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.

Birkhauser, 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. Birkhauser Boston

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 Kuchler. 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’Ete de Probabilites de 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. SIAM J. Sci. Stat. Comput., 3:357–366, 1982.

[64] J. Manuel de las Heras and J. Casanova. Spatially distinct downregulation of capicua repression

and tailless activation by the torso rtk pathway in the Drosophila. Mech. of Dev., 123:481–486,

2006.

[65] P. Diaconis, E. Mayer-Wolf, O. Zeitouni, and M.P.W. Zerner. The Poisson-Dirichlet law is the

unique invariant distribution for uniform split-merge transformations. Annals of Probability,

pages 915–938, 2004.

[66] Peter Donnelly. Partition structures, Polya urns, the Ewens sampling formula, and the ages

of alleles. Theoret. Population Biol., 30(2):271–288, 1986.

[67] Peter Donnelly and Paul Joyce. Continuity and weak convergence of ranked and size-biased

permutations on the infinite simplex. Stochastic Process. Appl., 31(1):89–103, 1989.

134

[68] Peter Donnelly and Simon Tavare. The ages of alleles and a coalescent. Adv. in Appl. Probab.,

18(1):1–19, 1986.

[69] W. Driever and C. Nusslein-Volhard. The bicoid protein determines position in the Drosophila

embryo. Cell, 54:95–104, 1988.

[70] W. Driever and C. Nusslein-Volhard. The bicoid protein determines position in the Drosophila

embryo in a concentration-dependent manner. Cell, 54:95–104, 1988.

[71] W. Driever and C. Nusslein-Volhard. A gradient of bicoid protein in Drosophila embryos. Cell,

54:83–93, 1988.

[72] W. Driever, G. Thoma, and Nusslein-Volhard C. Determination of spatial domains of zy-

gotic gene expression in the Drosophila embryo by the affinity of binding sites for the bicoid

morphogen. Nature, 340:363–368, 1989.

[73] D. Dubhnau and Richard Losick. Bistability in bacteria. Molecular Microbiology, 61:564–572,

2006.

[74] J. B. Duffy. Gal4 system in Drosophila: a fly geneticist’s swiss army knife. Genesis, 34(1-2):1–

15, 2002.

[75] J. B. Duffy and N. Perrimon. The torso pathway in Drosophila: lessons on receptor tyrosine

kinase signaling and pattern formation. Dev. Bio., 166:380–395, 1994.

[76] Francois Dufour and Oswaldo L. V. Costa. Stability of piecewise-deterministic Markov pro-

cesses. SIAM J. Control Optim., 37(5):1483–1502 (electronic), 1999.

[77] Vincent Dumas, Fabrice Guillemin, and Philippe Robert. A Markovian analysis of additive-

increase multiplicative-decrease algorithms. Adv. in Appl. Probab., 34(1):85–111, 2002.

[78] Vincent Dumas, Fabrice Guillemin, and Philippe Robert. A Markovian analysis of additive-

increase multiplicative-decrease algorithms. Adv. in Appl. Probab., 34(1):85–111, 2002.

[79] R. Durrett, B.L. Granovsky, and S. Gueron. The equilibrium behavior of reversible coagulation-

fragmentation processes. Journal of Theoretical Probability, 12(2):447–474, 1999.

[80] Richard Durrett. Probability: theory and examples. Duxbury Press, Belmont, CA, second

edition, 1996.

[81] H. Dym and H. P. McKean. Gaussian processes, function theory, and the inverse spectral

problem. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. Probability

and Mathematical Statistics, Vol. 31.

135

[82] M. Ebisuya, K. Kondoh, and E. Nishida. The duration, magnitude and compartmentaliza-

tion of erk map kinase activity: mechanisms for providing signaling specificity. J. Cell Sci,

118:2997–3002, 2005.

[83] J. Elf and M. Ehrenberg. Fast evaluation of fluctuations in biochemical networks with the

linear noise approximation, 2003.

[84] Iddo Eliazar. The M/G/∞ system revisited: finiteness, summability, long range dependence,

and reverse engineering. Queueing Syst., 55(1):71–82, 2007.

[85] M. Elowitz and S. Leibler. A synthetic oscillatory network of transcriptional regulators. Nature,

403:335–338, 2000.

[86] M. Elowitz and S. Leibler. A synthetic oscillatory network of transcrptional regulators. Nature,

403:335–338, 2000.

[87] M.B. Elowitz, A.J. Levine, E.D. Siggia, and P.S. Swain. Stochastic gene expression in a single

cell, 2002.

[88] A. M. Etheridge and D. R. E. Williams. A decomposition of the (1 + β)-superprocess condi-

tioned on survival. Proc. Roy. Soc. Edinburgh Sect. A, 133(4):829–847, 2003.

[89] A. M. Etheridge and D. R. E. Williams. A decomposition of the (1 + β)-superprocess condi-

tioned on survival. Proc. Roy. Soc. Edinburgh Sect. A, 133(4):829–847, 2003.

[90] Alison M. Etheridge. An introduction to superprocesses, volume 20 of University Lecture Series.

American Mathematical Society, Providence, RI, 2000.

[91] Stewart N. Ethier and Thomas G. Kurtz. Markov processes: Characterization and conver-

gence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical

Statistics. John Wiley & Sons Inc., New York, 1986.

[92] Steven N. Evans. The entrance space of a measure-valued Markov branching process condi-

tioned on nonextinction. Canad. Math. Bull., 35(1):70–74, 1992.

[93] Steven N. Evans. Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh

Sect. A, 123(5):959–971, 1993.

[94] Steven N. Evans. Elementary divisors and determinants of random matrices over a local field.

