* [68] S. Post and P. Winternitz, _‘General Nth order integrals of motion in the Euclidean plane’_ , J. Phys. A: Math. Gen. 48, 405201 (2015).
  * [69] C.R. Holt, _‘Construction of new integrable Hamiltonians in two degrees of freedom’_ , J. Math. Phys. 23(6), 1037 (1982).
  * [70] C. Daskaloyannis and K. Ypsilantis, _‘Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold’_ , J. Math. Phys. 47 042904 (2006).
  * [71] O.C. Pin, _‘Curvature and Mechanics’_ , Adv. Math. 15, 269 (1975).
  * [72] C. Uggla, _‘Geometrizing the dynamics of Bianchi cosmology’_ , K. Rosquist and R.T. Jantzen, Phys. Rev. D 42(2), 404 (1990).
  * [73] K. Rosquist and G. Pucacco, _‘Invariants at fixed and arbitrary energy. A unified geometric approach.’_ , J. Phys. A: Math. Gen. 28, 3235 (1995).
  * [74] M. Karlovini and K. Rosquist, _‘A unified treatment of cubic invariants at fixed and arbitrary energy’_ , J. Math. Phys. 41(1), 370 (2000).
  * [75] M. Karlovini, G. Pucacco, K. Rosquist and L. Samuelson, _‘A unified treatment of quartic invariants at fixed and arbitrary energy’_ , J. Math. Phys. 43(8), 4041 (2002).
  * [76] G.H. Katzin and J. Levine, _‘Time-dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time-dependent Kepler potentials’_ , J. Math. Phys. 23(4), 552 (1982).
  * [77] M. Tsamparlis and A. Paliathanasis, _‘Symmetries of Differential Equations in Cosmology’_ , Symmetry 10(7), 233 (2018).
  * [78] S.A. Hojman, _‘A new conservation law constructed without using either Lagrangians or Hamiltonians’_ , J. Phys. A: Math. Gen. 25, L291 (1992).
  * [79] S.H. Strogatz, _‘Nonlinear Dynamics and Chaos: with applications to physics, biology, chemistry, and engineering’_ , Perseus Books Publishing (1994).
  * [80] E. Gourgoulhon, _‘An introduction to relativistic hydrodynamics’_ , EAS Publ. Ser. 21, 43 (2006).
  * [81] S. Grozdanov, D.M. Hofman and N. Iqbal, _‘Generalized global symmetries and dissipative magnetohydrodynamics’_ , Phys. Rev. D 95, 096003 (2017).
  * [82] J. Hernandez and P. Kovtum, _‘Relativistic Magnetohydrodynamics’_ , JHEP 2017(5), 1 (2017).
  * [83] J. Armas and A. Jain, _‘Magnetohydrodynamics as superfluidity’_ , Phys. Rev. Lett. 122, 141603 (2019).
  * [84] P.S. Letelier, _‘Clouds of strings in general relativity’_ , Phys. Rev. D 20(6), 1294 (1979).
  * [85] L.L. Smalley and J.P. Krisch, _‘String fluid dynamics’_ , Class. Quantum Grav. 13, L19 (1996).
  * [86] P.S. Letelier, _‘Anisotropic fluids with two-perfect-fluid components’_ , Phys. Rev. D 22(4), 807 (1980).
  * [87] P.S. Letelier, _‘Fluids of strings in general relativity’_ , Nuovo Cimento B, 63(2), 519 (1981).
  * [88] P.S. Letelier, _‘String cosmologies’_ , Phys. Rev. D 28(10), 2414 (1983).
  * [89] M. Tsamparlis, _‘General symmetries of a string fluid space-time’_ , Gen. Relativ. Gravit. 38(2), 279 (2006).
  * [90] M. Tsamparlis and D.P. Mason, _‘Ricci collineation vectors in fluid space-times’_ , J. Math. Phys. 31(7), 1707 (1990).
  * [91] J.L. Synge and A. Schild, _‘Tensor Calculus’_ , Dover, New York, (1978).
  * [92] G.F.R. Ellis and H. van Elst, _‘Cosmological Models’_ , NATO Adv. Study Inst. Ser. C Math. Phys. Sci. 541, 1-116 (1998).
  * [93] E. Saridakis and M. Tsamparlis, _‘Symmetry Inheritance of conformal Killing vectors’_ , J. Math. Phys. 32(6), 1541 (1991).
  * [94] G.F.R. Ellis, R. Maartens and M.A.H. MacCallum, _‘Relativistic Cosmology’_ , Cambridge University Press, (2012).
  * [95] D.P. Mason and M. Tsamparlis, _‘Spacelike conformal Killing vectors and spacelike congruences’_ , J. Math. Phys. 26(11), 2881 (1985).
  * [96] V.N. Duarte and R.A. Clemente, _‘Electron inertia effect on floating plasma potential’_ , J. Phys.: Conf. Ser. 511, 012015 (2014).
  * [97] I. Yavuz and I. Yilmaz, _‘Inheriting conformal and special conformal Killing vectors in string cosmology’_ , Gen. Relativ. Gravit. 29(10), 1295 (1997).
  * [98] I. Yilmaz, _‘Timelike and spacelike Ricci collineation vectors in string cosmology’_ , Int. J. Mod. Phys. D 10(5), 681 (2001).
  * [99] H. Baysal, U. Camci, I. Tarhan, I. Yilmaz and I. Yavuz, _‘Conformal collineations in string cosmology’_ , Int. J. Mod. Phys. D 11(3), 463 (2002).
  * [100] H. Baysal and I. Yilmaz, _‘Spacelike Ricci inheritance vectors in a model of string cloud and string fluid stress tensor’_ , Class. Quantum Grav. 19, 6435 (2002).
  * [101] U. Camci, _‘Conformal collineations and Ricci inheritance symmetry in string cloud and string fluids’_ , Int. J. Mod. Phys. D 11(3), 353 (2002).
  * [102] M. Sharif and U. Sheikh, _‘Timelike and spacelike matter inheritance vectors in specific forms of energy-momentum tensor’_ , Int. J. Mod. Phys. A 21(15), 3213 (2006).
  * [103] M. Tsamparlis, _‘Special Relativity: An Introduction with 200 Problems and Solutions’_ , Springer, 2nd ed. (2019).
  * [104] K.A. Dunn and B.O.J Tupper, _‘Type I, II, and III spatially homogeneous cosmologies with electromagnetic field’_ , Astroph. J. 235(2), 307-334 (1980).
  * [105] L.K. Norris, L.H. Green and W.R. Davis, _‘Fluid space-times including electromagnetic fields admitting symmetry mappings belonging to the family of contracted Ricci collineations’_ , J. Math. Phys. 18(7), 1305 (1977).
  * [106] R. Maartens, D.P. Mason and M. Tsamparlis, _‘Kinematic and dynamic properties of conformal Killing vectors in anisotropic fluids’_ , J. Math. Phys. 27(12), 2987 (1986).
  * [107] A.A. Coley and B.O.J. Tupper, _‘Special conformal Killing vector space-times and symmetry inheritance’_ , J. Math. Phys. 30(11), 2616 (1989).
  * [108] K.C. Jacobs, _‘Cosmologies of Bianchi type I with a uniform magnetic field’_ , Astroph. J. 155, 379 (1969).
  * [109] M. Tsamparlis, A. Paliathanasis and L. Karpathopoulos, _‘Exact solutions of Bianchi I spacetimes which admit Conformal Killing Vectors’_ , Gen. Relativ. Gravit. 47, 15 (2015).
  * [110] P.E. Hydon, _‘Symmetry analysis of initial-value problems’_ , J. Math. Anal. Appl. 309, 103 (2005).
  * [111] R. Cherniha and S. Kovalenko, _‘Lie symmetries of nonlinear boundary value problems’_ , Commun. Nonlin. Sci. Numer. Simulat. 17(1), 71 (2012).
  * [112] L. Herrera and J. Ponce de Leon, _‘Anisotropic spheres admitting a one-parameter group of conformal motions’_ , J. Math. Phys. 26(8), 2018 (1985).
  * [113] L. Herrera and J. Ponce de Leon, _‘Isotropic and anisotropic charged spheres admitting a one-parameter group of conformal motions’_ , J. Math. Phys 26(9), 2302 (1985).
  * [114] A.A. Coley and B.O.J. Tupper, _‘Spacetimes admitting inheriting conformal Killing vector fields’_ , Class. Quantum Grav. 7(11), 1961 (1990).
  * [115] A.A. Coley and B.O.J. Tupper, _‘Spherically symmetric spacetimes admitting inheriting conformal Killing vector fields’_ , Class. Quantum Grav. 7(12), 2195 (1990).
  * [116] A.A. Coley and B.O.J. Tupper, _‘Affine conformal vectors in space-time_ , J. Math. Phys. 33(5), 1754 (1992).
  * [117] A.A. Coley and B.O.J. Tupper, _‘Spherically symmetric anisotropic fluid ICKV spacetimes’_ , Class. Quantum Grav. 11, 2553 (1994).
  * [118] R. Maartens, S.D. Maharaj and B.O.J. Tupper, _‘General solution and classification of conformal motions in static spherical spacetimes’_ , Class. Quantum Grav. 12, 2577 (1995).
