἱὲ διαβόλου. This designation is a major element in creating Paul’s direct style of address. Its position in the syntax of the sentence also emphasizes deceit and villainy and possibly implies that the label ‘son of the devil’ is dependent on or deducible from Bar-Jesus’ actions.The second theme, implicit in this instance, is the adversary’s loss of status in the community of God. Despite appearing to be a ᾽Ιουδαῖος, the text reveals that Bar-Jesus is in fact a ψευδοπροϕήτης (13.6), effectively undermining his legitimacy as a member of the house of Israel. In fact, by describing him as a μάγος Luke depicts Bar-Jesus as someone more akin to an antagonistic Gentile. In addition to identifying him as a μάγος, the opening verses (6-7) characterize him in a fashion strongly reminiscent of the court magicians in the Jewish scriptures who attend the palaces of Gentile kings . Such intertextual allusions contribute to the re-labelling of Bar-Jesus as an outsider in relation to the community, as impious and atheistic as a Gentile magician. Paul’s curse completes the stripping of Bar-Jesus’ status as a Jewish holy man when he challenges him (13.10): οὐ παύσῃ διαστρέϕων τὰς ὁδοὺς [τοῦ] κυρίου τὰς εὐθείας; Paul effectively accuses his opponent of perverting the nation, an accusation traditionally made against those who promoted idolatry and impurity in the OT  and for which dramatic punishments were commanded : 110. Pafull of all deceit and villainy’ (13.10). Paul’s searching gaze (13.9) reveals that Paul has insight into his adversary’s inner self, an ability commonly associated with God and his prophets in the OT, as well as with Jesus in the gospels.13The third theme is the impurity of the adversary’s heart. Paul describes Bar-Jesus as ‘Finally, the reaction of the witness to Paul’s miracle is also significant: ‘When the proconsul saw what had happened, he believed’ (13.12). Paul’s curse, like his more positive acts of power, leads to conversion or else a newly found respect for the Christian movement.defixiones, amulets, ostraca and inscriptions provide valuable insights into the practice of cursing in ancient Mediterranean societies.14 Whilst the dating of the Coptic Christian curses in Meyer and Smith’s collection varies widely (first century ce to eleventh or twelfth century), the texts in Preisendanz’s Papyri Graecae Magicae . Whereas binding curses from classical Athens are very formulaic, curses of the imperial era are far more elaborate and explicit. In addition to appeals’ : 13-14. defixiones in the way that they often address the local gods of the institutional cults, in their lack of magical formulae and their use of quasi-legal language, in their supplicatory posture, and in their desire for recompense on named enemies rather than anonymous ones , constitute a type of curse distinct from that of public cursing , suggesting the immediate fulfilment of Paul’s pronouncement.28At first glance, Paul’s judgment on Bar-Jesus appears the most curse-like because of his direct style of address and use of particular motifs. His use of the second person imperative (13.11a: καὶ νῦν ἰδοὺ) draws the comparison to the mode of Faraone’s first category of ‘direct curse’, ‘which is a performative utterance, that is, a form of incantation by which the tic way’ : 10. AltPGM IV.467-68 and IV.831-32 quote Homer’s Iliad 8.424 in the form of a question.Paul’s utterance also shares a number of stylistic features and motifs with the curse texts. Paul’s use of epiplexis in his curse on Bar-Jesus  mirrors the style of some Greek curse texts that employ epiplexis seemingly as part of their rhetorical strategy. τολμήσεις Διὸς ἄντα πελώριον ἔγχος ἀεῖραι;29Will you dare to raise your mighty spear against Zeus?Devices such as rhetorical questions serve to heighten the drama of the curse ritual and, if performed publicly, had the potential to intimidate one’s opponent.PGM XIVc.16-27 uses καταβάλλω to convey sudden, harmful action as well as a spatial dynamic of descent.The most obvious parallel between Paul’s curse and Greek and Coptic curse spells is the desire for the deity to strike the spell-caster’s adversary. ἧκέ μοι κ[αὶ] βάδισον καὶ κατάβαλε τὸν δεῖνα (ἢ τὴν δεῖνα) ῥίγει καὶ πυ|ρετῷ αὐτὸς ἠδ[ί]κησέν με καὶ τὸ ἇιμα τοῦ Τυϕῶνος ἐξέχυσεν παρ᾽ ἑαυ|τῷ (ἢ αὑτῇ). διὰ το[ῦ]το ταῦτα ποιῶPGM XIVc.25-27 = PDM xiv.690-94, trans. ‘Come to me and go and strike down him, NN  with chills and fever. That very person has wronged me and he (or she) has spilled the blood of Typhon in his own (or her own) house. For this reason I am doing this’ . Another implores: ‘You