Stochastic Process. Appl., 102(1):89–102, 2002.

[95] Steven N. Evans and Edwin Perkins. Measure-valued Markov branching processes conditioned

on nonextinction. Israel J. Math., 71(3):329–337, 1990.

136

[96] Steven N. Evans and Jim Pitman. Stationary Markov processes related to stable Ornstein-

Uhlenbeck processes and the additive coalescent. Stochastic Process. Appl., 77(2):175–185,

1998.

[97] Steven N. Evans, Jim Pitman, and Anita Winter. Rayleigh processes, real trees, and root

growth with re-grafting. Probab. Theory Related Fields, 134(1):81–126, 2006.

[98] Steven N. Evans and Peter L. Ralph. Dynamics of the time to the most recent common

ancestor in a large branching population, 2008. cite arxiv:0812.1302.

[99] Pignoni F, Steingrimsson E, and Lengyel JA. bicoid and the terminal system activate tailless

expression in the early Drosophila embryo. Development, 115:239–251, 1992.

[100] Pignoni F, Baldarelli RM, Steingrimsson E, Diaz RJ, Patapoutian A, Merriam JR, and Lengyel

JA. The Drosophila gene tailless is expressed at the embryonic termini and is a member of

the steroid receptor superfamily. Cell, 62:151–163, 1990.

[101] Sprenger F. and Nusslein-Volhard C. Torso receptor activity is regulated by a diffusible ligand

produced at the extracellular terminal regions of the Drosophila egg. Cell, 71:987–1001, 1992.

[102] William Feller. Diffusion processes in genetics. In Proceedings of the Second Berkeley Sym-

posium on Mathematical Statistics and Probability, 1950, pages 227–246, Berkeley and Los

Angeles, 1951. University of California Press.

[103] William Feller. The general diffusion operator and positivity preserving semi-groups in one

dimension. Ann. of Math. (2), 60:417–436, 1954.

[104] William Feller. On differential operators and boundary conditions. Comm. Pure Appl. Math.,

8:203–216, 1955.

[105] William Feller. On generalized Sturm-Liouville operators. In Proceedings of the conference

on differential equations (dedicated to A. Weinstein), pages 251–270. University of Maryland

Book Store, College Park, Md., 1956.

[106] William Feller. Generalized second order differential operators and their lateral conditions.

Illinois J. Math., 1:459–504, 1957.

[107] William Feller. On the intrinsic form for second order differential operators. Illinois J. Math.,

2:1–18, 1958.

[108] William Feller. The birth and death processes as diffusion processes. J. Math. Pures Appl.

(9), 38:301–345, 1959.

[109] William Feller. Differential operators with the positive maximum property. Illinois J. Math.,

3:182–186, 1959.

137

[110] William Feller. On the equation of the vibrating string. J. Math. Mech., 8:339–348, 1959.

[111] William Feller. An introduction to probability theory and its applications. Vol. I. Third edition.

John Wiley & Sons Inc., New York, 1968.

[112] William Feller. An introduction to probability theory and its applications. Vol. II. Second

edition. John Wiley & Sons Inc., New York, 1971.

[113] William Feller and Henry P. McKean, Jr. A diffusion equivalent to a countable Markov chain.

Proc. Nat. Acad. Sci. U.S.A., 42:351–354, 1956.

[114] Dominique Ferrandon, Jean-Luc Imler, Charles Hetru, and Jules A. Hoffmann. The drosophila

systemic immune response: sensing and signalling during bacterial and fungal infections. Na-

ture Reviews, 7:862–874, 2007.

[115] P. J. Fitzsimmons, Bert Fristedt, and L. A. Shepp. The set of real numbers left uncovered by

random covering intervals. Z. Wahrsch. Verw. Gebiete, 70(2):175–189, 1985.

[116] P. J. Fitzsimmons and Jim Pitman. Kac’s moment formula and the Feynman-Kac formula for

additive functionals of a Markov process. Stochastic Process. Appl., 79(1):117–134, 1999.

[117] P. J. Fitzsimmons and Jim Pitman. Kac’s moment formula and the feynman-kac formula for

additive functionals of a markov process. Stochastic Processes and their Applications, 79:117–

134, 1999.

[118] Philippe Flajolet and Fabrice Guillemin. The formal theory of birth-and-death processes,

lattice path combinatorics and continued fractions. Adv. in Appl. Probab., 32(3):750–778,

2000.

[119] U. Freiberg and M. Zahle. Harmonic calculus on fractals—a measure geometric approach. I.

Potential Anal., 16(3):265–277, 2002.

[120] Uta Freiberg. Analytical properties of measure geometric Krein-Feller-operators on the real

line. Math. Nachr., 260:34–47, 2003.

[121] Uta Freiberg. A survey on measure geometric Laplacians on Cantor like sets. Arab. J. Sci.

Eng. Sect. C Theme Issues, 28(1):189–198, 2003. Wavelet and fractal methods in science and

engineering, Part I.

[122] Uta Freiberg. Dirichlet forms on fractal subsets of the real line. Real Anal. Exchange, 30(2):589–

603, 2004/05.

[123] Uta Freiberg. Spectral asymptotics of generalized measure geometric Laplacians on Cantor

like sets. Forum Math., 17(1):87–104, 2005.

138

[124] Uta Freiberg and Jorg-Uwe Lobus. Zeros of eigenfunctions of a class of generalized second

order differential operators on the Cantor set. Math. Nachr., 265:3–14, 2004.

[125] A. Fujioka, K. Teral, R. E. Itoh, K. Aokl, T. Nakamura, S. Kuroda, E. Nishida, and M. Mat-

suda. Dynamics of ras/erk mapk cascade as monitored by fluorescent probes. J Bio. Chem.,

281:8917–8926, 2006.