  * [119] C. Frances, _‘Local dynamics of conformal vector fields’_ , Geom. Dedicata 158, 35 (2012).
  * [120] P. Joharinad and B. Bidabad, _‘Conformal vector fields on Finsler spaces’_ , Diff. Geom. Appl. 31, 33 (2013).
  * [121] R. Geroch, _‘A Method for Generating Solutions of Einstein’s Equations’_ , J. Math. Phys. 12(6), 918 (1971).
  * [122] R. Geroch, _‘A Method for Generating New Solutions of Einstein’s Equations. II’_ , J. Math. Phys. 13(3), 394 (1972).
  * [123] M. Tsamparlis and A. Paliathanasis, _‘Lie and Noether symmetries of geodesic equations and collineations’_ , Gen. Relativ. Gravit. 42, 2957 (2010).
  * [124] G.H. Katzin and J. Levine, _‘Related First Integral Theorem: A Method for Obtaining Conservation Laws of Dynamical Systems with Geodesic Trajectories in Riemannian Spaces Admitting Symmetries’_ , J. Math. Phys. 9(1), 8 (1968).
  * [125] A.V. Aminova, _‘Projective transformations and symmetries of differential equations’_ , Sbornik: Math. 186(12), 1711 (1995).
  * [126] A.V. Aminova and N.A.-M. Aminov, _‘Projective geometry of systems of differential equations: general conceptions’_ , Tensor N.S. 62(1), 65 (2000).
  * [127] M. Tsamparlis and A. Paliathanasis, _‘Lie symmetries of geodesic equations and projective collineations’_ , Nonlin. Dyn. 62, 203 (2010).
  * [128] Y. Bozhkov and I.L. Freire, _‘Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation’_ , J. Diff. Eq. 249, 872 (2010).
  * [129] A. Paliathanasis and M. Tsamparlis, _‘Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations’_ , J. Geom. Phys. 107, 45 (2016).
  * [130] A. Paliathanasis and M. Tsamparlis, _‘Lie symmetries for systems of evolution equations’_ , J. Geom. Phys. 124, 165 (2018).
  * [131] M. Tsamparlis, D. Nikolopoulos and P.S. Apostolopoulos, _‘Computation of the conformal algebra of $1+3$ decomposable spacetimes’_, Class. Quantum Grav. 15(9), 2909 (1998).
  * [132] P.S. Apostolopoulos and M. Tsamparlis, _‘The computation of the Conformal Killing Vectors of an $1+(n-1)$ decomposable metric’_, Tensor N.S. 61, 260-269 (1999).
  * [133] A. Karagiorgos, T. Pailas, N. Dimakis, P.A. Terzis and T. Christodoulakis, _‘Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field’_ , J. Cosm. Astropart. Phys. 03, 030 (2018).
  * [134] A. Krasiński, _‘Inhomogeneous Cosmological Models’_ , Cambridge University Press, (1997).
  * [135] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, _‘Exact Solutions to Einstein’s Field Equations’_ , Cambridge University Press, (2003).
  * [136] R.M. Wald, _‘General Relativity’_ , Chicago University Press, (1984).
  * [137] M.P. Ryan, Jr. and L.C. Shepley, _‘Homogeneous Relativistic Cosmologies’_ , Princeton University Press, (1975).
  * [138] P.S. Apostolopoulos and M. Tsamparlis, _‘Hypersurface homogeneous locally rotational symmetric spacetimes admitting conformal symmetries’_ , Class. Quantum Grav. 18, 3775 (2001).
  * [139] M. Tsamparlis and A. Paliathanasis, _‘The geometric nature of Lie and Noether symmetries’_ , Gen. Relativ. Gravit. 43, 1861 (2011).
  * [140] A. Paliathanasis and M. Tsamparlis, _‘The geometric origin of Lie point symmetries of the Schrödinger and the Klein-Gordon equations’_ , Int. J. Geom. Meth. Mod. Phys. 11(4), 1450037 (2014).
  * [141] M. Lutzky, _‘Symmetry groups and conserved quantities for the harmonic oscillator’_ , J. Phys. A: Math. Gen. 11(2), 249 (1978).
  * [142] W. Sarlet and F. Cantrijin, _‘Higher-order Noether symmetries and constants of the motion’_ , J. Phys. A: Math. Gen. 14, 479 (1981).
  * [143] T.M. Kalotas and B.G. Wybourne, _‘Dynamical Noether symmetries’_ , J. Phys. A: Math. Gen. 15, 2077 (1982).
  * [144] H. Kaplan, _‘The Runge-Lenz vector as an ‘extra’ constant of the motion’_ , Americ. J. Phys. 54(2), 157 (1986).
  * [145] L. Karpathopoulos, M. Tsamparlis and A. Paliathanasis, _‘Quadratic conservation laws and collineations: A discussion’_ , J. Geom. Phys. 133, 279 (2018).
  * [146] M. Tsamparlis and A. Paliathanasis, _‘Two-dimensional dynamical systems which admit Lie and Noether symmetries’_ , J. Phys. A: Math. Theor. 44, 175202 (2011).
  * [147] L.P. Eisenhart, _‘Riemannian Geometry’_ , 2nd ed., Princeton University Press, (1949).
  * [148] P.G.L. Leach and V.M. Gorringe, _‘A conserved Laplace-Runge-Lenz-like vector for a class of three-dimensional motions’_ , Phys. Lett. A 133(6), 289 (1988).
  * [149] G.H. Katzin and J. Levine, _‘Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity-dependent mappings of second-order differential equations’_ , J. Math. Phys. 26(12), 3080 (1985).
  * [150] G. Darboux, _‘Sur un probléme de mècanique’_ , Arch. Neerlandaises Sci. 6, 371 (1901).
  * [151] B. Dorizzi, B. Grammaticos and A. Ramani, _‘A new class of integrable systems’_ , J. Math. Phys. 24(9), 2282 (1983).
  * [152] T. Sen, _‘Lie symmetries and integrability’_ , Phys. Lett. A 122(6,7), 327 (1987).
  * [153] E.G. Kalnins, J.M. Kress, G.S. Pogosyan and W. Miller Jr., _‘Completeness of superintegrability in two-dimensional constant-curvature spaces’_ , J. Phys. A: Math. Gen. 34, 4705 (2001).
  * [154] G. Thompson, _‘Polynomial constants of motion in flat space’_ , J. Math. Phys. 25(12), 3474 (1984).
  * [155] G. Koenigs, _‘Sur les géodésiques a intégrales quadratiques’_. A note in G. Darboux, _‘Lecons sur la théorie générale des surfaces’_ , Vol IV, pp. 368-404, Paris: Gauthier-Villars (1896).
  * [156] C. Markakis, _‘Constants of motion in stationary axisymmetric gravitational fields’_ , MNRAS 441, 2974 (2014).
  * [157] C.M. Adlam, R.G. McLenaghan and R.G. Smirnov, _‘An orbit analysis approach to the study of superintegrable systems in Euclidean plane’_ , Phys. Atom. Nucl. 70(3), 486 (2007).
  * [158] B. Coll, J.J. Ferrando and J.A. Sáez, _‘On the geometry of Killing and conformal tensors’_ , J. Math. Phys. 47, 062503 (2006).
  * [159] M. Crampin, _‘Structural equations for a special class of conformal Killing tensors of arbitrary valence’_ , Rep. Math. Phys. 62(2), 241 (2008).
  * [160] P. Sommers, _‘On Killing tensors and constants of motion’_ , J. Math. Phys. 14(6), 787 (1973).
  * [161] V.P. Ermakov, _‘Second-order differential equations: conditions of complete integrability’_ , Appl. Anal. Discrete Math. 2, 123 (2008). (Translated from Russian by A.O. HHarin, original paper published in Universita Izvestia Kiev, Series III 9 (1880), 1-25)
  * [162] J.R. Ray and J.L. Reid, _‘More exact invariants for the time-dependent harmonic oscillator’_ , Phys. Lett. A 71(4), 317 (1979).
  * [163] C. Athorne, _‘Polyhedral Ermakov systems’_ , Phys. Lett. A 151(8), 407 (1990).
  * [164] P.G.L. Leach, _‘Generalized Ermakov systems’_ , Phys. Lett. A 158(3,4), 102 (1991).
  * [165] K.S. Govinder and P.G.L. Leach, _‘Ermakov systems: a group theoretic approach’_ , Phys. Lett. A 186, 391 (1994).
  * [166] J. Goedert and F. Haas, _‘On the Lie symmetries of a class of generalized Ermakov systems’_ , Phys. Lett. A 239, 348 (1998).
  * [167] P.G.L. Leach and K. Andriopoulos, _‘The Ermakov equation: A commentary’_ , Appl. Anal. Discrete Math. 2, 146 (2008).
  * [168] R. Naz, _‘Noether-type Hamiltonian symmetry classification, first integrals and exact solutions of two classes of the generalized Ermakov’s systems’_ , Eur. Phys. J. Plus 135, 641 (2020).