must strike Prestasia and Tnounte and Eboneh, quickly, deservedly’.30Such requests are common in Coptic spell texts. One speaker orders the deity to ‘strike Philadelphe and her children’ . The vivid image of descent conjured for the audience reflects the descent of Bar-Jesus’s social status as he submits to the apostle.Many curse texts also emphasize the spatial movement of descent. They speak of bringing their victims ‘down’ in some manner : 121-23,PGM III.494-611 appeals to a deity who ‘lowers darkness’.Another parallel between the two groups of texts is their choice of affliction. Multiple Greek and Coptic curse texts seek to inflict blindness upon their enemies. δεῦρό μοι,|| κύριε, ὁ ποτὲ τ[ὸ] ϕῶς ἀνά[γ]ων, ποτὲ τὸ σκότος κατά|γων <κατὰ> τὴν σεαυτοῦ δῦναμιν […]‘Come to me, / lord, you who sometimes raise the light, sometimes lower the darkness [with] your own power’ .33 One Coptic curse asks the deity to ‘cause their eyes to fog and come out’ . Another requests blindness to fall on its victim’s ‘two eyes’ .Again, this spell evokes a descending spatial dynamic through its use of κατά in κατάγων. Similarly, the prayer in PGM VII.764-65 describes the deity’s ability to ‘wane from light into darkness’ (trans. Grese).ACM 28 begins: ‘avenge me on the one who opposes me and on the one who has driven me from my place’ (trans. Meyer). Others present their speakers as vulnerable persons or victims of violence, gossip or slander (ACM 89-94). Paul’s curse depicts Bar-Jesus as an aggressor; he is an ἐχθρός, an adversarial term in curse texts.34 He is also ‘full of deceit (δόλος)’, a common term for unknown causes of death which Versnel includes as part of an ‘idiom of suspicion’ that characterizes ‘prayers for justice’ . However, Luke suggests that it is God who is the victim of Bar-Jesus’ aggression (13.10), which helps to portray Paul’s curse as an act of divine judgment rather than one of personal vengeance.Both Paul’s curse and the ‘magical’ curse texts also share a sense of victimization: many Greek and Coptic curse texts are characterized by their speaker’s desire for justice after being wronged. ACM 88, 92). Although the devil is not exactly the same as a demon in the NT, Paul shares the concept that his opponent is filled by a malevolent force. In Acts it seems especially important for the apostles to identify the satan as the force of evil in direct opposition to God , which appears to concur with the thought of the Bar-Jesus narrative.36 The curse texts also identify the demonic as both present in their enemies and as instruments of punishment. One curse includes the command: ‘Send to him an evil demon’, and another: ‘You must make a demon descend upon her’ . Another text requests that the deity brings physical pain and ‘demonic madness’ on its victim .The curse texts and Acts also propound the belief that the spell-caster’s opponent is possessed by a demon (n to God . Howeverlasting image of the scene simply because the curse is not relieved within the narrative of Acts (1989: 84). But whilst it is the final image, the words ἄχρι καιροῦ (13.11) prevent it from being the lasting one. At first glance, Paul’s seeming allowance for repentance contrasts with some of the Greek and Coptic curses that command enduring afflictions on their opponents . However, Faraone has highlighted how many of the Greek binding curses seek not to harm aggressively but to restrict defensively (1991: 8). ACM 106 includes an intriguing clause in its command: ‘Let them come and bring them down upon the body of N. child of N. … until I myself, N. child of N., have mercy on it’ (trans. Emmel). If Faraone’s argument is correct, namely, that Greek binding curses should be primarily characterized by their desire to control their opponents rather than by their will to harm, then Paul’s curse on Bar-Jesus fits on the more defensive end of a spectrum of curse traditions that span from temporarily incapacitating a person, to causing illness and even killing. Luke’s theological perspective does not place the apostle’s speech outside the field of cursing but rather complements the type of curse that is cast.The limited duration of Bar-Jesus’ blindness (13.11) has interested scholars and is often understood to be in keeping with the theological programme in Acts that desires sinners to repent and submit themselves to God’s mercy. Garrett takes an opposing view in interpreting Bar-Jesus’ blindness as the final and ACM 90, 90, 95). Luke–Acts also connects to OT miracle stories through allusions. Paul describes the χεὶρ κυρίου working against Bar-Jesus, a probable