[126] Jason Fulman. The Rogers-Ramanujan identities, the finite general linear groups, and the

Hall-Littlewood polynomials. Proc. Amer. Math. Soc., 128(1):17–25, 2000.

[127] Jason Fulman. A probabilistic proof of the Rogers-Ramanujan identities. Bull. London Math.

Soc., 33(4):397–407, 2001.

[128] Jason Fulman. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.),

39(1):51–85 (electronic), 2002.

[129] M. Furriols and J. Casanova. New embo member’s review: in and out of torso rtk signaling.

EMBO J., 22:1947–1952, 2003.

[130] M. Furriols, F. Sprenger, and J. Casanova. Variation in the number of activated torso receptors

correlates with differential gene expression. Development, 122:2313–2317, 1996.

[131] C.W. Gardiner. Handbook of Stochastic Methods, Second Edition. Springer, New York, 1985.

[132] George Gasper and Mizan Rahman. Basic hypergeometric series, volume 96 of Encyclopedia

of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition,

2004. With a foreword by Richard Askey.

[133] Jeffrey S. Geronimo and Theodore P. Hill. Necessary and sufficient condition that the limit of

Stieltjes transforms is a Stieltjes transform. J. Approx. Theory, 121(1):54–60, 2003.

[134] C. Ghiglione, N. Perrimon, and Perkins L. A. Quantitative variations in the level of mapk

activity control patterning of the embryonic termini in Drosophila. Dev. Bio., 205:181–193,

1999.

[135] S. Gilbert. Developmental Biology, Eight Edition. Sinauer Associates Inc., Sunderland, MA,

2006.

[136] D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.,

81:2340–2361, 1977.

[137] Lea Goentoro, Gregory Reeves, Craig Kowal, Luigi Martinelli, Trudi Schupbach, and Shvarts-

man Stanislav. Quantifying the shape of the gurken gradient in Drosophila oogenesis. Devel-

opmental Cell, 11:263–272, 2006.

139

[138] S. Greenwood and G. Struhl. Different levels of ras activity can specify distinct transcriptional

and morphological consequences in early Drosophila embryos. Development, 124:4879–4886,

1997.

[139] T. Gregor, W. Bialek, R. Steveninck, D. Tank, and E. Wieschaus. Diffusion and scaling during

early embryonic pattern formation. PNAS, 102:18403–18407, 2005.

[140] Thomas Gregor, William Bialek, David W. Tank, and Eric F. Wieschaus. Probing the limits

to positional information. Cell, in Press, 2007.

[141] Thomas Gregor, Eric F. Wieschaus, Alistair P. McGregor, William Bialek, and David W.

Tank. Stability and nuclear dynamics of the bicoid morphogen gradient. Cell, in Press, 2007.

[142] D. J. Griffiths. Intro. to Quantum Mechanics; 2nd ed. Pearson Education, Inc., Upper Saddle

River, NJ, 2005.

[143] Fabrice Guillemin and Didier Pinchon. Excursions of birth and death processes, orthogonal

polynomials, and continued fractions. J. Appl. Probab., 36(3):752–770, 1999.

[144] D. P. Gupta, M. E. H. Ismail, and D. R. Masson. Contiguous relations, basic hypergeometric

functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials.

J. Comput. Appl. Math., 68(1-2):115–149, 1996.

[145] William W. Hager. Updating the inverse of a matrix. SIAM Rev., 31(2):221–239, 1989.

[146] M. D. Hirschhorn. A continued fraction. Duke Math. J., 41:27–33, 1974.

[147] G. Hognas and A. Mukherjea. Probability measures on semigroups. Plenum New York, 1995.

[148] K. J. Hollenbeck. Invlap.m: A matlab function for numerical inversion of laplace transforms

by the de hoog algorithm, 1998.

[149] Richard Holley and Thomas M. Liggett. Generalized potlatch and smoothing processes. Z.

Wahrsch. Verw. Gebiete, 55(2):165–195, 1981.

[150] Joung-Woo Hong, David A Hendrix, and Michael S. Levine. Shadow enhancers as a source of

evolutionary novelty. Science, 321:1314, 2008.

[151] Joung-Woo Hong, David A Hendrix, Dmitri Papatesnko, and Michael S. Levine. How the

dorsal gradient works: Insights from postgenome technologies. PNAS, 105:20072–20076, 2008.

[152] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cam-

bridge, 1990. Corrected reprint of the 1985 original.

[153] X. S. Hou, Chou T. B., Melnick M. B., and Perrimon N. The torso receptor tyrosine kinase

can actvate raf in a ras-independent pathway. Cell, 81:63–71, 1995.

140

[154] B. Houchmandzadeh, E. Wieschaus, and S. Leibler. Establishment of developmental precision

and proportions in the early drosophila embryo. Nature, 415:798–402, 2002.

[155] B. Houchmandzadeh, E. Wieschaus, and S. Leibler. Precise domain specification in the devel-

oping drosophila embryo. Physical Review E., 72:061920, 2005.

[156] B. Houchmandzedah. Filtering the noise of embryonic development. J. Phys.: Condens.

Matter, 17:S1245–S1258, 2005.

[157] M. Howard and P. R. ten Wolde. Finding the center reliably: robust patterns of developmental

gene expression. Phys. Rev. Letters, 95:208103, 2005.

[158] Audrey M. Huang, Jannette Rusch, and Michael Levine. An anteroposterior dorsal gradient

in the Drosophila embryo. Genes and Dev., 11:1963–1973, 1997.

[159] C. Y. Huang and J. E. Ferrell. Ultrasensitivity in the mitogen-activated protein kinase cascade.