  * [169] H.R. Lewis, _‘Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators’_ , J. Math. Phys. 9(11), 1976 (1968).
  * [170] M. Tsamparlis, A. Paliathanasis and L. Karpathopoulos, _‘Autonomous three-dimensional Newtonian systems which admit Lie and Noether point symmetries’_ , J. Phys. A: Math. Theor. 45, 275201 (2012).
  * [171] M. Tsamparlis, _‘Geometrization of Lie and Noether symmetries and applications’_ , Int. J. Mod. Phys.: Conf. Ser. 38, 1560078 (2015).
  * [172] G.H. Katzin, J. Levine and R.N. Sane, _‘Time-dependent dynamical symmetry mappings and associated constants of motion for classical particle systems. II’_ , J. Math. Phys. 18(3), 424 (1977).
  * [173] G.H. Katzin and J. Levine, _‘Time-dependent dynamical symmetries and constants of motion. III. Time-dependent harmonic oscillator’_ , J. Math. Phys. 18(6), 1267 (1977).
  * [174] J.R. Ray and J.L. Reid, _‘Noether’s theorem, time-dependent invariants and nonlinear equations of motion’_ , J. Math. Phys. 20(10), 2054 (1979).
  * [175] G.E. Prince and C.J. Eliezer, _‘Symmetries of the time-dependent $N$-dimensional oscillator’_, J. Phys. A: Math. Gen. 13, 815 (1980).
  * [176] J.R. Ray, _‘Noether’s theorem and exact invariants for time-dependent systems’_ , J. Phys. A: Math. Gen. 13, 1969 (1980).
  * [177] L. Karpathopoulos, A. Paliathanasis and M. Tsamparlis, _‘Lie and Noether point symmetries for a class of non-autonomous dynamical systems’_ , J. Math. Phys. 58, 082901 (2017).
  * [178] N.J. Günther and P.G.L. Leach, _‘Generalized invariants for the time-dependent harmonic oscillator’_ , J. Math. Phys. 18(4), 572 (1977).
  * [179] G.E. Prince and C.J. Eliezer, _‘On the Lie symmetries of the classical Kepler problem’_ , J. Phys. A: Math. Gen. 14, 587 (1981).
  * [180] V. Rosenhaus and G.H. Katzin, _‘On symmetries, conservation laws, and variational problems for partial differential equations’_ , J. Math. Phys. 35(4), 1998 (1994).
  * [181] M. Tsamparlis and A. Paliathanasis, _‘Symmetries of second-order PDEs and conformal Killing vectors’_ , J. Phys.: Conf. Ser. 621, 012014 (2015).
  * [182] G.H. Katzin and J. levine, _‘Timedependent vector constants of motion, symmetries, and orbit equations for the dynamical system $\ddot{\mathbf{r}}=\hat{\mathbf{i}}_{r}\left\\{\left[\ddot{U}(t)/U(t)\right]r-\left[\mu_{0}/U(t)\right]r^{-2}\right\\}$’_, J. Math. Phys. 24(7), 1761 (1983).
  * [183] N.H. Ibragimov, A.H. Kara and F.H. Mahomed, _‘Lie-Bäcklund and Noether Symmetries with Applications’_ , Nonlinear Dyn. 15, 115 (1998).
  * [184] M.R.M. Crespo da Silva, _‘A transformation approach for finding first integrals of motion of dynamical systems’_ , Int. J. Non-Linear Mech. 9(4), 241 (1974).
  * [185] W. Sarlet and L.Y. Bahar, _‘A direct construction of first integrals for certain non-linear dynamical systems’_ , Int. J. Non-Linear Mech. 15, 133 (1980).
  * [186] H. Stephani, _‘A new interior solution of Einstein’s field equations for a spherically symmetric perfect fluid in shear-free motion’_ , J. Phys. A: Math. Gen. 16, 3529 (1983).
  * [187] D.C. Srivastana, _‘Exact solutions for shear-free motion of spherically symmetric perfect fluid distributions in general relativity’_ , Class. Quantum Grav. 4, 1093 (1987).
  * [188] P.G.L Leach and S.D. Maharaj, _‘A first integral for a class of time-dependent anharmonic oscillators with multiple anharmonicities’_ , J. Math. Phys. 33(6), 2023 (1992).
  * [189] P.G.L. Leach, R. Maartens and S.D. Maharaj, _‘Self-similar solutions of the generalized Emden-Fowler equation’_ , Int. J. Non-Linear Mech. 27(4), 575 (1992).
  * [190] S.D. Maharaj, P.G.L. Leach and R. Maartens, _‘Expanding spherically symmetric models without shear’_ , Gen. Relativ. Grav. 28(1), 35 (1996).
  * [191] B. Muatjetjeja and C.M. Khalique, _‘Exact solutions of the generalized Lane-Emden equations of the first and second kind’_ , Pramana J. Phys. 77(3), 545 (2011).
  * [192] C.M. Khalique, F.M. Mahomed and B. Muatjetjeja, _‘Lagrangian formulation of a generalized Lane-Emden equation and double reduction’_ , J. Nonlin. Math. Phys. 15(2), 152 (2008).
  * [193] A.G. Riess et al., _‘Observational evidence from supernovae for an accelerating universe and a cosmological constant’_ , Astron. J. 116(3), 1009 (1998).
  * [194] S. Perlmutter et al., _‘Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae’_, Astrophys. J. 517(2), 565 (1999).
  * [195] A.G. Riess et al., _‘New Hubble space telescope discoveries of type Ia supernovae at $z\geq 1$: narrowing constraints on the early behavior of dark energy’_, Astrophys. J. 659(1), 98 (2007).
  * [196] N. Suzuki et al., _‘The Hubble space telescope cluster supernova survey. V. Improving the dark-energy constraints above $z>1$ and building an early-type-hosted supernova sample’_, Astrophys. J. 746, 85 (2012).
  * [197] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D.F. Mota, A.G. Riess and J. Silk, _‘In the realm of the Hubble tension – a review of solutions’_ , Class. Quantum Grav. 38, 153001 (2021).
  * [198] A.A. Starobinsky, _‘A new type of isotropic cosmological models without singularity’_ , Phys. Lett. B 91(1), 99 (1980).
  * [199] A.H. Guth, _‘Inflationary universe: A possible solution to the horizon and flatness problems’_ , Phys. Rev. D 23(2), 347 (1981).
  * [200] V. Müller, H.-J. Schmidt and A.A. Starobinsky, _‘The stability of the De Sitter space-time in fourth order gravity’_ , Phys. Lett. B 202(2), 198 (1988).
  * [201] L.A Kofman, A.D. Linde and A.A. Starobinsky, _‘Inflationary universe generated by the combined action of a scalar field and gravitational vacuum polarization’_ , Phys. Lett. B 157(5,6), 361 (1985).
  * [202] R.M. Wald, _‘Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant’_ , Phys. Rev. D 28(8), 2118 (1983).
  * [203] G.W. Gibbons and S.W Hawking, _‘Cosmological event horizons, thermodynamics, and particle creation’_ , Phys. Rev. D 15(10), 2738 (1977).
  * [204] S.W. Hawking and I.G. Moss., _‘Supercooled phase transitions in the very early universe’_ , Phys. Lett. B 110(1), 35 (1982).
  * [205] K. Sato, _‘First-order phase transition of a vacuum and the expansion of the Universe’_ , MNRAS 195, 467 (1981).
  * [206] J.D. Barrow and A.C. Ottewill, _‘The stability of general relativistic cosmological theory’_ , J. Phys. A: Math. Gen. 16, 2757 (1983).
  * [207] B. Ratra and P.J.E Peebles, _‘Cosmological consequences of a rolling homogeneous scalar field’_ , Phys. Rev. D 37(12), 3406 (1988).
  * [208] J.D. Barrow and P. Saich, _‘Scalar-field cosmologies’_ , Class. Quantum Grav. 10(2), 279 (1993).
  * [209] C. Brans and R.H. Dicke, _‘Mach’s Principle and a Relativistic Theory of Gravitation’_ , Phys. Rev. 124(3), 925 (1961).
  * [210] V. Faraoni, _‘Cosmology in Scalar-Tensor Gravity’_ , Fundamental Theories of Physics vol. 139, Springer, Dordrecht, (2004).
  * [211] G.W. Horndeski, _‘Second-order scalar-tensor field equations in a four-dimensional space’_ , Int. J. Theor. Phys. 10(6), 363 (1974).
  * [212] J. O’Hanlon, _‘Intermediate-Range Gravity: A Generally Covariant Model’_ , Phys. Rev. Lett. 29(2), 137 (1972).
