Bayesian Maximum Entropy (Two-Dimensional) Lifetime Distribution Reconstruction from Time-Resolved Spectroscopic Data

Abstract

Time-resolved spectroscopy is often used to monitor the relaxation processes (or reactions) of physical, chemical, and biochemical systems after some fast physical or chemical perturbation. Time-resolved spectra contain information about the relaxation kinetics, in the form of macroscopic time constants of decay and their decay associated spectra. In the present paper we show how the Bayesian maximum entropy inversion of the Laplace transform (MaxEnt-iLT) can provide a lifetime distribution without sign-restrictions (or two-dimensional (2D)-lifetime distribution), representing the most probable inference given the data. From the reconstructed (2D) lifetime distribution it is possible to obtain the number of exponentials decays, macroscopic rate constants, and exponential amplitudes (or their decay associated spectra) present in the data. More importantly, the obtained (2D) lifetime distribution is obtained free from pre-conditioned ideas about the number of exponential decays present in the data. In contrast to the standard regularized maximum entropy method, the Bayesian MaxEnt approach automatically estimates the regularization parameter, providing an unsupervised and more objective analysis. We also show that the regularization parameter can be automatically determined by the L-curve and generalized cross-validation methods, providing (2D) lifetime reconstructions relatively close to the Bayesian best inference. Finally, we propose the use of MaxEnt-iLT for a more objective discrimination between data-supported and data-unsupported quantitative kinetic models, which takes both the data and the analysis limitations into account. All these aspects are illustrated with realistic time-resolved Fourier transform infrared (FT-IR) synthetic spectra of the bacteriorhodopsin photocycle.

Keywords

Exponential analysis Time-resolved spectroscopy Maximum entropy Bayesian inference Inverse problems Lifetime distribution Bacteriorhodopsin

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References

Slayton

R. M.

and Anfinrud

P. A

, Curr. Opin. Struct. Biol. 7, 717 (1997).

Kötting

and Gerwert

, Chem. Phys. Chem. 6, 881 (2005).

Eaton

W. A.

Muñoz

Hagen

S. J.

Jas

G. S.

Lapidus

L. J.

Henry

E. R.

and Hofrichter

, Annu. Rev. Biophys. Biomol. Struct. 29, 327 (2000).

Van Stokkum

I. H.

Larsen

D. S.

and Van Grondelle

, Biochim. Biophys. Acta 1657, 82 (2004).

Nagle

J. F.

Parodi

L. A.

and Lozier

R. H

, Biophys. J. 38, 161 (1982).

Demchenko

A. P.

, FEBS Lett. 310, 211 (1992).

Ross

and Vlad

M. O

, Annu. Rev. Phys. Chem. 50, 51 (1999).

Włodarczyk

and Kierdaszuk

, Biophys. J. 85, 589 (2003).

Nagle

J. F.

, Biophys. J. 59, 476 (1991).

10.

Dioumaev

A. K.

, Biophys. Chem. 67, 1 (1997).

11.

Szundi

Lewis

J. W.

and Kliger

D. S

, Biophys. J. 73, 688 (1997).

12.

Henry

E. R.

, Biophys. J. 72, 652 (1997).

13.

Zimányi

, J. Phys. Chem. B 108, 4199 (2004).

14.

Bonneau

Wirz

and Zubergühler

A. D

, Pure Appl. Chem. 69, 979 (1997).

15.

McWhirte

J. G.

and Pike

E. R

, J. Phys. A: Math. Gen. 11, 1729 (1978).

16.

Press

W. H.

Teukolsky

S. A.

Vetterling

W. T.

and Flannery

B. P

, Numerical Recipes in C. The Art of Scientific Computing (Cambridge University Press, New York, 1992).

17.

Gull

S. F.

, “Bayesian Inductive Inference and Maximum Entropy”, in Maximum Entropy and Bayesian Methods in Science and Engineering, Erickson

G. J.

and Smith

C. R

, Eds. (Kluwer Academic Press, New York, 1988), pp. 37–52.

18.

Jaynes

E. T.

, Proc. IEEE 70, 939 (1982).

19.

Sibisi

, Nature (London) 301, 134 (1983).

20.

Gull

S. F.

and Daniell

G. J

, Nature (London) 272, 686 (1978).

21.

Gull

S. F.

and Skilling

, Quantified Maximum Entropy MemSys5 User's Manual (Maximum Entropy Data Consultants Ltd, Suffolk, 1990).

22.