Proc. Natl. Acad. Sci. USA, pages 10078–10083, 1996.

[160] Max Ingman, Henrik Kaessmann, Svante Paabo, and Ulf Gyllensten. Mitochondrial genome

variation and the origin of modern humans. Nature, 408(6813):708–713, 2000.

[161] Y. Tony Ip, Ronald E. Park, David Kosman, Ethan Bier, and Michael Levine. The dorsal gra-

dient morphogen regulates stripes of rhomboid expression in tile presumptive neuroectoderm

of the drosophila embryo. Genes and Dev., 6:1728–1739, 1992.

[162] Y. Tony Ip, Ronald E Park, David Kosman, Karina Yazdanbakhsh, and Michael Levine. dorsal-

twist interactions establish snail expression in the presumptive mesoderm of the Drosophila

embryo. Genes and Development, 6:1518–1530, 1992.

[163] Kiyosi Ito and Henry P. McKean, Jr. Diffusion processes and their sample paths. Springer-

Verlag, Berlin, 1974. Second printing, corrected, Die Grundlehren der mathematischen Wis-

senschaften, Band 125.

[164] Casanova J. and G. Struhl. The torso receptor localizes as well as transduces the spatial signal

specifying terminal body pattern in Drosophila. Nature, 362:152–155, 1993.

[165] J. Jacod and A. V. Skorokhod. Jumping Markov processes. Ann. Inst. H. Poincare Probab.

Statist., 32(1):11–67, 1996.

[166] A. B. Jaffe and A. Hall. Rho gtpases: Biochemistry and biology. Annu. Rev. Cell. Dev. Biol.,

21:247–269, 2005.

[167] Jens Carsten Jantzen. Lectures on quantum groups, volume 6 of Graduate Studies in Mathe-

matics. American Mathematical Society, Providence, RI, 1996.

141

[168] Jin Jiang, Timothy Hoey, and Michael Levine. Autoregulation of a segmentation gene in

Drosophila: combinatorial interaction of the even-skipped homeo box protein with a distal

enhancer element. Genes and Dev., 5:265–277, 1991.

[169] Jin Jiang, David Kosman, Y. Tony Ip, and Michael Levine. The dorsal morphogen gradient

regulates the mesoderm determinant twist in early Drosophila embryos. Genes and Dev.,

5:1881–1891, 1991.

[170] G. Jimenez, A. Guichet, A. Ephrussi, and J. Casanova. Relief of gene repression by torso rtk

signaling: role of capicua in Drosophila terminal and dorso-ventral patterning. Genes Dev.,

14:224–231, 2000.

[171] Paul Joyce and Simon Tavare. Cycles, permutations and the structure of the Yule process

with immigration. Stochastic Process. Appl., 25(2):309–314, 1987.

[172] Horace Judson. The Eighth Day of Creation. Simon and Schuster Touchstone Books, New

York, 1979.

[173] Victor Kac and Pokman Cheung. Quantum calculus. Universitext. Springer-Verlag, New York,

2002.

[174] Olav Kallenberg. An extension of the basic Palm measure correspondence. Probab. Theory

Related Fields, 117(1):113–131, 2000.

[175] K. Kammerle. Looking forwards and backwards in a bisexual Moran model. Journal of Applied

Probability, pages 880–885, 1989.

[176] K. Kammerle. The extinction probability of descendants in bisexual models of fixed population

size. Journal of Applied Probability, pages 489–502, 1991.

[177] NG Van Kampen. Stochastic processes in physics and chemistry. North Holland, 2007.

[178] S. Karlin and J. McGREGOR. The number of mutant forms maintained in a population.

4:415–438, 1967.

[179] Samuel Karlin and James McGregor. The classification of birth and death processes. Trans.

Amer. Math. Soc., 86:366–400, 1957.

[180] Samuel Karlin and James McGregor. Linear growth birth and death processes. J. Math.

Mech., 7:643–662, 1958.

[181] Samuel Karlin and James McGregor. Many server queueing processes with Poisson input and

exponential service times. Pacific J. Math., 8:87–118, 1958.

142

[182] M. Karlsson, M. Mandl and S. M. Keyse. Spatio-temporal regulation of mitogen-activated

protein kinase (mapk) signalling by protein phosphatases. Biochem. Soc. Trans., 34:842–845,

2006.

[183] Christian Kassel. Quantum groups, volume 155 of Graduate Texts in Mathematics. Springer-

Verlag, New York, 1995.

[184] I. S. Kats. The spectral theory of a string. Ukraın. Mat. Zh., 46(3):155–176, 1994.

[185] I. S. Kats. The spectral theory of a string and pathological birth and death processes. Funkt-

sional. Anal. i Prilozhen., 39(2):74–78, 2005.

[186] Marie Keaveney and Kevin Struhl. Activator-mediated recruitment of the rna polymerase ii

machinery is the predominant mechanism for transcriptional activation in yeast. Mol. Cell,

1:917–924, 1998.

[187] Adrienne W. Kemp. Heine-Euler extensions of the Poisson distribution. Comm. Statist. Theory

Methods, 21(3):571–588, 1992.

[188] Adrienne W. Kemp. Absorption sampling and the absorption distribution. J. Appl. Probab.,

35(2):489–494, 1998.

[189] John T. Kent. Eigenvalue expansions for diffusion hitting times. Z. Wahrsch. Verw. Gebiete,

52(3):309–319, 1980.

[190] JFC Kingman. Random discrete distributions. Journal of the Royal Statistical Society. Series

B (Methodological), pages 1–22, 1975.

[191] Frank B. Knight. Characterization of the Levy measures of inverse local times of gap diffusion.

In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981), volume 1 of Progr. Prob.