  * [213] A. Nicolis, R. Rattazzi and E. Trincherini, _‘Galileon as a local modification of gravity’_ , Phys. Rev. D 79, 064036 (2009).
  * [214] C. Deffayet, G. Esposito-Farèse and A. Vikman, _‘Covariant Galileon’_ , Phys. Rev. D 79(8), 084003 (2009).
  * [215] J.A. Belinchón, T. Harko and M.K. Mak, _‘Exact scalar-tensor cosmological models’_ , Int. J. Mod. Phys. D 26, 1750073 (2017).
  * [216] I.V Fomin and S.V. Chervon, _‘Inflation with explicit parametric connection between general relativity and scalar-tensor gravity’_ , Mod. Phys. Lett. A 33(28), 1850161 (2018).
  * [217] I.V. Fomin, S.V. Chervon and S.D. Maharaj, _‘A new look at the Schrödinger equation in exact scalar field cosmology’_ , Int. J. Geom. Meth. Mod. Phys. 16(2), 1950022 (2019).
  * [218] V.N. Shapovalov, _‘Stäckel spaces’_ , Sib. Math. J. 20, 790 (1979).
  * [219] E. Osetrin and K. Osetrin, _‘Pure radiation in space-time models that admit integration of the eikonal equation by the separation of variables method’_ , J. Math. Phys. 58, 112504 (2017).
  * [220] M. Tsamparlis and A. Paliathanasis, _‘The generic model of general relativity’_ , Arab. J. Math. 8, 201 (2019).
  * [221] V. Obukhov, _‘Separation of variables in Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle in the Stäckel spaces of type (1.1)’_ , Int. J. Geom. Meth. Mod. Phys. 18(3), 2150036 (2021).
  * [222] L. Amendola and S. Tsujikawa, _‘Dark Energy: Theory and Observations’_ , Cambrdige University Press, (2010).
  * [223] M. Demianski, R. de Ritis, G. Marmo, G. Platania, C. Rubano, P. Scudellaro and C. Stornaiolo, _‘Scalar field, nonminimal coupling, and cosmology’_ , Phys. Rev. D 44(10), 3136 (1991).
  * [224] N. Dimakis, A. Giacomini and A. Paliathanasis, _‘Integrability from point symmetries in a family of cosmological Horndeski Lagrangians’_ , Eur. Phys. J. C 77, 458 (2017).
  * [225] N. Dimakis, P.A. Terzis and T. Christodoulakis, _‘Integrability of the mixmaster model’_ , Phys. Rev. D 99(2), 023536 (2019).
  * [226] G. Papagiannopoulos, J.D. Barrow, S. Basilakos, A. Giacomini and, A. Paliathanasis, _‘Dynamical symmetries in Brans-Dicke Cosmology’_ , Phys. Rev. D 95, 024021 (2017).
  * [227] P. Mukherjee and S. Chakrabarti, _‘Exact solutions and accelerating universe in modified Brans-Dicke theories’_ , Eur. Phys. J. C 79, 681 (2019).
  * [228] A. Giacomini, G. Leon, A. Paliathanasis and S. Pan, _‘Cosmological evolution of two-scalar fields cosmology in the Jordan frame’_ , Eur. Phys. J. C 80, 184 (2020).
  * [229] M. Jamil, D. Momeni and R. Myrzakulov, _‘Noether symmetry of $F(T)$ cosmology with quintessence and phantom scalar fields’_, Eur. Phys. J. C 72, 2137 (2012).
  * [230] A. Cid, G. Leon and Y. Leyva, _‘Intermediate accelerated solutions as generic late-time attractors in a modified Jordan-Brans-Dicke theory’_ , J. Cosm. Astropart. Phys. 02, 027 (2016).
  * [231] A. Paliathanasis, _‘Dynamics of chiral cosmology’_ , Class. Quantum Grav. 37, 195014 (2020).
  * [232] A.A. Coley and R.J. van den Hoogen, _‘Dynamics of multi-scalar-field cosmological models and assisted inflation’_ , Phys. Rev. D 62, 023517 (2000).
  * [233] Y.-F. Cai, E.N. Saridakis, M.R. Setare and J.-Q. Xia, _‘Quintom cosmology: Theoretical implications and observations’_ , Phys. Reports 493(1), 1 (2010).
  * [234] A.R. Brown, _‘Hyperbolic Inflation’_ , Phys. Rev. Lett. 121, 251601 (2018).
  * [235] N. Dimakis and A. Paliathanasis, _‘Crossing the phantom divide line as an effect of quantum transitions’_ , Class. Quantum Grav. 38, 075016 (2021).
  * [236] D.H. Lyth, _‘Generating the curvature perturbation at the end of inflation’_ , J. Cosm. Astropart. Phys. 11, 006 (2005).
  * [237] K.-Y. Choi, S.A. Kim and B. Kyae, _‘Primordial curvature perturbation during and at the end of multi-field inflation’_ , Nucl. Phys. B 861(3), 271 (2012).
  * [238] D. Langlois and S. Renaux-Petel, _‘Perturbations in generalized multi-field inflation’_ , J. Cosm. Astropart. Phys. 04, 017 (2008).
  * [239] S. Nojiri, S.D. Odintsov and V.K. Oikonomou, _‘Ghost-free $F(R)$ gravity with Lagrange multiplier constraint’_, Phys. Lett. B 775, 44 (2017).
  * [240] S. Capozziello, J. Matsumoto, S. Nojiri and S.D Odintsov, Phys. Lett. B 693(2), 198 (2010).
  * [241] S.V. Chervon, I.V. Fomin, E.O. Pozdeeva, M. Sami and S.Y. Vernov, Phys. Rev. D 100, 063522 (2019).
  * [242] L.G.S. Duarte, I.C. Moreira, N. Euler and W.-H. Steeb, _‘Invertible point transformations, Lie symmetries and the Painlevé test for the equation $\frac{d^{2}x}{dt^{2}}+f_{1}(t)\frac{dx}{dt}+f_{2}(t)x+f_{3}(t)x^{n}=0$’_, Physica Scripta 43, 449 (1991).
  * [243] A.R. Liddle and R.J. Scherrer, _‘Classification of scalar field potentials with cosmological scaling solutions’_ , Phys. Rev. D 59, 023509 (1999).
  * [244] J.-P. Uzan, _‘Cosmological scaling solutions of nonminimally coupled scalar fields’_ , Phys. Rev. D 59, 123510 (1999).
  * [245] P.G.L. Leach, _‘Applications of the Lie theory of extended groups in Hamiltonian Mechanics: The oscillator and the Kepler problem’_ , J. Austral. Math. Soc. B 23, 173 (1981).
  * [246] H.R. Lewis and P.G.L. Leach, _‘A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians’_ , J. Math. Phys. 23(12), 2371 (1982).
  * [247] M. Abramowitz and I.A. Stegun, _‘Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables’_ , Dover, NY (1965).
  * [248] J.D. Hadjidemetriou, _‘Two-Body Problem with Variable Mass: A New Approach’_ , Icarus 2, 440 (1963).
  * [249] A.R. Plastino and J.C. Muzzio, _‘On the use and abuse of Newton’s second law for variable mass problems’_ , Celest. Mech. Dyn. Astron. 53, 227 (1992).
  * [250] H. Gyldén, _‘Die Bahnbewegungen in einem Systeme von zwei Körpern in dem Falle, dass die Massen Veränderungen unterworfen sind’_ , Astron. Nachr. 109(2593), 1 (1884).
  * [251] J. Mestschersky, _‘Ein Specialfall des Gyldén’schen Problems (A.N. 2593)’_ Astron. Nachr. 132(3153), 8 (1893).
  * [252] J. Mestschersky, _‘Ueber die Integration der Bewegungsgleichungen im Probleme zweier Körper von veränderlicher Masse’_ Astron. Nachr. 159(3807), 229 (1902).
  * [253] C. Prieto and J.A. Docobo, _‘Analytic solution of the two-body problem with slowly decreasing mass’_ , Astron. Astrophys. 318, 657 (1997).
  * [254] W.A. Rahoma, F.A. Abd El-Salam and M.K. Ahmed, _‘Analytical Treatment of the Two-Body Problem with Slowly Varying Mass’_ , J. Astrophs. Astr. 30, 187 (2009).
  * [255] P. Debye and E. Hückel, _‘Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen’_ , Physikalische Zeitschrift 24(9), 185 (1923).
  * [256] P.M. Bellan, _‘Fundamentals of Plasma Physics’_ , Cambridge University Press (2006).
  * [257] H. Yukawa, _‘On the Interaction of Elementary Particles. I’_ , Proc. Phys. Math. Soc. Jap. 17, 48 (1935); Prog. Theor. Phys. Suppl. 1, 1 (1955).
  * [258] J.E. Jones, _‘On the determination of molecular fields. I. From the variation of the viscocity of a gas with temperature’_ , Proc. Roy. Soc. Lon. Ser. A 106(738), 441 (1924).
  * [259] J.E. Jones, _‘On the determination of molecular fields. II. From the equation of state of a gas’_ , Proc. Roy. Soc. Lon. Ser. A 106(738), 463 (1924).
  * [260] J.E. Lennard-Jones, _‘Cohesion’_ , Proc. Phys. Soc. 43(5), 461 (1931).

