129 Bibliography [1] Stathopoulos A. and Levine M. Dorsal gradient networks in the Drosophila embryo. Dev. Bio, 246:57–67, 2002. [2] Khaled A. S. Abdel-Ghaffar. The determinant of random power series matrices over finite fields. Linear Algebra Appl., 315(1-3):139–144, 2000. [3] J. Alappattu and J. Pitman. Coloured loop-erased random walk on the complete graph. Combinatorics, Probability and Computing, 17(06):727–740, 2008. [4] Bruce Alberts, Dennis Bray, Karen Hopkin, Alexander Johnson, Julian Lewis, Martin Raff, Keith Roberts, and Peter Walter. essential cell biology; second edition. Garland Science, New York, 2004. [5] David Aldous. Stopping times and tightness. II. Ann. Probab., 17(2):586–595, 1989. [6] David Aldous. Probability distributions on cladograms. 76:1–18, 1996. [7] L. Ambrosio, A. P. Mahowald, and N. Perrimon. l(1)polehole is required maternally for patter formation in the terminal regions of the embryo. Development, 106:145–158, 1989. [8] George E. Andrews, Richard Askey, and Ranjan Roy. Special functions, volume 71 of Ency- clopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999. [9] S. Astigarraga, R. Grossman, J. Diaz-Delfin, C. Caelles, Z. Paroush, and G. Jimenez. A mapk docking site is crtical for downregulation of capicua by torso and egfr rtk signaling. EMBO, 26:668–677, 2007. [10] K.B. Athreya and PE Ney. Branching processes. Dover Publications, 2004. [11] Julien Berestycki. Exchangeable fragmentation-coalescence processes and their equilibrium measures. Electron. J. Probab., 9:no. 25, 770–824 (electronic), 2004. [12] A. M. Berezhkovskii, L. Batsilas, and S. Y. Shvarstman. Ligand trapping in epithelial layers and cell cultures. J. Biophys. Chem., 107:221–227, 2004. 130 [13] A. M. Berezhkovskii, M. I. Monine, C. B. Muratov, and S. Y. Shvarstman. Boundary homog- enization for trapping by patchy surfaces. J. Chem. Phys., 121:11390–4, 2004. [14] A. M. Berezhkovskii, M. I. Monine, C. B. Muratov, and S. Y. Shvarstman. Homogenization of boundary conditions for surfaces with regular arrays of traps. J. Chem. Phys., 121:036103, 2006. [15] Christian Berg. On a generalized gamma convolution related to the q-calculus. In Theory and applications of special functions, volume 13 of Dev. Math., pages 61–76. Springer, New York, 2005. [16] H. Berg and E. Purcell. Physics of chemoreception. Bio. Phys. J., 20:193–219, 1977. [17] Jean Bertoin. Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes. Combin. Probab. Comput., 17(3):329–337, 2008. [18] Jean Bertoin, Philippe Biane, and Marc Yor. Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. In Seminar on Stochastic Analysis, Random Fields and Applications IV, volume 58 of Progr. Probab., pages 45–56. Birkh¨auser,Basel, 2004. [19] Shankar Bhamidi, Steven N. Evans, Ron Peled, and Peter Ralph. Brownian motion on dis- connected sets, basic hypergeometric functions, and some continued fractions of Ramanujan. In Probability and statistics: essays in honor of David A. Freedman, volume 2 of Inst. Math. Stat. Collect., pages 42–75. Inst. Math. Statist., Beachwood, OH, 2008. [20] S. Bhargava and Chandrashekar Adiga. On some continued fraction identities of Srinivasa Ramanujan. Proc. Amer. Math. Soc., 92(1):13–18, 1984. [21] W. Bialek. Physics 562 course notes, spring 2006. Princeton University, Graduate Physics Course. [22] W. Bialek and S. Setayeshgar. Cooperativity, sensitivity and noise in biochemical signaling. 2005. [23] W. Bialek and S. Setayeshgar. Physical limits to biochemical signaling. PNAS, 102:10040– 10045, 2005. [24] Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. A Wiley-Interscience Publication. [25] Alistair N. Boettiger and Michael S. Levine. Transcriptional synchrony in the early drosophila embryo. Manuscript in prep, 2009. 131 [26] Alistair N. Boettiger, Peter Ralph, Michael S. Levine, and Steven Evans. Effects of network topology on noise in transcriptional regulation. Manuscript in prep, 2009. [27] Martin Bohner and Allan Peterson. Dynamic equations on time scales. Birkh¨auserBoston Inc., Boston, MA, 2001. An introduction with applications. [28] K. Borovkov and D. Vere-Jones. Explicit formulae for stationary distributions of stress release processes. J. Appl. Probab., 37(2):315–321, 2000. [29] Onno Boxma, David Perry, Wolfgang Stadje, and Shelemyahu Zacks. A Markovian growth- collapse model. Adv. in Appl. Probab., 38(1):221–243, 2006. [30] W. E. Boyce and R. C. DiPrima. Elementary Differential Equations and Boundary Value Problems; 7th ed. John Wiley & Sons, Inc., Hoboken, NJ, 2003. [31] Otto. Bretscher. Linear Algebra with Applications, Second Edition. Prentice Hall, Upper Saddle River, NJ, 2001. [32] Wlodzimierz Bryc. Classical versions of q-Gaussian processes: conditional moments and Bell’s inequality. Comm. Math. Phys., 219(2):259–270, 2001. [33] W lodzimierz Bryc, Wojciech Matysiak, and Pawe lJ. Szab lowski. Probabilistic aspects of