Statist., pages 53–78. Birkhauser Boston, Mass., 1981.

[192] Frank B. Knight. Essentials of Brownian motion and diffusion, volume 18 of Mathematical

Surveys. American Mathematical Society, Providence, R.I., 1981.

[193] T. H. Koornwinder. q-special functions, a tutorial. In Velleda Baldoni and Massimo A.

Picardello, editors, Representations of Lie groups and quantum groups. Proceedings of the

European School of Group Theory and the Congress on Advances in Representation Theory

of Lie Groups and Quantum Groups held in Trento, July 19–30, 1993, volume 311 of Pitman

Research Notes in Mathematics Series, pages 46–128, Harlow, 1994. Longman Scientific &

Technical.

[194] Uwe Kuchler. Some asymptotic properties of the transition densities of one-dimensional qua-

sidiffusions. Publ. Res. Inst. Math. Sci., 16(1):245–268, 1980.

143

[195] Uwe Kuchler. Exponential families of Markov processes. I. General results. Math. Operations-

forsch. Statist. Ser. Statist., 13(1):57–69, 1982.

[196] Uwe Kuchler. Exponential families of Markov processes. II. Birth-and-death processes. Math.

Operationsforsch. Statist. Ser. Statist., 13(2):219–230, 1982.

[197] Uwe Kuchler. Quasidiffusions, sojourn times and spectral measures. C. R. Acad. Bulgare Sci.,

38(11):1445–1448, 1985.

[198] Uwe Kuchler. On sojourn times, excursions and spectral measures connected with quasidiffu-

sions. J. Math. Kyoto Univ., 26(3):403–421, 1986.

[199] Uwe Kuchler. On Ito’s excursion law, local times and spectral measures for quasidiffusions. In

Probability theory and mathematical statistics, Vol. II (Vilnius, 1985), pages 161–165. VNU

Sci. Press, Utrecht, 1987.

[200] Uwe Kuchler. A limit theorem for the excursion of quasidiffusions straddling t. In Markov pro-

cesses and control theory (Gaußig, 1988), volume 54 of Math. Res., pages 100–103. Akademie-

Verlag, Berlin, 1989.

[201] Uwe Kuchler and Paavo Salminen. On spectral measures of strings and excursions of quasi

diffusions. In Seminaire de Probabilites, XXIII, volume 1372 of Lecture Notes in Math., pages

490–502. Springer, Berlin, 1989.

[202] H. Kunita. Absolute continuity of Markov processes and generators. Nagoya Math. J, 36:1–26,

1969.

[203] Boris A. Kupershmidt. q-probability. I. Basic discrete distributions. J. Nonlinear Math. Phys.,

7(1):73–93, 2000.

[204] T. C. Lacalli and L. G. Harrison. From gradient to segments: models for pattern formation in

early Drosophila. Semin. Dev. Biol., 2:107–117, 1991.

[205] Tze Leung Lai. Summability methods for independent identically distributed random variables.

Proc. Amer. Math. Soc., 45:253–261, 1974.

[206] Amaury Lambert. Coalescence times for the branching process. Adv. in Appl. Probab.,

35(4):1071–1089, 2003.

[207] M. Levine. personal communication.

[208] Michael Levine and Eric H. Davidson. Gene regulatory networks for development. PNAS,

102:4936–4942, 2005.

[209] J. Li, F. Xia, and W. Li. Coactivation of stat and ras is required for germ cell proliferation

and invasive migration in Drosophila. Dev Cell, 5:787–798, 2003.

144

[210] W. Li. Tor extra-copy flies. Personal communication.

[211] W. X. Li. Functions and mechansims of receptor tyrosine kinase torso siganling: lessons from

the Drosophila embryonic terminal development. Dev. Dynamics, 232:656–672, 2005.

[212] Thomas M. Liggett and Frank Spitzer. Ergodic theorems for coupled random walks and other

systems with locally interacting components. Z. Wahrsch. Verw. Gebiete, 56(4):443–468, 1981.

[213] T.M. Liggett. Interacting particle systems. Springer Berlin Heidelberg, 2005.

[214] J. T. Lis. Nature Progress, in review, 2007.

[215] Thomas E. Lloyd, Richard Atkinson, Mark N. Wu, Yi Zhou, Giuseppa Pennetta, and Hugo J.

Bellen. Hrs regulates endosome membrane invagination and tyrosine kinase receptor signaling

in Drosophila. Cell, 108:261–269, 2002.

[216] Jorg-Uwe Lobus. Generalized second order differential operators. Math. Nachr., 152:229–245,

1991.

[217] Jorg-Uwe Lobus. Constructions and generators of one-dimensional quasidiffusions with appli-

cations to self-affine diffusions and Brownian motion on the Cantor set. Stochastics Stochastics

Rep., 42(2):93–114, 1993.

[218] Harvey Lodish, Arnold Berk, Chris A. Kaiser, Monty Krieger, Matthew P. Scott, Anthony

Bretscher, Hidde Ploegh, and Paul T. Matsudaira. Molecular Cell Biology. W.H.Freeman and

Co Ltd., New York, 2007.

[219] Lisa Lorentzen and Haakon Waadeland. Continued fractions with applications, volume 3 of

Studies in Computational Mathematics. North-Holland Publishing Co., Amsterdam, 1992.

[220] Benoit B. Mandelbrot. Renewal sets and random cutouts. Z. Wahrscheinlichkeitstheorie und

Verw. Gebiete, 22:145–157, 1972.

[221] N. I. Markevich, J. B. Hoek, and B. N. Kholodenko. Signaling switches and bistability arising

from multisite phosphorylation in protein kinase cascades. J. Cell Bio., 164:353–359, 2004.