### Index

  * Acceleration era §13.1
  * Action §1.2
    * absolutely invariant §1.2
    * gauge invariant §1.2
  * Algebra
    * conformal §2.3
    * homothetic §10.3
  * Approximation
    * RMHD §5.3.3
  * Binary system §14.3.5
  * Central motion §14.2
  * Charge density §5.4.3
  * Christoffel symbols §2.4
  * Collineation §2.1
    * affine §2.5
    * generic §2.1
    * geometric §2.1
    * projective §2.5
    * proper affine §2.5
    * special projective §2.5
  * Condition
    * Cartan §3.5.2
    * Noether Proposition 1.2.3, §1.4
    * weak Noether §1.5, §3.4
  * Conformal factor §2.2
  * Connection §2.4
    * Riemannian §2.4
  * Coordinate
    * cyclic §3.2
  * Coordinates
    * action-angle §3.3
    * polar §10.3.3
  * Current
    * conduction §5.4.3
    * Hall §5.4.4
  * Dark energy §13.1
  * Degrees of freedom §3.1
  * Dust §5.3.1
  * Dynamical system §3.1
    * generalized Ermakov §10.1
    * integrable §3.1
    * Liouville integrable §3.3
    * superintegrable §3.3
    * Whittaker §9.6.2
  * Eccentricity §12.14.2
  * Effect
    * Hall §5.4.4
  * Eigendirection
    * fast §4.5.1
    * slow §4.5.1
  * Electric field §5.4.3
  * Energy (mass) density §5.3.1
  * Equation
    * auxiliary §10.2, §12.11.1
    * Bertrand-Darboux §7.2.1, §8.3, §9.3
    * characteristic §4.5
    * continuity §5.4.3
    * Generalized Killing §1.7
    * generalized Lane-Emden §12.15.1
    * Hamilton-Jacobi §3.5.4
    * Killing §1.6
    * Klein-Gordon §13.3.3
    * NBH §1.5
    * Scrödinger §14.3.4
    * state §13.2
    * wave §6.5.3, §8.5
    * weak generalized Killing §1.7
  * Equations
    * constraint §5.3.1, §5.4.3
    * Einstein field §5.4.1
    * Euler-Lagrange Proposition 1.2.2
    * generalized E-L Proposition 1.2.4
    * geodesic §7.3, §9.6.1
    * Maxwell §5.4.3
    * propagation §5.3.1, §5.4.3
  * Equilibrium point §4.1
  * Expansion §5.3.1
    * screen §5.3.2
  * Expansion era §13.1
  * First integral §3.1
    * autonomous §3.3
    * Ermakov §10.1
    * fundamental §9.5
    * Hojman §3.5.6
    * $m$th-order §11.2
    * Noether Theorem 1.2.1, §1.4
    * non-Noetherian Remark 3.5.2
    * quadratic §7.1
    * time-dependent §3.3
  * First prolongation
    * weak §1.5
  * Fixed point §4.1, §4.5.4
    * asymptotically stable §4.3
    * attractor §4.5.4
    * borderline §4.5.4
    * center §4.5.3
    * degenerate sink §4.5.2
    * degenerate source §4.5.2
    * exponentially stable §4.3
    * hyperbolic §4.5.4
    * isolated §4.5.4
    * non-hyperbolic §4.5.4
    * non-isolated §4.5.4
    * repeller §4.5.4
    * saddle §4.5.1
    * sink §4.5.1
    * source §4.5.1
    * spiral §4.5.3
    * stable §4.3
    * star §4.5.2
  * Fluid
    * anisotropic §5.3.1
    * electromagnetic string §5.1
    * heat conducting §5.3.1
    * isolated anisotropic gravitating §5.1
    * perfect §5.3.1
    * perfect string §5.3.3
    * shear-free perfect §12.15
    * string §5.1
  * Four-current §5.4.3
  * Function
    * Hubble §13.2
    * Noether §1.4
    * projection §2.5
  * Functional §1.2
    * action-like §1.2
  * Gas
    * ideal §13.2
  * Generator §1.1, §1.2
    * induced §1.1
    * space §1.2
    * time §1.2
  * Geometrical object §2.1
  * Hamilton’s equations §3.2
  * Hamilton’s least action principle Proposition 1.2.2
  * Hamiltonian §3.2
  * Hamiltonian vector field Example 1.4.1
  * Hypersurface Proposition 1.2.4
    * natural Proposition 1.2.4
  * Identity
    * Ricci §5.3.1
  * Inflation §13.1
  * Inflaton §13.1
  * Invariant
    * Ermakov-Lewis §8.5
    * function §1.1, §1.3
    * Lewis §10.2, §12.11.1
    * ODE §1.3, §1.4
  * Isotropic strain §5.3.1
  * Kronecker delta
    * generalized §5.2.2
  * Lagrangian
    * linearized §4.2.1
    * regular §3.2, §3.4
  * Laguerre polynomial §14.3.4
  * Law
    * Ohm §5.4.4
  * Levi-Civita symbols §5.2.2
  * Lie bracket Remark 7.2.1
  * Lnearization matrix §4.2
  * Magnetic field §5.4.3
    * frozen §5.4.4
  * Magnetohydrodynamics
    * relativistic §5.1
  * Manifold
    * flat §2.3
  * Method
    * algebraic §3.5.5
    * basis §12.1, §12.7
    * direct §3.5.5
    * Euler-Duarte-Moreira §13.3
    * geometric §3.5.5
    * polynomial §12.1, §12.4
  * Metric
    * conformally flat §6.2
    * conformally related §6.2
    * flat §2.3
    * Jacobi §3.5.5
    * kinetic §3.5.5
  * Momentum
    * angular §10.3.3, §12.12, §12.9.1, §7.4
    * conjugate §3.2, §9.6.1
    * generalized §10.3.3
  * Motion §2.2
    * affine §2.5
    * conformal §2.2
    * homothetic §2.2
    * special conformal §2.2
  * Oscillator
    * 1d time-dependent §10.1, §10.2
    * 3d harmonic §7.4
    * 3d time-dependent §12.11
    * coupled harmonic §9.6.3
    * time-dependent Proposition 12.11.1, §12.12, Proposition 12.12.1, §12.15, §12.9.2, §14.3.1
  * Pair
    * Lax §3.5.3
  * Parameter
    * Brans-Dicke §13.2
  * Path Proposition 1.2.2
    * natural Remark 1.1.1, Proposition 1.2.2
  * Phase plane §4.4, §4.5
  * Phase portrait §4.5
  * Plasma §5.1
  * Potential
    * Class I §8.4
    * Class II §8.5
    * Class III §8.6
    * generalized Kepler §12.8, §7.4
    * interatomic pair §14.3.6
    * Kepler §14.3.2, §7.4, §7.4
    * Lennard-Jones §14.3.6
    * Newton-Cotes §12.9.2
    * second order integrable Chapter 8
    * separable §8.5
    * time-dependent §12.1, Proposition 12.10.1
    * time-dependent central §14.1, Theorem 14.2.1
    * time-dependent generalized Kepler §12.8, §12.9.2, Proposition 12.9.2, §14.3.2
    * time-dependent interatomic pair §14.3.6
    * time-dependent Kepler §12.9.2, §14.3.5
    * time-dependent Yukawa §14.3.6
    * Yukawa §14.3.6
  * Pressure
    * isotropic §5.3.1
  * Pseudotensor §5.2.2
    * Levi-Civita §5.2.2
  * Representation
    * Lax §3.5.3
  * Scalar field §13.1
    * Brans-Dicke §13.2
    * quintessence §13.2
  * Scale factor §13.1, §13.2, §5.7.3
  * Shear §5.3.1
  * Solution
    * Darboux §8.5
  * Space
    * screen §5.3.2
  * Spacetime §5.2.1
    * Bianchi §6.1
    * Bianchi III Proposition 6.3.1
    * Bianchi V Proposition 6.4.1
    * decomposable §6.2
    * FLRW §13.1, §13.2
    * FRW §5.7.3
    * LRS §6.1
    * Minkowski Example 2.3.1
    * orientable §5.2.1
  * Symmetry §1.1, §1.3, §1.5
    * condition §1.3
    * contact §1.4
    * dynamical Lie §1.4
    * form-invariance §3.5.6
    * gauged dynamical Lie §1.4
    * generalized §1.4
    * geometric §2.1
    * hidden §2.6
    * Lie §1.4, §3.5.1
    * Lie point §1.4, §6.5.3
    * Lie-Bäcklund §1.4
    * point Noether §1.4, §10.3, Theorem 10.3.1
    * weak Noether §1.5, §3.4
  * System
    * linear §4.1
    * linearized §4.2
    * non-linear §4.1
  * Tensor
    * density §5.2.2
    * Einstein §13.2, §5.4.1
    * electromagnetic §5.3.3, §5.4.3
    * EMSF energy momentum §5.4
    * energy momentum §13.2, §5.1
    * Jauch-Hill-Fradkin §12.11.2, §12.8, §14.3.1, §7.4
    * Killing §2.6
    * Levi-Civita §5.2.2
    * Minkowski energy momentum §5.3.3
    * oriented §5.2.2
    * plane projection §5.3.2
    * projection §5.1, §5.1, §5.2.3
    * reducible Killing §2.6
    * relative §5.2.1
    * Ricci §5.4.1
    * Riemann §2.4
    * screen §5.3.2
    * screen projection §5.1
    * screen rotation §5.3.2
    * screen shear §5.3.2
    * shear stress §5.3.1
    * stress §5.3.1
    * vorticity §5.3.1
  * Theorem
    * Inverse Noether Theorem 3.4.1
    * Noether’s Theorem 1.2.1
    * Poincaré-Bendixson §4.4
  * Time
    * cosmic §13.2
  * Trajectory Remark 1.1.1, §12.14.2, §3.1
  * Transformation
    * canonical §3.2
    * infinitesimal §1.1
    * Legendre §3.2
  * Variable mass §14.3.5
  * Variables
    * action-angle §3.2
    * dynamical §5.1
    * kinematic §5.1
  * Variation
    * general §1.1
  * Vector
    * conformal Killing §2.2
    * energy (heat) flux §5.3.1
    * gradient §2.2
    * Greenberg §5.3.2
    * homothetic §2.2
    * Killing §2.2
    * proper Conformal Killing §2.2
    * Runge-Lenz §12.10.1, §12.8, §7.4
    * screen rotation §5.3.2
    * special conformal Killing §2.2
    * state §4.1.1
    * vorticity §5.3.1
  * Wavefunction §14.3.4
  * $\kappa$-prolongation §1.1, §1.3, §1.3


# Haplotype-based variant detection from short-read sequencing

Erik Garrison and Gabor Marth

###### Abstract

The direct detection of haplotypes from short-read DNA sequencing data
requires changes to existing small-variant detection methods. Here, we develop
a Bayesian statistical framework which is capable of modeling multiallelic
loci in sets of individuals with non-uniform copy number. We then describe our
implementation of this framework in a haplotype-based variant detector,
FreeBayes.