allusion to Moses bringing the plagues against the Egyptians (δάκτυλος θεοῦ in LXX Exod. 8.15), as well as to Jesus whose exorcisms are described as ἐν δακτύλῳ θεοῦ (Lk. 11.20). Paul’s curse does not employ historiola, but it does allude to OT intertexts that, as well as framing the scene within certain narrative expectations, function to suggest that both Paul and the text of Luke–Acts carry a similar authority to that of the OT. Both Paul’s curse and the spell texts appeal to holy scriptures to acquire some form of status or power. In summary, a comparison of Paul’s curse with Greek and Coptic curse texts has shown that they share a number of characteristics such as demonic possession and punishment, the desire for immediacy, the use of spatial imagery, and connections to historical demonstrations of divine power.A final parallel between Acts and the curse texts can be found in their common appeal to well-known miracle stories. A number of the spell texts from late Antiquity employ historiola as a means of accessing divine powers described in stories of past events. The curse texts often appeal to OT stories of God’s judgment against figures such as Cain, Pharaoh or the Assyrian army , in the same way that they ‘listened eagerly’ to Simon (8.11), suggesting that the apostle competes for their attention as Paul did with Sergius Paulus. But it also firmly distinguishes between them by clarifying their contrasting sources of power. Philip demands attention through spirit-empowered signs, whilst Simon amazes the people by his ‘magic’. As with Bar-Jesus, Simon’s association with μαγεύω may have reminded audiences of notorious adversaries in the Jewish scriptures.In this episode, the bite of the curse comes at the beginning of the speech in the form of a sharp rebuke, followed by a series of verdicts and accusations.Like Acts 13.10-11, Peter’s curse maintains the drama through its direct style of address. The second person pronoun is again heavily repeated, making use of σου, σοί and σε eight times in 8.20-23. This focuses the audience’s attention firmly on Peter’s opponent and indicates that Peter has likewise fixed his gaze on him. In contrast, the double use of the first person pronoun in Simon’s response to Peter shows Simon submitting to the direction of Peter’s attention and verdict.The phonaesthetic features of Peter’s curse also function to heighten the drama of its performance. The text employs sibilance in σου σὺν σοὶ and εἰς. It also creates assonant effects with repeated ει and η sounds (εἴη εἰς ἀπώλειαν). Such devices give a fluid, poetic quality to phrases like τὸ ἀργύριόν σου σὺν σοὶ εἴη εἰς ἀπώλειαν (8.20), at the same time functioning to hold the whole speech together as a textual unit.The thematic content of the speech itself also indicates continuity with Paul’s own speech. Verse 20 suggests that Simon’s crime is one involving greed and deceit. The text implies that Simon desires the ability to bestow the Holy Spirit because of its profiteering potential. Whilst Strelan interprets Simon as a sincere insider who mistakenly thinks he can become more fully integrated in the new movement by means of his money (2004: 213), there is better support for Klauck’s interpretation that Simon is slipping back into old ways (2000: 20). Peter’s insight into the state of Simon’s heart not only implies that Simon’s understanding of the Spirit is misplaced, but that his motivation is wrong.As with Acts 13.4-12, v. 21 shows that Peter, like Paul, has the ability to see into a person’s heart. The repetition of καρδία draws attention to both Peter’s power and Simon’s true nature. That Simon’s heart is ‘not right before God’ and that his intent requires forgiveness suggest a deep-rooted opposition to the apostles, rather than an earnestly misplaced one (contra Strelan).The element of excommunication that underlies Acts 13.4-12 is explicit in Acts 8.21. Because Simon is a member of the growing community of believers in Samaria, his chastisement by the community’s leaders has to be publicly demonstrated in order to protect the community’s purity and reputation. Verses 18-19 describe Simon’s attempt to purchase the Holy Spirit, a gift often understood by early Christians as a sign of the community’s inheritance (cf. 2 Cor. 1.22). As a result, Peter declares: ‘You have no part or share in this’ (Acts 8.21). Despite having claimed to have joined the Christian community, Simon’s greed shows him to be an impostor. Peter’s curse functions to publicly reverse Simon’s reputation before the eyes of those who had previously listened eagerly