[222] David R. Masson. Difference equations, continued fractions, Jacobi matrices and orthogonal

polynomials. In Nonlinear numerical methods and rational approximation (Wilrijk, 1987),

volume 43 of Math. Appl., pages 239–257. Reidel, Dordrecht, 1988.

[223] Tapani Matala-aho. On the values of continued fractions: q-series. J. Approx. Theory,

124(2):139–153, 2003.

[224] Frederick A. Matsen and Steven N. Evans. To what extent does genealogical ancestry imply

genetic ancestry? Theor. Popul. Biol., 74:182–190, 2008.

145

[225] E. Mayer-Wolf, O. Zeitouni, and M.P.W. Zerner. Asymptotics of certain coagulation-

fragmentation processes and invariant poisson-dirichlet measures. Electr. Journ. Prob, 7:1–25,

2002.

[226] Harley H McAdams and Adam Arkin. Stochastic mechanisms in gene expression. PNAS,

94:814–819, 1997.

[227] H.H. McAdams and A. Arkin. Stochastic mechanisms in gene expression, 1997.

[228] S. P. Meyn and R. L. Tweedie. Markov chains and stochastic stability. Communications and

Control Engineering Series. Springer-Verlag London Ltd., London, 1993.

[229] M. Mohle. Forward and backward processes in bisexual models with fixed population sizes. J.

Appl. Probab., 31(2):309–332, 1994.

[230] M. Mohle. Fixation in bisexual models with variable population sizes. J. Appl. Probab.,

34(2):436–448, 1997.

[231] M. Mohle and S. Sagitov. Coalescent patterns in diploid exchangeable population models.

Journal of Mathematical Biology, 47(4):337–352, 2003.

[232] Frank Morgan. Geometric measure theory. Elsevier/Academic Press, Amsterdam, fourth

edition, 2009. A beginner’s guide.

[233] B. Moussian and S. Roth. Dorsoventral axis formation in the Drosophila embryo – shaping

and transducing a morphogen gradient. Current Bio., 15:R887–R899, 2005.

[234] Ginger W Muse. Nature Genetics, 39:1507, 2007.

[235] S. H. Northrup. Brownian dynamics simulation of diffusion-influenced biomolecular reactions.

J. Phys. Chem., 80:1517–1524, 1984.

[236] A. Novick and M. Weiner. Enzyme induction as an all-or-none phenomenon. Proceedings of

the National Academy of Sciences, 43(7):553–566, 1957.

[237] Neil O’Connell. The genealogy of branching processes and the age of our most recent common

ancestor. Adv. in Appl. Probab., 27(2):418–442, 1995.

[238] Z. Paroush, S. M. Wainwright, and D. Ish-Horowicz. Torso signalling regulates terminal pat-

terning in Drosophila by antagonising groucho-mediated repression. Development, 124:3827–

3834, 1997.

[239] J. Paulsson. Summing up the noise in gene networks. Nature, 427:415–418, 2004.

146

[240] J. Paulsson and M. Ehrenberg. Random signal fluctuations can reduce random fluctuations in

regulated components of chemical regulatory networks. Physical Review Letters, 84(23):5447–

5450, 2000.

[241] J.M. Pedraza and J. Paulsson. Effects of molecular memory and bursting on fluctuations in

gene expression. Science, 319(5861):339, 2008.

[242] S. L. Pelech. Networking with protein kinases. Curr. Biol., 3:513–515, 1993.

[243] L. A. Perkins, I. Larsen, and Perrimon N. corkscrew encodes a putative protein tyrosine

phosphotase that functions to transduce the terminal signal from the receptor tyrosine kinase

torso. Cell, 70:225–236, 1992.

[244] M. Perman, J. Pitman, and M. Yor. Size-biased sampling of Poisson point processes and

excursions. Probability Theory and Related Fields, 92(1):21–39, 1992.

[245] N. Perrimon. Signalling pathways initiated by receptor protein tyrosine kinases in Drosophila.

Curr. Opin. Cell Biol., 6:260–266, 1994.

[246] N. Perrimon, L. Engstrom, and Mahowald A. P. Zygotic lethals with specific materinal efffect

phenotypes in Drosophila melanogaster. i. loci on the x chromosome. Genetics, 121:333–352,

1989.

[247] N. Perrimon, D. Mohler, Engstrom L., and Mahowald A. P. X-linked female-sterile loci in

Drosophila melanogaster. Genetics, 113:695–712, 1986.

[248] P. Pfaffelhuber and A. Wakolbinger. The process of most recent common ancestors in an

evolving coalescent. Stochastic Process. Appl., 116(12):1836–1859, 2006.

[249] P. Pfaffelhuber and A. Wakolbinger. The process of most recent common ancestors in an

evolving coalescent. Stochastic Process. Appl., 116(12):1836–1859, 2006.

[250] J. Pitman. Random discrete distributions invariant under size-biased permutation. Advances

in Applied Probability, pages 525–539, 1996.

[251] J. Pitman. Poisson–Dirichlet and GEM invariant distributions for split-and-merge transfor-

mations of an interval partition. Combinatorics, Probability and Computing, 11(05):501–514,

2002.

[252] J. Pitman. Poisson-Kingman partitions. Lecture Notes-Monograph Series, pages 1–34, 2003.

[253] J. Pitman and M. Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable

subordinator. Annals of Probability, 25(2):855–900, 1997.

[254] Jim Pitman. Cyclically stationary Brownian local time processes. Probab. Theory Related

Fields, 106(3):299–329, 1996.

147

[255] Jim Pitman and Marc Yor. Bessel processes and infinitely divisible laws. In Stochastic integrals

(Proc. Sympos., Univ. Durham, Durham, 1980), volume 851 of Lecture Notes in Math., pages

285–370. Springer, Berlin, 1981.