##  1 Motivation

While _statistical phasing_ approaches are necessary for the determination of
large-scale haplotype structure [Browning and Browning, 2007, Li et al., 2010,
Delaneau et al., 2012, Howie et al., 2011], sequencing traces provide short-
range phasing information that may be employed directly in primary variant
detection to establish phase between proximal alleles. Present read lengths
and error rates limit this _physical phasing_ approach to variants clustered
within tens of bases, but as the cost of obtaining long sequencing traces
decreases [Branton et al., 2008, Clarke et al., 2009], physical phasing
methods will enable the determination of larger haplotype structure directly
using only sequence information from a single sample.

Haplotype-based variant detection methods, in which short haplotypes are read
directly from sequencing traces, offer a number of benefits over methods which
operate on a single position at a time. Haplotype-based methods ensure
semantic consistency among described variants by simultaneously evaluating all
classes of alleles in the same context. The use of locally-phased genotype
data can lower the computational burden of genotype imputation by reducing the
possible space of haplotypes which must be considered. Locally phased
genotypes can be used to improve genotyping accuracy in the context of rare
variations that can be difficult to impute due to sparse linkage information.
Similarly, they can assist in the design of genotyping assays, which can fail
in the context of undescribed variation at the assayed locus. Provided
sequencing errors are independent, the use of longer haplotypes in variant
detection can improve detection by increasing the signal to noise ratio of the
genotype likelihood space that is used in analysis. This follows from the fact
that the space of possible erroneous haplotypes expands dramatically with
haplotype length, while the space of true variation remains constant, with the
number of true alleles less than or equal to the ploidy of the sample at a
given locus.

The direct detection of haplotypes from alignment data presents several
challenges to existing variant detection methods. As the length of a haplotype
increases, so does the number of possible alleles within the haplotype, and
thus methods designed to detect genetic variation over haplotypes in a unified
context must be able to model multiallelism. However, most variant detection
methods establish estimates of the likelihood of polymorphism at a given loci
using statistical models which assume biallelism [Li, 2011, Marth et al.,
1999] and uniform, typically diploid, copy number [DePristo et al., 2011].
Moreover, improper modeling of copy number impedes the accurate detection of
small variants on sex chromosomes, in polyploid organisms, or in locations
with known copy-number variations, where called alleles, genotypes, and
likelihoods should reflect local copy number and global ploidy.

To enable the application of population-level inference methods to the
detection of haplotypes, we generalize the Bayesian statistical method
described by Marth et al. [1999] to allow multiallelic loci and non-uniform
copy number across the samples under consideration. We have implemented this
model in FreeBayes [Garrison, 2012].

##  2 Generalizing variant detection to multiallelic loci and non-uniform copy
number

###  2.1 Definitions

At a given genetic locus we have $n$ samples drawn from a population, each of
which has a copy number or multiplicity of $m$ within the locus. We denote the
number of copies of the locus present within our set of samples as
$M=\sum_{i=1}^{n}m_{i}$. Among these $M$ copies we have $K$ distinct alleles,
$b_{1},\ldots,b_{K}$ with allele frequencies $f_{1},\ldots,f_{K}$. Each
individual has an unphased genotype $G_{i}$ comprised of $k_{i}$ distinct
alleles $b_{i_{1}},\ldots,b_{k_{i}}$ and corresponding genotype allele
frequencies $f_{i_{1}},\ldots,f_{i_{k_{i}}}$, which may be equivalently
expressed as a multiset of alleles $B_{i}:|B_{i}|=m_{i}$. For the purposes of
our analysis, we assume that we cannot accurately discern phasing information
outside of the haplotype detection window, so our $G_{i}$ are unordered and
all $G_{i}$ containing equivalent alleles and frequencies are regarded as
equivalent. Assume a set of $s_{i}$ sequencing observations
$r_{i_{1}},\ldots,r_{i_{s_{i}}}=R_{i}$ over each sample in our set of $n$
samples such that there are $\sum_{i=1}^{n}s_{i}$ reads at the genetic locus
under analysis. $Q_{i}$ denotes the mapping quality, or probability that the
read $r_{i}$ is mis-mapped against the reference.

###  2.2 A Bayesian approach

To genotype the samples at a specific locus, we could simply apply a Bayesian
statistic relating $P(G_{i}|R_{i})$ to the likelihood of sequencing errors in
our reads and the prior likelihood of specific genotypes. However, this
maximum-likelihood approach limits our ability to incorporate information from
other individuals in the population under analysis, which can improve
detection power.

Given a set of genotypes $G_{1},\ldots,G_{n}$ and observations observations
$R_{1},\ldots,R_{n}$ for all individuals at the current genetic locus, we can
use Bayes’ theorem to related the probability of a specific combination of
genotypes to both the quality of sequencing observations and _a priori_
expectations about the distribution of alleles within a set of individuals
sampled from the same population:

$P(G_{1},\ldots,G_{n}|R_{1},\ldots,R_{n})={P(G_{1},\ldots,G_{n})P(R_{1},\ldots,R_{n}|G_{1},\ldots,G_{n})\over
P(R_{1},\ldots,R_{n})}\\\ $ (1)

$P(G_{1},\ldots,G_{n}|R_{1},\ldots,R_{n})={P(G_{1},\ldots,G_{n})\prod_{i=1}^{n}P(R_{i}|G_{i})\over\sum_{\forall{G_{1},\ldots,G_{n}}}(P(G_{1},\ldots,G_{n})\prod_{i=1}^{n}P(R_{i}|G_{i}))}$
(2)

Under this decomposition,
$P(R_{1},\ldots,R_{n}|G_{1},\ldots,G_{n})=\prod_{i=1}^{n}P(R_{i}|G_{i})$
represents the likelihood that our observations match a given genotype
combination (our data likelihood), and $P(G_{1},\ldots,G_{n})$ represents the
prior likelihood of observing a specific genotype combination. We estimate the
data likelihood as the joint probability that the observations for a specific
individual support a given genotype. We use a neutral model of allele
diffusion conditioned on an estimated population mutation rate to estimate the
prior probability of sampling a given collection of genotypes.

Except for situations with small numbers of samples and alleles, we avoid the
explicit evaluation of the posterior distribution as implied by (2), instead
using a number of optimizations to make the algorithm tractable to apply to
very large datasets (see section 3.3).

###  2.3 Estimating the probability of a sample genotype given sequencing
observations, $P(R_{i}|G_{i})$

Given a set of reads $R_{i}=r_{i_{1}},\ldots,r_{i_{s_{i}}}$ of a sample at a
given locus, we can extract a set of $k_{i}$ observed alleles
$B^{\prime}_{i}=b^{\prime}_{1},\ldots,b^{\prime}_{k_{i}}$ which encapsulate
the potential set of represented variants at the locus in the given sample,
including erroneous observations. Each of these observed alleles
$b^{\prime}_{l}$ has a frequency $o^{\prime}_{f}$ within the observations of
the individual sample $:\sum_{j=1}^{k_{i}}o^{\prime}_{j}=s_{i}$ and
corresponds to a true allele $b_{l}$.

If we had perfect observations of a locus, $P(R_{i}|G_{i})$ for any individual
would approximate the probability of sampling observations $R_{i}$ out of
$G_{i}$ with replacement. This probability is given by the multinomial
distribution in $s_{i}$ over the probability $P(b_{l})$ of obtaining each
allele from the given genotype, which is ${f_{i_{j}}\over m_{i}}$ for each
allele frequency in the frequencies which define $G_{i}$,
$f_{i_{1}},\ldots,f_{i_{k_{i}}}$.

$P(R_{i}|G_{i})\approx
P(B^{\prime}_{i}|G_{i})={s_{i}!\over{\prod_{j=1}^{k_{i}}o^{\prime}_{j}!}}\prod_{j=1}^{k_{i}}{\left({f_{i_{j}}\over
m_{i}}\right)^{o^{\prime}_{j}}}$ (3)

Our observations are not perfect, and thus we must account for the probability
of errors in the reads. We can use the per-base quality scores provided by
sequencing systems to develop the probability that an observed allele is drawn
from an underlying true allele, $P(B^{\prime}_{i}|R_{i})$. We assume a mapping
between sequencing quality scores and allele qualities such that each observed
allele $b^{\prime}_{l}$ has a corresponding quality $q_{l}$ which approximates
the probability that the observed allele is incorrect.