to him (8.10). For the audience of Acts, Simon’s ‘true’ status as an impostor should come as no surprise in the wake of Luke’s characterization of Simon as one who practised the arts of the μάγοι, a label that may have invited suspicion in connection to LXX Dan. 2.2.40 Like Bar-Jesus, Simon is perceived to be opposed to righteousness . Whilst the standard labels for demonic powers are not used in the story of Simon magus, lexical and thematic links with Acts 13.4-12 and other exorcism narratives in Luke–Acts suggest that readers should understand Simon, like Bar-Jesus, to be under the influence of demonic forces and in opposition to God; this in turn invites the reader to recognize a theological battle between God and evil underlying the narrative.Like Acts 13.10, the demonic is also present in Peter’s opponent despite the absence of labels like δαιμόνιον, διάβολος, or σατανᾶς. The absence of such terms leads Peter’s speech shares a number of motifs with Greek and Coptic curses and also with some crucial elements of Faraone’s second category of binding curses: ‘the prayer formula’ (1991: 10). Although it does not match the category’s most defining characteristic – it is not addressed to God – it does manifest a number of its other aspects. It employs the third person optative, ‘may your silver perish with you’. Its disclosure that it is God who is the true victim of Simon’s crime and its desire for confession also evoke Versnel’s related category of ‘prayers for justice’ or ‘judicial prayers’ : 192.In ‘prayers for justice’ the speaker typically seeks vindication for some harm, real or perceived, often relating to stolen goods and/or to potential gossip or slander. In these prayers, the petitioner appears to ‘transfer’ the crime onto the deity, employing quasi-legal vocabulary : 175. Th41I consecrate to the mother of the gods the gold pieces that I have lost, all of them, so that the goddess will track them down and bring everything to light and will punish the guilty in accordance with her power and in this way will not be made a laughing stock.The act of ‘handing over’ the culprit to the deity or to some lesser power is the natural extension of the transferral of victimization onto the deity. In the eyes of the spell-caster, divine punishment functions as either irrevocable punishment or as a ‘conditional and temporary means of pressure’ which might bring the culprit to confession or the perceived damage to restitution : 80.This ‘transfer’ terminology is clearly seen in other words attributed to the apostles; Paul ‘hands over’ (παραδίδωμι) troublesome figures to Satan for limited periods . In the case of Peter’s speech, the transfer is implicit, as the apostle simply states that Simon’s crime is against God and the Holy Spirit, using quasi-legal terminology such as ἀδικία. Simon’s response (8.24) acknowledges that he has been moved into the sphere of divine influence.42 In many curse texts, a public confession of guilt, and the humiliation it would bring, is as desirable as the return of stolen possessions because of the honour it would restore to the deity and to the wronged person .d person : 153.PGM IV.1227-64.43In addition to these aspects of ‘prayers for justice’, Peter’s pronouncement shares other motifs with the curse texts. A number of scholars have recognized the lexical overlap of Peter’s curse (8.20: τὸ ἀργύριόν σου σὺν σοὶ εἴη εἰς ἀπώλειαν) with ἔξελθε, δαῖμον,|| ἐπεί σε δεσμεύω δεσμοῖς ἀδαμαντίνοις | ἀλύτοις, καὶ παραδίδωμί σε εἰς τὸ μέ|λαν χάος ἐν ταῖς ἀπωλείαις.Come out, daimon, since I bind you with unbreakable adamantine fetters, and I deliver you into the black chaos in perdition .PGM IV.1227-64 is an exorcism spell rather than a curse text. However, this should not deter one from drawing a comparison with Peter’s curse since the exorcism simply involves the client aiming the curse towards the demon rather than towards an individual. Both texts use ἀπώλεια as an appropriate punishment for their adversary and favour sibilant phonetics in their speech.44 The sense of a demonic presence in Peter’s reference to ‘bonds’ also parallels many of the spell texts that describe their opponents as being possessed by a demonic power. The belief that a person could be overpowered by evil is present in both groups of texts.It should be acknowledged that PGM XL.1-18 curses her daughter’s father who has apparently robbed her daughter of her funeral gifts.The style of Peter’s curse on Simon and his money also parallels Greek curses against an individual’s property as well as against their person. The client of τῆς δὲ | καταβοιῆς ἐνθῦτα κειμένης, κακῶς ἀπολλύοιτο κἐγ γῆι κἐν θαλάσσηι καὐτὸς | καὶ τὰ αὐτοῦ ὑπὸ τοῦ Ὀσερ[ά]πιος καὶ τῶν θεῶν τῶν ἀμπ᾽ Ὀσερᾶπι καθημένων […]As long as my cry for help is deposited here, he and what belongs to him should be utterly destroyed badly, both on earth and on sea, by Oserapis and the gods who sit together with Oserapis […] .The speaker of this curse adapts the punishment to the theme of the crime; her adversary is a thief, therefore his possessions are involved in his retribution. A similar style is employed in the first line of Peter’s curse.Like historiola in the curse spells, Acts 8.4-25 consciously associates itself with historic instances of divine power. Both Peter’s instruction and Simon’s subsequent petition are pregnant with allusions to the OT. Peter’s instruction to repent and Simon’s positive reaction parallel the interaction between Jonah and the city of Nineveh (LXX Jon. 3.1-9), another narrative in which a man of God is sent to a Gentile land to proclaim the news of God’s judgment. Connections include ἀπώλεια , μετανοέω  and κακία . The second part of Simon’s petition is also evocative of the words of the narrator in Jon. 3.10. For an audience familiar with the Septuagint, these parallels frame Peter as a prophet in the mould of Jonah, who is sent by God to a non-Jewish audience. These OT allusions function to associate the apostles with the authority of the Jewish scriptures, an important strategy for presenting the apostles’ curses as legitimate acts of power. The text presents Peter in line with the prophets, authoritative agents commissioned to communicate God’s will to his people. Peter’s curses are therefore legitimate in the same way that the prophets were authorized to curse on behalf of the Lord . Such strategies for presenting the apostolic curses as carrying divine authorization are vital for defending the apostles’ actions against charges of magic.45 However, by comparing Peter’s words with Greek curse texts from the ‘magical’ papyri, another dimension to Simon’s reaction can be offered. Peter’s use of χολή connects his speech to a number of texts in the PGM where χολή is frequently employed in lists of ‘magical’ ingredients to denote the gall or bile of animals required for the performances of spells.46PGM XII.401-44 even acknowledges that such ingredients were inscribed on religious statutes by temple scribes so that common people would not use them to practise magic. The use of χολή, when combined with δεσμός, a term that also has ‘magical’ connotations (binding spells were known as κατάδεσμοι), suggests that Peter’s curse could involve a knowing slur on Simon’s association with magic (LSJ: 380).An obvious problem with interpreting Peter’s words to Simon as a curse is that its effects do not appear to materialize. Strelan, one of the few scholars to interpret Peter’s words as a curse, argues that Simon’s request for a prayer represents a request for a remedial blessing in a fashion reminiscent of Judg. 17.1-4 (2004: 215). Strelan’s reading is convincing, strengthened by the thematic and lexical parallels between these two narratives.PGM IX.1-14, a text focused on defeating an opponent’s anger.However, Peter’s use of χολή also brings it into comparison with θυμοῦ σε πα<ύ>σω καί σε πραΰνω χολῆς. ἐλθὲ καὶ διακράτει σιγῇ σιγὴν ϕέρων| τε πα<ῦ>σιν καὶ θυμοὺ<ς> στῆσον ψυχῶν πάντων ὀργάς τε πάσας σβέσον, ϕρένας ὁρκρίσας […]I’ll give you rest from wrathAnd soothe your raging.Come silently and bringSilence and keep it.Stop ev’ry wrath in soulsAnd melt all angerOf those with temper […] .Spells against anger in the ‘magical’ papyri combine positive sentiments that express the desire to cure an opponent with negative sentiments aimed towards their anger.κατυπόταξον, καταδοῦλωσον, ϕίμωσον τὴν ψυχήν, τὸν θυμὸν <τοῦ δεῖνα>PGM IX.9, trans. Hock).47Bring into subjection, enslave, and put to silence the soul, the wrath  […]  establishes Peter in a dominant position over his adversary.Again, Peter’s speech against his adversary is direct. The second person pronoun and the possessive adjective are employed five times in the forms of σου, σε, σοι, σου and σῇ, focusing attention firmly on his opponent. He addresses Ananias by name, which functions to isolate Peter and Ananias from amongst the onlookers. That it is Peter who actively names Ananias . The three rhetorical questions function to communicate Peter’s judgment on Ananias; he is relabelled as demonic, his deviancy is identified, and his status as an impostor is revealed.The final, declarative sentence does not issue a punitive command, but