[256] Jim Pitman and Marc Yor. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete,

59(4):425–457, 1982.

[257] Jim Pitman and Marc Yor. A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete,

59(4):425–457, 1982.

[258] Jim Pitman and Marc Yor. Hitting, occupation and inverse local times of one-dimensional

diffusions: martingale and excursion approaches. Bernoulli, 9(1):1–24, 2003.

[259] Boris Pittel. Note on exact and asymptotic distributions of the parameters of the loop-erased

random walk on the complete graph. In Mathematics and computer science, II (Versailles,

2002), Trends Math., pages 423–440. Birkhauser, Basel, 2002.

[260] T. D. Pollard and W. C. Earnshaw. Cell Biology. Saunders, Philadelphia, 2002.

[261] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C.

Cambridge University Press, Cambridge, 1992.

[262] A. Raj and A. van Oudenaarden. Nature, nurture, or chance: stochastic gene expression and

its consequences. Cell, 135(2):216–226, 2008.

[263] Arjun Raj and Alexander van Oudenaarden. Nature, nurture, or chance: Stochastic gene

expression and its consequences. Cell, 135:216–226, 2008.

[264] Don Rawlings. Limit formulas for q-exponential functions. Discrete Math., 126(1-3):379–383,

1994.

[265] Don Rawlings. Absorption processes: models for q-identities. Adv. in Appl. Math., 18(2):133–

148, 1997.

[266] Don Rawlings. A probabilistic approach to some of Euler’s number-theoretic identities. Trans.

Amer. Math. Soc., 350(7):2939–2951, 1998.

[267] A. Renyi. Treating chemical reactions using the theory of stochastic processes. Magyar Tud.

Akad. Alkalm. Mat. Int. Kzl., 2:83–101, 1953.

[268] Frigyes Riesz and Bela Sz.-Nagy. Functional analysis. Dover Books on Advanced Mathematics.

Dover Publications Inc., New York, 1990. Translated from the second French edition by Leo

F. Boron, Reprint of the 1955 original.

148

[269] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol.

1. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Founda-

tions, Reprint of the second (1994) edition.

[270] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2.

Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Ito calculus,

Reprint of the second (1994) edition.

[271] Douglas L. T. Rohde, Steve Olson, and Joseph T. Chang. Modelling the recent common

ancestry of all living humans. Nature, 431(7008):562–566, sep 2004.

[272] Wiley S. and P. Berke. Regulation of receptor tyrosine kinase signaling by endocytosis. Traffic,

2:12–18, 2001.

[273] Abbie Saunders, Leighton J. Core, and John T. Lis. Breaking barriers to transcription elon-

gation. Nature Reviews, 7:557–567, 2006.

[274] G. Schupbach and E. Wieschaus. Female sterile mutations on the second chromosome 2 of

drosophila melanogaster: Maternal effect mutations. Genetics, 121:101–117, 1989.

[275] L. A. Shepp. Covering the line with random intervals. Z. Wahrscheinlichkeitstheorie und Verw.

Gebiete, 23:163–170, 1972.

[276] Stanislav Shvartsman. Hypersensitivity, oscillations, and bistability in the mapk cascade.

[277] Damien Simon and Bernard Derrida. Evolution of the most recent common ancestor of a

population with no selection. J. Stat. Mech., page P05002 (electronic), 2006.

[278] M. A. Simon, Dodson G. S., and Rubin G. M. An sh3-sh2-sh3 protein is required for p21ras1

activation and binds to sevenless and sos proteins in vitro. Cell, 73:169–177, 1993.

[279] M. V. Smoluchowski. Versuch einer mathematischen theorie der koagulationskinetik kolloider

losungen. Z. Phys. Chem., 92:129–168, 1917.

[280] F. Sprenger, Stevens L. M., and Nusslein-Volhard C. The Drosophila gene torso encodes a

putative receptor tyrosine kinase. Nature, 338:478–483, 1989.

[281] Angelike Stathopolous and Michael S. Levine. Linear signaling in the toll-dorsal pathway of

drosophila: activated pelle kinase specifies all threshold outputs of gene expression while the

bhlh protein twist specifies a subset. Development, 129:3411–3419, 2002.

[282] D. Stein and L. M. Stevens. The torso ligand, unmasked? Sci. STKE, 98:1–3, 2001.

[283] L. M. Stevens, H. G. Frohnhofer, M. Klinger, and C. Nusslein-Volhard. Localized requirement

for torso-like expression in follicle cells for development of terminal anlagen of the Drosophila

embryo. Nature, 346:660–663, 1990.

149

[284] L. M. Stevens, H. G. Frohnhofer, M. Klingler, and Nusslein-Volhard C. Localized requirement

for torso-like expression in follicle cells for development of terminal anlagen of the Drosophila

embryo. Nature, 346:660–663, 2003.

[285] G. Struhl, K. Struhl, and P. M. Macdonald. The gradient morphogen bicoid is a concentration

dependent transcriptional activator. Cell, 57:1259–1273, 1989.

[286] Lajos Takacs. Introduction to the theory of queues. University Texts in the Mathematical

Sciences. Oxford University Press, New York, 1962.

[287] Simon Tavare. The birth process with immigration, and the genealogical structure of large

populations. J. Math. Biol., 25(2):161–168, 1987.

[288] Mukund Thattai and Alexander van Oudenaarden. Intrinsic noise in gene regulatory networks.

PNAS, 98:8614–8619, 2001.

[289] Mukund Thattai and Alexander van Oudenaarden. Attenuation of noise in ultrasensitive

signaling cascades. Biophysical Journal, 82:2943–2950, 2002.