Meier

J. E.

and Marshall

A. G

, Anal. Chem. 63, 551 (1991).

23.

Marshall

P. J.

Hobson

M. P.

Gull

S. F.

and Bridle

S. L

, Mon. Not. R. Astron. Soc. 335, 1037 (2002).

24.

Potton

J. A.

Daniell

G. J.

and Melville

, J. Phys. D: Appl. Phys. 17, 1567 (1984).

25.

Livesey

A. K.

and Bronchon

J. C

, Biophys. J. 52, 693 (1987).

26.

Livesey

A. K.

Licinio

and Delaye

, J. Chem. Phys. 84, 5102 (1986).

27.

Bryan

R. K.

, Eur. Biophys. J. 18, 165 (1990).

28.

Brochon

J. C.

, Methods Enzymol. 240, 262 (1994).

29.

Bronchon

J. C.

Pouget

and Valeur

, J. Fluoresc. 5, 193 (1995).

30.

Lórenz-Fonfría

V. A.

and Kandori

, Appl. Spectrosc. 60, 407 (2006).

31.

Delsuc

M. A.

and Malliavin

T. E

, Anal. Chem. 70, 2146 (1998).

32.

Lanyi

J. K.

and Váró

, Isr. J. Chem. 35, 365 (1995).

33.

Váró

and Lanyi

J. K

, Biochemistry 29, 2241 (1990).

34.

Váró

and Lanyi

J. K

, Biochemistry 30, 5008 (1991).

35.

Váró

and Lanyi

J. K

, Biochemistry 30, 5016 (1991).

36.

Ames

J. B.

and Mathies

R. A

, Biochemistry 29, 7181 (1990).

37.

Nagle

J. F.

, Photochem. Photobiol. 54, 897 (1991).

38.

Lozier

R. H.

Xie

Hofrichter

and Clore

G. M

, Proc. Natl. Acad. Sci. U.S.A. 89, 3610 (1992).

39.

Souvignier

and Gerwert

, Biophys. J. 63, 1393 (1992).

40.

Váró

and Lanyi

J. K

, Biochemistry 34, 12161 (1995).

41.

Chizhov

Chernavskii

D. S.

Engelhard

Mueller

K. H.

Zubov

B. V.

and Hess

, Biophys. J. 71, 2329 (1996).

42.

Ludmann

Gergely

and Váró

, Biophys. J. 75, 3110 (1998).

43.

Hendler

R. W.

Shrager

R. I.

and Bose

, J. Phys. Chem. B 105, 3319 (2001).

44.

Van Stokkum

I. H. M.

and Lozier

R. H

, J. Phys. Chem. B 106, 3477 (2002).

45.

Lukács

and Papp

, J. Photochem. Photobiol. B 77, 1 (2004).

46.

Shrager

R. I.

and Hendler

R. W

, J. Phys. Chem. B 107, 1708 (2003).

47.

Hofrichter

Henry

E. R.

and Lozier

R. H

, Biophys. J. 56, 693 (1989).

48.

Borucki

Otto

and Heyn

M. P

, J. Phys. Chem. B 103, 6371 (1999).

49.

Zscherp

and Heberle

, J. Phys. Chem. B 101, 10542 (1997).

50.

Gergely

Zimányi

and Váró

, J. Phys. Chem. B 101, 9390 (1997).

51.

Ruckebusch

Duponchel

Huvenne

J. P.

and Saurina

, Vib. Spectrosc. 35, 21 (2004).

52.

Mauer

Vogel

and Schneider

, Photochem. Photobiol. 46, 247 (1987).

53.

Müller

K.-H.

Butt

H. J.

Bamberg

Fendler

Hess

Siebert

and Engelhard

, Eur. Biophys. J. 19, 241 (1991).

54.

Xie

A. H.

Nagle

J. F.

and Lozier

R. H

, Biophys. J. 51, 627 (1987).

55.

Hendler

R. W.

, J. Phys. Chem. B 109, 16515 (2005).

56.

Groma

G. I.

Bogomolni

R. A.

and Stoeckenius

, Biochim. Biophys. Acta 1319, 59 (1997).

57.

Zimányi

Saltiel

Brown

L. S.

and Lanyi

J. K

, J. Phys. Chem. A 110, 2318 (2006).

58.

Klink

B. U.

Winter

Engelhard

and Chizhov

, Biophys. J. 83, 3490 (2002).

59.

Rammelsberg

Huhn

Lübben

and Gerwert

, Biochemistry 37, 5001 (1998).