it is an incrimination that serves to pass Ananias over for divine judgment. The assonance of repeated ω sounds in the words οὐκ ἐψεύσω ἀνθρώποις ἀλλὰ τῷ θεῷ gives the statement its musical quality. Through the structure and style of the text, Acts establishes the potency of Peter’s words; his narration in v. 5 demonstrates its effects.Like his speech to Simon, Peter’s speech to Ananias employs consonance and sibilance to enhance its rhetorical force. Repeated sigmas across σου, ψεύσασθαί σε and νοσϕίσασθαι ἀπὸ τῆς τιμῆς (5.3), combined with the alliteration of τῆς τιμῆς τοῦ χωρίου, serve to hold the speech together as a distinct unit of text: the repeated phonetics increase its fluidity as it crescendos towards its climax.The three themes of the two other episodes examined in this article are also present in Peter’s rhetorical questions. First, Ananias is filled by the satan (5.3: reminiscent of Judas in Lk. 22.3), which parallels Peter who is ‘filled with the Holy Spirit’ (4.8). The presence of the demonic also places Peter’s speech in line with the Coptic and Greek curse spells that identify their enemies with demonic activity and wish demonic punishment upon them.filled … to lie’ by the satan. The third aspect shared with the other speeches is its excommunicatory function.49 Ananias is described as laying ‘only a part’ (μέρος) of his proceeds at the apostle’s feet (5.2). Ironically, the text implies that Ananias dies at Peter’s feet and makes it explicit that Sapphira ends up there too (5.10), the same place where the money was laid. In this way, the punishment – as in some curse texts – mirrors the crime. In bringing only a part of the proceeds to the apostles, Ananias and Sapphira lose their part in the community of believers.Secondly, impure motives are also present. καρδία is repeated three times in the pericope. Just as Bar-Jesus was said to be ‘full of all deceit and villainy’ (13.10), so Ananias’s heart is ‘50 The text emphasizes this effect through the repetition of πίπτω in the description of Ananias and Sapphira immediately falling down .Whilst Peter’s speech itself does not include any explicit demand for divine striking, the effect of his words is one that suggests that his adversaries have been struck by the invisible hand of God.The curse-like nature of Peter’s words is more explicit in his speech to Sapphira. With regard to the narrative of the episode, Peter’s utterance comes at a point when he has seen the effects of his words on Ananias and has waited three hours for Sapphira’s return (5.7). Peter incriminates her for the same crime as Ananias, but his verdict carries an even stronger tone of judgment.Peter again expresses his incrimination of Sapphira as a rhetorical question (5.9b). He identifies God as the victim of his opponent’s deceit. Beginning with an emphatic ἰδοὺ̀, Peter’s verdict employs notable sibilance in the words καὶ ἐξοίσουσίν σε to end his curse with a sharp, staccato flourish. The speech is then immediately followed by its fulfilment, as Luke portrays the effect by employing the modifier ἔπεσεν δὲ παραχρῆμα (5.10). Such emphasis on immediacy characterizes Peter’s speech as words of power that cause Sapphira’s death.If the audience understands Peter’s words to Sapphira as a curse, then it follows that Peter’s words to Ananias should be understood in a similar light. The first incident mirrors the second; both husband and wife are accused of the same crime, both suffer the same punishment, and both are carried out by the young men. The effects of Peter’s words are immediate in both encounters: in v. 5 the text’s use of participles  draws the audience’s attention to the aspects of cause and effect.defixiones are written, or the senselessness of an accompanying figurine.51 Often, the characteristics of the corpse by which a curse was buried were also invoked: ‘Just as this corpse lies useless, so too may everything be useless for NN’.52 Jordan explains: ‘That the living should be affected by the conditions of the dead with whom curse tablets are deposited is a common idea in Greek magic’ (1999: 118). In Acts 5.9c Peter declares, in dramatic fashion, that the death of Ananias will be the model for Sapphira’s own fate. Grimly, she falls down just as he fell down, becoming as useless as he is. This parallelism could simply be attributed to Luke’s use of irony or accomplished literary style . In this context, Peter’s proclamation would represent a curse at the harsher end of the spectrum.Peter’s ‘verdict’ on Sapphira is even more intriguing when it is compared with ‘persuasive analogy’, the third style of curse in Faraone’s typology. These curses reflect a belief that a ritual action might ‘urge’ the victim to become like something to which it is dissimilar : 8. AttrThe image of Ananias and Sapphira falling at the feet of Peter , juxtaposed with the upwards movement of the young members of the community (5.6), also evokes motifs from the Greek curse texts.