[290] Russell Thomson, Jonathan K. Pritchard, Peidong Shen, Peter J. Oefner, and Marcus W.

Feldman. Recent common ancestry of human Y chromosomes: Evidence from DNA sequence

data. Proceedings of the National Academy of Science, 97(13):7360–7365, June 2000.

[291] G. Tkacik, C.G. Callan, and W. Bialek. Information flow and optimization in transcriptional

regulation. Proceedings of the National Academy of Sciences, 105(34):12265, 2008.

[292] G. Tkacik, T. Gregor, and W. Bialek. The role of input noise in transcriptional regulation.

PLoS ONE, 3(7), 2008.

[293] G. Tkacik, C.G. Callan Jr, and W. Bialek. Information capacity of genetic regulatory elements.

Physical Review E, 78(1), 2008.

[294] Simon A. A. Travers, Jonathan P. Clewley, Judith R. Glynn, Paul E. M. Fine, Amelia C.

Crampin, Felix Sibande, Dominic Mulawa, James O. McInerney, and Grace P. McCormack.

Timing and reconstruction of the most recent common ancestor of the subtype C clade of

Human Immunodeficiency Virus type 1. J. Virol., 78(19):10501–10506, 2004.

[295] S. Traverse, K. Seedorf, H. Paterson, C. J. Marshall, P. Cohen, and A. Ullrich. Egf triggers

neuronal differentitation of pc12 cells that overexpress the egf receptor. Curr. Biol., 4:694–701,

1994.

[296] Erik A. van Doorn. Birth-death processes and associated polynomials. J. Comput. Appl.

Math., 153(1-2):497–506, 2003.

150

[297] JS. van Zon, Marco J. Morelli, Sorin Tanase-Nicola, and Pieter Rein ten Woldey. Diffusion of

transcription factors can drastically enhance the noise in gene expression. Phys. Rev. Letters,

94:128103, 2005.

[298] Sarah K. Volkman, Alyssa E. Barry, Emily J. Lyons, Kaare M. Nielsen, Susan M. Thomas,

Mehee Choi, Seema S. Thakore, Karen P. Day, Dyann F. Wirth, and Daniel L. Hartl. Recent

origin of Plasmodium falciparum from a single progenitor. Science, 293(5529):482 – 484, 2001.

[299] Hans Volkmer. Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relation-

ship. Electron. J. Differential Equations, pages No. 48, 15 pp. (electronic), 2005.

[300] John Wakeley. Coalescent Theory, an Introduction. Roberts and Company, Greenwood Village,

CO, 2005. Chapters 1-3 at www.coalescenttheory.com.

[301] John B. Walsh. On the Chacon-Jamison theorem. Z. Wahrsch. Verw. Gebiete, 68(1):9–28,

1984.

[302] Wolpert. The triumph of the embryo. Oxford University Press, Oxford, 1991.

[303] Makoto Yamazato. Hitting times of single points for 1-dimensional generalized diffusion pro-

cesses. In Stability problems for stochastic models (Sukhumi, 1987), volume 1412 of Lecture

Notes in Math., pages 352–359. Springer, Berlin, 1989.

[304] Makoto Yamazato. Hitting time distributions of single points for 1-dimensional generalized

diffusion processes. Nagoya Math. J., 119:143–172, 1990.

[305] Makoto Yamazato. Characterization of the class of hitting time distributions of 1-dimensional

generalized diffusion processes. In Probability theory and mathematical statistics (Kiev, 1991),

pages 422–428. World Sci. Publishing, River Edge, NJ, 1992.

[306] Makoto Yamazato. Hitting time distributions of 1-dimensional generalized diffusions. In Trends

in probability and related analysis (Taipei, 1996), pages 325–338. World Sci. Publishing, River

Edge, NJ, 1997.

[307] Jie Yao, Katherine M. Munson3, Watt W. Webb, and John T. Lis. Dynamics of heat shock

factor association with native gene loci in living cells. Nature, 442:1050–1053, 2006.

[308] Christopher B. Yohn, Leslie Pusateri, Vitor Barbosa, and Ruth Lehmann. l(3)malignant

brain tumor and three novel genes are required for Drosophila germ-cell formation. Genetics,

165:1889–1900, 2003.

[309] J. Zeitlinger, Robert P. Zinzen, Alexander Stark, Manolis Kellis, Hailan Zhang, Richard A.

Young, and Michael Levine. Whole-genome chip-chip analysis of dorsal, twist, and snail

suggests integration of diverse patterning processes in the drosophila embryo. Genes and

Development, 21:385–390, 2007.

151

[310] Julia Zeitlinger, Alexander Stark, Manolis Kellis, Joung-Woo Hong, Sergei Nechaev, Karen

Adelman, Michael Levine, and Richard A Young. Rna polymerase stalling at developmental

control genes in the drosophila melanogaster embryo. Nature Genetics, 39:1512, 2007.

[311] Anton Zettl. Sturm-Liouville theory, volume 121 of Mathematical Surveys and Monographs.

American Mathematical Society, Providence, RI, 2005.

[312] Robert P Zinzen and Dmitri Papatsenko. Enhancer responses to similarly distributed antag-

onistic gradients in development. PLoS Computational Biology, 3:826–835, 2007.

[313] Robert P. Zinzen, Kate Senger, Mike Levine, and Dmitri Papatsenko. Computational models

for neurogenic gene expression in the Drosophila embryo. Current Biology, 16:1358–1365, 2006.

[314] Robert Zwanzig. Diffusion-controlled ligand binding to spheres partially covered by receptors:

An effective medium treatment. Proc. Natl. Acad. Sci., 87:5856–5857, 1990.