Furthermore, we must sum $P(R_{i}|G_{i})$ for all possible $R_{i}$
combinations $\forall(R_{i}\in G_{i}:|R_{i}|=k_{i})$ drawn from our genotype
to obtain the joint probability of $R_{i}$ given $G_{i}$, as each
$\prod_{l=1}^{s_{i}}{P(b^{\prime}_{l}|b_{l})}$ only accounts for the marginal
probability of the a specific $R_{i}$ given $B^{\prime}_{i}$.

This extends $P(R_{i}|G_{i})$ as follows:

$P(R_{i}|G_{i})=\sum_{\forall(R_{i}\in
G_{i})}\left({s_{i}!\over{\prod_{j=1}^{k_{i}}o^{\prime}_{j}!}}\prod_{j=1}^{k_{i}}{\left({f_{i_{j}}\over
m_{i}}\right)^{o^{\prime}_{j}}}\prod_{l=1}^{s_{i}}{P(b^{\prime}_{l}|b_{l})}\right)$
(4)

In summary, the probability of obtaining a set of reads given an underlying
genotype is proportional to the probability of sampling the set of
observations from the underlying genotype, scaled by the probability that our
reads are correct. As $q_{l}$ approximates the probability that a specific
$b_{l}$ is incorrect, $P(b^{\prime}_{l}|b_{l})=1-q_{l}$ when $b_{l}\in G_{i}$
and $P(b^{\prime}_{l}|b_{l})=q_{l}$ when $b_{l}\notin G_{i}$.

###  2.4 Priors for unphased genotype combinations, $P(G_{1},\ldots,G_{n})$

####  2.4.1 Decomposition of prior probability of genotype combination

Let $G_{1},\ldots,G_{n}$ denote the set of genotypes at the locus and
$f_{1},\ldots,f_{k}$ denote the set of allele frequencies which corresponds to
these genotypes. We estimate the prior likelihood of observing a specific
combination of genotypes within a given locus by decomposition into resolvable
terms:

$P(G_{1},\ldots,G_{n})=P(G_{1},\ldots,G_{n}\cap f_{1},\ldots,f_{k})$ (5)

The probability of a given genotype combination is equivalent to the
intersection of that probability and the probability of the corresponding set
of allele frequencies. This identity follows from the fact that the allele
frequencies are derived from the set of genotypes and we always will have the
same $f_{1},\ldots,f_{k}$ for any equivalent $G_{1},\ldots,G_{n}$.

Following Bayes’ Rule, this identity further decomposes to:

$P(G_{1},\ldots,G_{n}\cap
f_{1},\ldots,f_{k})=P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})P(f_{1},\ldots,f_{k})$
(6)

We now can estimate the prior probability of $G_{1},\ldots,G_{n}$ in terms of
the genotype combination sampling probability,
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})$, and the probability of observing a
given allele frequency in our population, $P(f_{1},\ldots,f_{k})$.

####  2.4.2 Genotype combination sampling probability
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})$

The multinomial coefficient ${M\choose f_{1},\ldots,f_{k}}$ gives the number
of ways which a set of alleles with frequencies $f_{1},\ldots,f_{k}$ may be
distributed among $M$ copies of a locus. For phased genotypes $\hat{G_{i}}$
the probability of sampling a specific $\hat{G_{1}},\ldots,\hat{G_{n}}$ given
allele frequencies $f_{1},\ldots,f_{k}$ is thus provided by the inverse of
this term:

$P(\hat{G_{1}},\ldots,\hat{G_{n}}|f_{1},\ldots,f_{k})={M\choose
f_{1},\ldots,f_{k}}^{-1}$ (7)

However, our model is limited to unphased genotypes because our primary data
only allows phasing within a limited context. Consequently, we must adjust (7)
to reflect the number of phased genotypes which correspond to the unphased
genotyping $G_{1},\ldots,G_{n}$. Each unphased genotype corresponds to as many
phased genotypes as there are permutations of the alleles in $G_{i}$. Thus,
for a given unphased genotyping $G_{1},\ldots,G_{n}$, there are
$\prod_{i=1}^{n}{m_{i}\choose f_{i_{1}},\ldots,f_{i_{k_{i}}}}$ phased
genotypings.

In conjunction, these two terms provide the probability of sampling a
particular unphased genotype combination given a set of allele frequencies:

$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})={M\choose
f_{1},\ldots,f_{k}}^{-1}\prod_{i=1}^{n}{m_{i}\choose
f_{i_{1}},\ldots,f_{i_{k_{i}}}}=\frac{1}{M!}\prod_{l=1}^{k}f_{l}!\prod_{i=1}^{n}\frac{m_{i}!}{\prod_{j=1}^{k_{i}}f_{i_{j}}!}$
(8)

In the case of a fully diploid population, the product of all possible
multiset permutations of all genotypes reduces to $2^{h}$, where $h$ is the
number of heterozygous genotypes, simplifying (8) to:

$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})=2^{h}{M\choose
f_{1},\ldots,f_{k}}^{-1}$ (9)

####  2.4.3 Derivation of $P(f_{1},\ldots,f_{k})$ by Ewens’ sampling formula

Provided our sample size $n$ is small relative to the population which it
samples, and the population is in equilibrium under mutation and genetic
drift, the probability of observing a given set of allele frequencies at a
locus is given by Ewens’ sampling formula [Ewens, 1972]. Ewens’ sampling
formula is based on an infinite alleles coalescent model, and relates the
probability of observing a given set of allele frequencies to the number of
sampled chromosomes at the locus ($M$) and the population mutation rate
$\theta$.

The application of Ewens’ formula to our context is straightforward. Let
$a_{f}$ be the number of alleles among $b_{1},\ldots,b_{k}$ whose allele
frequency within our set of samples is $f$. We can thus transform our set of
frequencies $f_{1},\ldots,f_{k}$ into a set of non-negative frequency counts
$a_{1},\ldots,a_{M}:\sum_{f=1}^{M}{fa_{f}}=M$. As many $f_{1},\ldots,f_{k}$
can map to the same $a_{1},\ldots,a_{M}$, this transformation is not
invertible, but it is unique from $a_{1},\ldots,a_{M}$ to
$f_{1},\ldots,f_{k}$.

Having transformed a set of frequencies over alleles to a set of frequency
counts over frequencies, we can now use Ewens’ sampling formula to approximate
$P(f_{1},\ldots,f_{k})$ given $\theta$:

$P(f_{1},\ldots,f_{k})=P(a_{1},\ldots,a_{M})={M!\over\theta\prod_{z=1}^{M-1}(\theta+z)}\prod_{j=1}^{M}{\theta^{a_{j}}\over
j^{a_{j}}a_{j}!}$ (10)

In the bi-allelic case in which our set of samples has two alleles with
frequencies $f_{1}$ and $f_{2}$ such that $f_{1}+f_{2}=M$:

$P(a_{f_{1}}=1,a_{f_{2}}=1)={M!\over\prod_{z=1}^{M-1}(\theta+z)}{\theta\over
f_{1}f_{2}}$ (11)

While in the monomorphic case, where only a single allele is represented at
this locus in our population, this term reduces to:

$P(a_{M}=1)={(M-1)!\over\prod_{z=1}^{M-1}(\theta+z)}$ (12)

In this case, $P(f_{1},\ldots,f_{k})=1-\theta$ when $M=2$. This is sensible as
$\theta$ represents the population mutation rate, which can be estimated from
the pairwise heterozygosity rate of any two chromosomes in the population
[Watterson, 1975, Tajima, 1983].

##  3 Direct detection of phase from short-read sequencing

By modeling multiallelic loci, this Bayesian statistical framework provides
the foundation for the direct detection of longer, multi-base alleles from
sequence alignments. In this section we describe our implementation of a
haplotype-based variant detection method based on this model.

Our method assembles haplotype observations over minimal, dynamically-
determined, reference-relative windows which contain multiple segregating
alleles. To be used in the analysis, haplotype observations must be derived
from aligned reads which are anchored by reference-matching sequence at both
ends of the detection window. These haplotype observations have derived
quality estimations which allow their incorporation into the general
statistical model described in section 2. We then employ a gradient ascent
method to determine the maximum _a posteriori_ estimate of a mutual genotyping
over all samples under analysis and establish an estimate of the probability
that the loci is polymorphic.

###  3.1 Parsing haplotype observations from sequencing data

In order to establish a range of sequence in which multiple polymorphisms
segregate in the population under analysis, it is necessary to first determine
potentially polymorphic windows in order to bound the analysis. This
determination is complicated by the fact that a strict windowing can
inappropriately break clusters of alleles into multiple variant calls. We
employ a dynamic windowing approach that is driven by the observation of
multiple proximal reference-relative variations (SNPs and indels) in input
alignments.