καὶ ϕίμωσον, | ὑπόταξον, καταδούλωσον τὸν δεῖνα τῷ δεῖνα καὶ ποίη|σον αὐτόν, ὑπὸ τοὺς πόδας μοι ἔλθῃ. |PGM VII.966-68, trans. Hock).Put to silence, subordinate, enslave him, NN, to him, NN, and cause him to come under my feet .The common themes that can be identified in the apostles’ speeches strongly suggest that the confrontations with Bar-Jesus, Simon magus and Ananias and Sapphira should be read in continuity with each other. As a group of texts, they reveal important information concerning the attitude of the early church towards deviant and threatening behaviour. In each of these three narratives, the opponents’ crimes are associated with deceit and greed, a tendency that appears to complement Luke’s concern for the poor and destitute. Taken together, the texts also shed light on the nature of accusations of magic: Simon and Bar-Jesus’ involvement with magic is associated with deceit and greed, but it also functions as an ethno-classification that emphasizes the foreign territory in which the apostles find themselves. Literary analysis of the three narratives suggests that Luke crafts each text in such a way that audiences would have heard the apostles’ speeches as dramatic words of power. In contrast to interpreters who downplay the apostles’ responsibility in the drama, this article has argued that the text emphasizes the relation between the apostles’ pronouncements and the subsequent afflictions.voces magicae,55 whereas they do employ common styles of performance, showing elements of ‘direct curses’, ‘prayers for justice’ and ‘persuasive analogies’. Luke also frames the curses in agonistic contexts – the usual context for binding curses – as Paul and Peter compete with other wonder-workers and benefactors. Their accusations against impure motives mirror the concern for ‘evil thoughts’ and Schadenfreude in the curse texts , portraying Paul and Peter as genuine men of God who are comparable to the prophets of the Jewish scriptures. Their Spirit-powered signs are superior to anything practised by magicians, and they have the authority to judge those inside the Christian community. Rather than explicitly petitioning the deity for action in the manner of the supplicants of the curse spells, the apostles speak for and about the deity in their capacity as his agents. However, for the reasons stated above, Peter’s curse on Sapphira strains these strategies to their limits, which is reflected in the ‘great fear’ of those who hear about the encounter (5.11). In this episode, Luke must rely on the sheer authority of the charismatic leader within the Christian community (4.30-31) to legitimate his actions.Smith’s topography of religions  offers a


Edition of December 2017, vol. 109 (6), Supl. 2, p. 1-34In the “Atualização das Diretrizes Brasileiras de Valvopatias: Abordagem das LesõesAnatomicamente Importantes”, published as a supplement to the Arquivos Brasileiros deCardiologia , the following corrections shouldbe considered:In table 18, column 5, line 6: replacement of “IIb” with “IIb C”.In Table 24, line 4: the items “IB, IB, IC” should be aligned with the phrase”Fração de ejeção < 50%”. In line 5, items “IIa, IIa B, I C” should be alignedwith the phrase “ Ausência de reserva inotrópica no teste ergométrico e/ou baixacapacidade funcional”. On line 6, remove item “B”.In table 24, section “Tratamento cirúrgico convencional”, change from “Semreserva contrátil” to “Sem reserva contrátil + escore de cálcio valvarelevado”.In table 43, section “ TAVI ritmo sinusal “, change of recommendations asfollows:Varfarina – Correct: ESC - IIb B*.AAS + clopidogrel  –Correct: SBC - IIa C; AHA - IIb C; ESC - IIa C. NOACs – Correct: SBC –III.Inclusion of the name of Dr. Samira Kaissar Nasr Ghorayeb in the document.Edition of April 2016, vol. 106 (4), Supl. 1, p. 1-23In the “III Diretrizes da Sociedade Brasileira de Cardiologia sobre Análise e Emissão deLaudos Eletrocardiográficos”, published as a supplement to the Arquivos Brasileiros deCardiologia , the following corrections shouldbe considered:On page 1, section 2.1., text correction “O eixo de P pode variar entre -30º e +90º” to“O eixo de P pode variar entre 0º e +90º”.