Al- Salam-Chihara polynomials. Proc. Amer. Math. Soc., 133(4):1127–1134 (electronic), 2005. [34] W lodzimierz Bryc and Jacek Weso lowski. Conditional moments of q-Meixner processes. Probab. Theory Related Fields, 131(3):415–441, 2005. [35] G. Burkhardt and Uwe K¨uchler. The semimartingale decomposition of one-dimensional qua- sidiffusions with natural scale. Stochastic Process. Appl., 25(2):237–244, 1987. [36] Wolfgang J. ER Bhler. Generations and degree of relationship in supercritical markov branch- ing processes. Probability Theory and Related Fields, 18(2):141–152, June 1971. [37] Maria-Emilia Caballero, Amaury Lambert, and Geronimo Uribe Bravo. Proof(s) of the lam- perti representation of continuous-state branching processes, 2008. [38] A. Casali and J. Casanova. The spatial control of torso rtk activation: a c-terminal fragment of the trunk protein acts as a signal for torso receptor in the Drosophila embryo. Development, 128:1709–1715, 2001. [39] J. Casanova, M. Furriols, C. A. McCormick, and G. Strhul. Similarties between trunk and spatzle, putative extracellular ligands specifying body pattern in Drosophila. Genes Dev., 9:2539–2544, 1995. [40] J. Casanova, M. Llimargas, S. Greenwood, and G. Struhl. An oncogenic form of human raf can specify terminal body pattern in Drosophila. Mech. Dev., 48:59–64, 1994. 132 [41] J. Casanova and G. Struhl. Localized surface activity of torso, a receptor tyrosine kinase, specifies terminal body pattern in Drosophila. Genes Dev, 3:2025–2038, 1989. [42] R. V. Chacon and B. Jamison. A fundamental property of Markov processes with an appli- cation to equivalence under time changes. Israel J. Math., 33(3-4):241–269 (1980), 1979. A collection of invited papers on ergodic theory. [43] Joseph Chang. Recent common ancestors of all present-day individuals. Advances in Applied Probability, 31:1002–1026, 1999. [44] Joseph Chang. Recent common ancestors of all present-day individuals. Advances in Applied Probability, 31:1002–1026, 1999. [45] Vyjayanthi Chari and Andrew Pressley. A guide to quantum groups. Cambridge University Press, Cambridge, 1994. [46] M. L. Chaudhry. On computations of the mean and variance of the number of renewals: A unified approach. The Journal of the Operational Research Society, 46(11):1352–1364, 1995. [47] T. S. Chihara. An introduction to orthogonal polynomials. Gordon and Breach Science Pub- lishers, New York, 1978. Mathematics and its Applications, Vol. 13. [48] E. Cinnamon, D. Gur-Whanon, A. Helman, D. St. Johnston, G. Jimenez, and Z. Paroush. Capicua integrates input from two maternal systems in Drosophila terminal patterning. EMBO, 23:4571–4582, 2004. [49] F. C. Collins and G. E. Kimball. Diffusion-controlled reaction rates. J. Colloid Sci., 4:425–437, 1949. [50] G. Colombo and P. Dai Pra. A class of piecewise deterministic Markov processes. Markov Process. Related Fields, 7(2):251–287, 2001. [51] Mathieu Coppey, Alistair N. Boettiger, Alexander M. Berezhkovskii, and Stanislav Y. Shvarts- man. Nuclear trapping shapes the terminal gradient in the Drosophila embryo. Current Biol- ogy, 18:915–919, 2008. [52] Mathieu Coppey, Alexander M. Berezhkovskii Yoosik Kim, Alistair N. Boettiger, and Stanislav Y. Shvartsman. Modeling the bicoid gradient: Diffusion and reversible nuclear trap- ping of a stable protein. Developmental Biology, 312:623–630, 2007. [53] L. J. Core, J. J. Waterfall, and J. T. Lis. Nascent rna sequencing reveals, widespread pausing and divergent initiation at human promoters. Science, 322:1845–1848, 2008. 133 [54] M. Costa, M. Marchi, F. Cardarelli, A. Roy, F. Beltram, L. Maffei, and G. M. Ratto. Dynamic regulation of erk2 nuclear translocation and mobility in living cells. J. Cell Sci., 119:4952–4963, 2006. [55] O. L. V. Costa and F. Dufour. Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optim., 47(2):1053–1077, 2008. [56] Richard Cowan and S. N. Chiu. A stochastic model of fragment formation when DNA repli- cates. J. Appl. Probab., 31(2):301–308, 1994. [57] John Cowden and Michael Levine. Ventral dominance governs sequential patterns of gene expression across the dorsal-ventral axis of the neuroectoderm in the Drosophila embryo. Developmental Biology, 262:335–349, 2003. [58] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988. [59] M. H. A. Davis. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B, 46(3):353–388, 1984. With discussion. [60] M. H. A. Davis. Markov models and optimization, volume 49 of Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1993. [61] Donald A. Dawson. Measure-valued Markov processes. In Ecole´ d’Et´ede´ Probabilit´esde Saint- Flour XXI—1991, volume 1541 of Lecture Notes in Math., pages 1–260. Springer, Berlin, 1993. [62] Donald A. Dawson and Edwin A. Perkins. Historical processes, volume 93. 1991. [63] F. R. De Hoog, J. H. Knight, and Stokes A. N. An improved method for numerical inversion of laplace transforms.