Where reference-relative variations are separated by less than a configurable
number of non-polymorphic bases in an aligned sequence trace, our method
combines them into a single haplotype allele observation, $H_{i}$. The
observational quality of these haplotype alleles is given as
$\min(q_{l}\,\forall\,b^{\prime}_{i}\in H_{i},\,Q_{i})$, or the minimum of the
supporting read’s mapping quality and the minimum base quality of the
haplotype’s component variant allele observations.

###  3.2 Determining a window over which to assemble haplotype observations

At each position in the reference, we collect allele observations derived from
alignments as described in 3.1. To improve performance, we apply a set of
input filters to exclude alleles from the analysis which are highly unlikely
to be true. These filters require a minimum number of alternate observations
and a minimum sum of base qualities in a single sample in order to incorporate
a putative allele and its observations into the analysis.

We then determine a haplotype length over which to genotype samples by a
bounded iterative process. We first determine the allele passing the input
filters which is longest relative to the reference. For instance, a longer
allele could be a multi-base indel or a composite haplotype observation
flanked by SNPs. Then, we parse haplotype observations from all the alignments
which fully overlap this window, finding the rightmost end of the longest
haplotype allele which begins within the window. This rightmost position is
used to update the haplotype window length, and a new set of haplotype
observations are assembled from the reads fully overlapping the new window.
This process repeats until the rightmost end of the window is not partially
overlapped by any haplotype observations which pass the input filters. This
method will converge given reads have finite length and the only reads which
fully overlap the detection window are used in the analysis.

###  3.3 Detection and genotyping of local haplotypes

Once a window for analysis has been determined, we parse all fully-overlapping
reads into haplotype observations which are anchored at the boundaries of the
window. Given these sets of sequencing observations
$r_{i_{1}},\ldots,r_{i_{s_{i}}}=R_{i}$ and data likelihoods $P(R_{i}|G_{i})$
for each sample and possible genotype derived from the putative alleles, we
then determine the probability of polymorphism at the locus given the Bayesian
model described in section 2.

To establish a maximum _a posteriori_ estimate of the genotype for each
sample, we employ a convergent gradient ascent approach to the posterior
probability distribution over the mutual genotyping across all samples under
our Bayesian model. This process begins at the genotyping across all samples
$G_{1},\ldots,G_{n}$ where each sample’s genotype is the maximum-likelihood
genotype given the data likelihood $P(R_{i}|G_{i})$:

$G_{1},\ldots,G_{n}=\underset{G_{i}}{\operatorname{argmax}}\;P(R_{i}|G_{i})$
(13)

The posterior search then attempts to find a genotyping $G_{1},\ldots,G_{n}$
in the local space of genotypings which has higher posterior probability under
the model than this initial genotyping. In practice, this step is done by
searching through all genotypings in which a single sample has up to the $N$th
best genotype when ranked by $P(R_{i}|G_{i})$, and $N$ is a small number (e.g.
2). This search starts with some set of genotypes $G_{1},\ldots,G_{n}=\\{G\\}$
and attempts to find a genotyping $\\{G\\}^{\prime}$ such that:

$P(\\{G\\}^{\prime}|R_{1},\ldots,R_{n})>P(\\{G\\}|R_{1},\ldots,R_{n})$ (14)

$\\{G\\}^{\prime}$ is then used as a basis for the next update step. This
search iterates until convergence, but in practice must be bounded at a fixed
number of steps in order to ensure optimal performance. As the quality of
input data increases in coverage and confidence, this search will converge
more quickly because the maximum-likelihood estimate will lie closer to the
maximum _a posteriori_ estimate under the model.

This method incorporates a basic form of genotype imputation into the
detection method, which in practice improves the quality of raw genotypes
produced in primary allele detection and genotyping relative to methods which
only utilize a maximum-likelihood method to determine genotypes. Furthermore,
this method allows for the determination of marginal genotype likelihoods via
the marginalization of assigned genotypes for each sample over the posterior
probability distribution.

###  3.4 Probability of polymorphism

Provided a maximum _a posteriori_ estimate of the genotyping of all the
individuals in our sample, we might like establish an estimate of the quality
of the genotyping. For this, we can use the probability that the locus is
polymorphic, which means that the number of distinct alleles at the locus,
$K$, is greater than 1. While in practice the space of possible genotypings is
too great to integrate over, it is possible to derive the probability that the
loci is polymorphic in our samples by summing across the monomorphic cases:

$P(K>1|R_{1},\ldots,R_{n})=1-P(K=1|R_{1},\ldots,R_{n})$ (15)

Equation (15) thus provides the probability of polymorphism at the site, which
is provided as a quality estimate for each evaluated locus in the output of
FreeBayes.

###  3.5 Marginal likelihoods of individual genotypes

Similarly, we can establish a quality estimate for a single genotype by
summing over the marginal probability of that specific genotype and sample
combination under the model. The marginal probability of a given genotype is
thus:

$P(G_{j}|R_{i},\ldots,R_{n})=\sum_{\forall(\\{G\\}:G_{j}\in\\{G\\})}P(\\{G\\}|R_{i},\ldots,R_{n})$
(16)

In implementation, the development of this term is more complex, as we must
sample enough genotypings from the posterior in order to obtain well-
normalized marginal likelihoods. In practice, we marginalize from the local
space of genotypings in which each individual genotype is no more than a small
number of steps in one sample from the maximum _a posteriori_ estimate of
$G_{i},\ldots,G_{n}$. This space is similar to that used during the posterior
search described in section 3.3. We apply (16) to it to estimate marginal
genotype likelihoods for the most likely individual genotypes, which are
provided for each sample at each site in the output of our implementation.

## References

  * Branton et al. [2008] D. Branton, D. W. Deamer, A. Marziali, H. Bayley, S. A. Benner, T. Butler, M. Di Ventra, S. Garaj, A. Hibbs, X. Huang, S. B. Jovanovich, P. S. Krstic, S. Lindsay, X. S. Ling, C. H. Mastrangelo, A. Meller, J. S. Oliver, Y. V. Pershin, J. M. Ramsey, R. Riehn, G. V. Soni, V. Tabard-Cossa, M. Wanunu, M. Wiggin, and J. A. Schloss.  The potential and challenges of nanopore sequencing.  _Nat. Biotechnol._ , 26(10):1146–1153, Oct 2008\.
  * Browning and Browning [2007] S. R. Browning and B. L. Browning.  Rapid and accurate haplotype phasing and missing-data inference for whole-genome association studies by use of localized haplotype clustering.  _Am. J. Hum. Genet._ , 81(5):1084–1097, Nov 2007\.
  * Clarke et al. [2009] J. Clarke, H. C. Wu, L. Jayasinghe, A. Patel, S. Reid, and H. Bayley.  Continuous base identification for single-molecule nanopore DNA sequencing.  _Nat Nanotechnol_ , 4(4):265–270, Apr 2009.
  * Delaneau et al. [2012] O. Delaneau, J. Marchini, and J. F. Zagury.  A linear complexity phasing method for thousands of genomes.  _Nat. Methods_ , 9(2):179–181, Feb 2012.
  * DePristo et al. [2011] M. A. DePristo, E. Banks, R. Poplin, K. V. Garimella, J. R. Maguire, C. Hartl, A. A. Philippakis, G. del Angel, M. A. Rivas, M. Hanna, A. McKenna, T. J. Fennell, A. M. Kernytsky, A. Y. Sivachenko, K. Cibulskis, S. B. Gabriel, D. Altshuler, and M. J. Daly.  A framework for variation discovery and genotyping using next-generation DNA sequencing data.  _Nat. Genet._ , 43(5):491–498, May 2011.
  * Ewens [1972] W. J. Ewens.  The sampling theory of selectively neutral alleles.  _Theor Popul Biol_ , 3:87–112, Mar 1972.
  * Garrison [2012] E. Garrison.  FreeBayes source repository.  https://github.com/ekg/freebayes, 2012.
  * Howie et al. [2011] B. Howie, J. Marchini, and M. Stephens.  Genotype imputation with thousands of genomes.  _G3 (Bethesda)_ , 1(6):457–470, Nov 2011.
  * Li [2011] H. Li.  A statistical framework for SNP calling, mutation discovery, association mapping and population genetical parameter estimation from sequencing data.  _Bioinformatics_ , 27(21):2987–2993, Nov 2011\.
  * Li et al. [2010] Y. Li, C. J. Willer, J. Ding, P. Scheet, and G. R. Abecasis.  MaCH: using sequence and genotype data to estimate haplotypes and unobserved genotypes.  _Genet. Epidemiol._ , 34(8):816–834, Dec 2010\.
  * Marth et al. [1999] G. T. Marth, I. Korf, M. D. Yandell, R. T. Yeh, Z. Gu, H. Zakeri, N. O. Stitziel, L. Hillier, P. Y. Kwok, and W. R. Gish.  A general approach to single-nucleotide polymorphism discovery.  _Nat. Genet._ , 23:452–456, Dec 1999.
  * Tajima [1983] F. Tajima.  Evolutionary relationship of DNA sequences in finite populations.  _Genetics_ , 105(2):437–460, Oct 1983.
  * Watterson [1975] G. A. Watterson.  On the number of segregating sites in genetical models without recombination.  _Theor Popul Biol_ , 7(2):256–276, Apr 1975.