Page 1; line 22: “was found within the pNPD4_2 plasmid” should read “was found within the chromosome.”Volume 5, no. 29, e00375-17, 2017,


Property ζs = f(Jvm) for Jvm > 0 and for Jvm ≈ 0 and property ζs = f(ΔC1) are calculated. Moreover, results of a simultaneous influence of ΔP and Δπ on a value of coefficient ζs when Jvm = 0 and Jvm ≠ 0 are investigated and a graphical representation of the dependences obtained in the research is provided. Also, mathematical relationships between the coefficient ζs and a concentration Rayleigh number (RC) were studied providing a relevant graphical representation. In an experimental test, aqueous solutions of glucose and ethanol were used.In this paper, the authors investigate the membrane transport of aqueous non-electrolyte solutions in a single-membrane system with the membrane mounted horizontally. The purpose of the research is to analyze the influence of volume flows on the process of forming concentration boundary layers (CBLs). A mathematical model is provided to calculate dependences of a concentration polarization coefficient ( Lp), the reflection coefficient (σm) and the diffusive permeability coefficient (ωm). Usefulness of the classical as well as a modified form of the Kedem–Katchalsky equations has been confirmed repeatedly (δlδh)−1,φ1 = [2LpΔP(Dhδl – Dlδh) – 4DlDh]δl−1δh−1,φ2 = 4LpDlDh (ΔP – σmΔπ)δl−1δh−1,μ0 = 4Lpσm2RTDlδh + Dhδl)δl−1δh−1,μ1 = 4LpσmΔπωmRT(Dlδh + Dhδl) δl−1δh−1.Including Eq.  and 13)13) in EqLp, σm and ωm), solutions , volume flux (Jvm) and thicknesses of CBL   + 2(Δπ)2 (Dhδl – Dlδh) – 4RTDhδl +ZDlδh)σmΔπ ]δl−1δh−13 = [Lp3σmδlδhΔπ(ΔP)2 – 2(Dhδl – Dlδh)Lp2σmΔπΔP – 2Lp3σm2δlδhΔP(Δπ)2 –ZDlDhLpσmΔπ + 4Lp2σm2RTDhδl + Dlδh)ΔP –4LpσmωmRT(Dhδl + Dlδh)Δπ]δl−1δh−144 = [4LpDlDhσmΔπ + Lp3δlδh(ΔP)2σmΔπ ]δl−1δh−1ZIncluding Eq.  in Eq. , we obtaLp = 5 × 10−12 m3N−1s−1. The values of the reflection coefficient and diffusive permeability coefficient of the membrane for the glucose and ethanol are respectively σm1 = 0.068, ωm1 = 8 × 10−10 mol N−1s−1, σm2 = 0.025 and ωm2 = 14.3 × 10−10 mol N−1s−1. The diffusion of each individual component in the solution is characterized by the following coefficients: D1 = 0.69 × 10−9 m2s−1 and D2 = 1.57 × 10−9 m2s−1. The volumes of the compartments  were the same and equal to 200 cm3. For the low glucose and ethanol concentration, we have ρh = ρl(1 + α1Ch1 + α2Ch2), νh = νl(1 + γ1Ch1 + γ2Ch2) with the coefficients α1 = ρl−1∂ρ/∂C1 = 6.01 × 10−5 m3 mol−1, γ1 = νl−1∂ν/∂C1 = 3.95 × 10−4 m3 mol−1, α2 = ρl−1∂ρ/∂C2 = −9.02 × 10−6 m3 mol−1and γ2 = ρl−1∂ν/∂C2 = 1.82 × 10−5 m3 mol−1  }α2 = −0,5JvmRClDlνlρlα3 = −RClDl2νlρlβ1 = g(∂ρ/∂C){ζsωmΔπ – Jvm[0.5 (Ch – Cl) + ζsσm(Ch + Cl)]}β2 = 0.5JvmRChDhνhρhβ3 = −RChDh2νhρh.Taking into account Eq.  and 13)13) in EqP = 0 is Jvm = −LpσmζsDΔπ, consequently Δπ = −Jvm(LpσmζsD)−1. Assuming that Cl = 0, we have Ch–Cl = Ch + Cl = Ch = −Jvm(LpσmζsDRT)−1. Moreover, assuming that the CBL thickness is δl = Dl(2RTωm)−1(ζs−1 – 1) and ρl = ρh = ρ0, then Eq. . The spatial graph of this function presents the dependence of concentration polarization coefficient ζs on the Rayleigh number (RCl) and the volume flux (Jvm). It is not essential to present Eq.  was made for the Nephrophan membrane and aqueous ethanol solution  and the volume flux (Jvm). The graph also shows the significant influence of the two parameters mentioned earlier on the value of the concentration polarization coefficient.A graph of dependence ion Fig. . The shaion Fig.  proves tζs = f(Jvm), ζs = f(ΔC), ζs = f and it is possible to evaluate the influence of osmotic flux (Jvm) and/or the simultaneous operation of osmotic forces (Δπ) and hydrostatic forces ((ΔP) on the value of the concentration polarization coefficient (ζs). Equations , allowing the evaluation of the numerical relations between the concentration polarization coefficient (ζs), the osmotic flux (Jvm) and the concentration Rayleigh number (RC). The results of the research carried out confirmed the significant role of concentration boundary layers in osmotic and diffusive transport, in particular their applicative aspect in technology and medicine, as mentioned in the Introduction [Equations  and 17)17), deriquations -22) and andζs =oduction , 32–34. oduction , 31, 33.


Scientific Reports7:10141; doi:10.1038/s41598-017-09415-7; Article published online 04 September 2017This Article contains errors in Figure 4, where “Eremis acutirostris” should read “Eremias acutirostris”, “Acananthodactylus cantoris” should read “Acanthodactylus cantoris”, and “Pedioplais burchelli” should read “Pedioplanis burchelli”. The correct Figure 4 appears below as Figure