60.

Rödig

Chizhov

Weidlich

and Siebert

, Biophys. J. 76, 2687 (1999).

61.

MacKay

D. J. C.

, Neural Comp. 4, 415 (1992).

62.

MacKay

D. J. C.

, Neural Comp. 4, 698 (1992).

63.

MacKay

D. J. C.

, Neural Comp. 11, 1035 (1999).

64.

Gregory

P. C.

, Bayesian Logical Data Analysis for the Physical Science (Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, 2005).

65.

Jaynes

E. T.

, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, 2003).

66.

MacKay

D. J. C.

, Information Theory, Inference and Learning Algorithms (Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, 2003).

67.

Jaynes

E. T.

, “Where do we stand on maximum entropy?”, in The Maximum Entropy Formalism, Levine

R. D.

and Tribus

, Eds. (MIT Press, Cambridge, 1978), pp. 15–118.

68.

Loredo

T. J.

, “From Laplace to supernova SN 1987A: Bayesian inference in astrophysics”, in Maximum Entropy and Bayesian Methods, Fougère

P. F.

Ed. (Kluwer Academic Publishers, Dordrecht, 1990), pp. 81–142.

69.

Caticha

and Preuss

, Phys. Rev. E 70, 046127(1–12) (2004).

70.

Hobson

M. P.

and Lasenby

A. N

, Mon. Not. R. Astron. Soc. 298, 905 (1998).

71.

Kandori

, Biochim. Biophys. Acta 1460, 177 (2000).

72.

Lórenz-Fonfría

V. A.

and Kandori

, Appl. Spectrosc. 61, 74 (2007).

73.

Skilling

and Bryan

R. K

, Mon. Not. R. Astron. Soc. 211, 111 (1984).

74.

Steinbach

P. J.

Ionescu

and Matthews

C. R

, Biophys. J. 82, 2244 (2002).

75.

Kumar

A. T. N.

Zhu

Christian

J. F.

Demidov

A. A.

and Champion

P. M

, J. Phys. Chem. B 105, 7847 (2001).

76.

Lavalette

Tetreau

Brochon

J. C.

and Livesey

, Eur. J. Biochem. 196, 591 (1991).

77.

Shaver

J. M.

and McGown

L. B

, Anal. Chem. 68, 9 (1996).

78.

Sengupta

Garai

Balaji

Periasamy

and Maiti

, Biophys. J. 84, 1977 (2003).

79.

Galatsanos

N. P.

and Katsaggelos

A. K

, IEEE Trans. Image Proc. 1, 322 (1992).

80.

Hansen

P. C.

, SIAM Rev. 34, 561 (1992).

81.

Hansen

P. C.

, SIAM J. Sci. Comput. 14, 1487 (1993).

82.

Hansen

P. C.

, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (Society for Industrial and Applied Mathematics, Philadelphia, 1998).

83.

Golub

G. H.

Heath

and Wahba

, Technometrics 21, 215 (1979).

84.

Ruhe

and Wedin

P. A

, SIAM Rev. 22, 318 (1980).

85.

Gerwert

, “Molecular Reaction Mechanism of Proteins Monitored by Time-Resolved FT-IR Difference Spectroscopy”, in Handbook of Vibrational Spectroscopy, Chalmers

J. M.

and Griffiths

P. R

, Eds. (John Wiley and Sons Ltd, Chichester, 2002).

86.

Bretthorst

G. L.

Hutton

W. C.

Garbow

J. R.

and Ackerman

J. J. H

, Concepts Magn. Reson. Part A 27, 64 (2005).

87.

Shrager

R. I.

and Hendler

R. W

, J. Biochem. Biophys. Methods 36, 157 (1998).

88.

Straume

and Johnson

M. L

, Methods Enzymol. 210, 87 (1992).

89.

James

D. R.

and Ware

W. R

, Chem. Phys. Lett. 120, 455 (1985).

90.

Marshall

D. B.

, Anal. Chem. 61, 660 (1989).

91.

Vix

and Lami

, Biophys. J. 68, 1145 (1995).

92.

Istratov

A. A.

and Vyvenko

O. F

, Rev. Sci. Instrum. 70, 1233 (1999).

93.

Bajzer

Ž.

Therneau

T. M.

Sharp

J. C.

and Prendergast

F. G

, Eur. Biophys. J. 20, 247 (1991).

94.

Lüdeke

Fonfría

Lórenz V. A.

Siebert

and Vogel

, Biopolymers 83, 159 (2006).