The Complexity of the Immune System: Scaling Laws Alan S. Perelson* Theoretical Division Los Alamos National Laboratory Los Alamos, NM 87545 Jason G. Bragg Department of Biology University of New Mexico Albuquerque, NM 87131 Frederik W. Wiegel Institute of Theoretical Physics University of Amsterdam Valckenierstraat 65 1018 XE Amsterdam The Netherlands *Corresponding author. Email: asp@lanl.gov, phone 505-667-6829, fax 505-665-3493 1
INTRODUCTION The immune system is a complex system responsible for protection against pathogenic agents. Pathogens can reside in any tissue of the body and the immune system needs to find and respond to them. By necessity the immune system is distributed and the cells and molecules that make up the system move or are transported throughout the body. The cells of the adaptive immune system are a type of white blood cell called lymphocytes. These cells are transported by the blood but can exit the blood stream and crawl through the tissues of body, returning to the blood in the lymph fluid that bathes cells and which is collected in a system of ducts called lymphatics, which ultimately connect with the blood stream. The best-studied immune systems are those of the mouse and the human. The mouse has about 10 8 lymphocytes while a human has about 10 12. Immune systems are found in all vertebrates and thus organisms as small as a tadpole (body mass of the order of 10 1 g ) and as large as an elephant (10 6 g) or whale (10 8 g) have immune systems. Here we address the question of whether there exist any scaling principles that can guide our understanding of the operation of the immune system in organisms that differ by nine orders of magnitude in mass. For much of its history immunology has been a subfield of microbiology and closely linked with medicine. As such, studies of the immune systems of diverse species have not attracted much attention or much funding, and there is a paucity of data about the immune system of most species. We know of only one paper that has theoretically addressed the question of the scaling of the immune system (Langman & Cohn, 1987). 2
Thus the goal of this chapter is to raise questions, and provide a brief overview of what is known about scaling laws in the immune system. The Protecton Hypothesis Langman and Cohn (1987) suggested that the immune system has a modular structure and is built of basic units called "protectons. Each protecton guarantees an adequate immune response in some volume element of the animal. Thus, in the Langman-Cohn view, a big animal simply has more protectons in its immune system than a small animal. Langman and Cohn also estimated, based on the concentration of antibody needed to protect an animal, that a protecton contains about 10 7 B cells in a volume of about 1 ml. In terms of scaling, the protecton idea suggests that the size of the immune system scales as the mass of the organism. While modularity is a desirable property of any large complex system, we argue against this strict point of view and the simple scaling proportional to mass (~M) for a number of reasons: 1) Transport of lymphocytes depends on the circulatory system. As shown by West, Brown and Enquist (1997) properties of the circulatory system do not scale ~M, but rather as ~M 3/4. 2) The immune system has architectural features, which include a single spleen and a single thymus in mammals of differing size. Thus, at least some components of the immune system are not modular. 3) A larger animal lives longer than a smaller animal. Hence its immune system has to do a better job of protecting it. 3
SCALING LAWS IN IMMUNOLOGY It has been observed that the average lifespan (T 0 ) of a mammal seems to scale with its body mass ( M) according to the scaling law T 0 ~ M 1/4, cf. West, Brown and Enquist (1997). A scaling law between a biological variable Y and body mass M is written in the form Y ~ M b, and b is called the scaling exponent. This is shorthand for an approximate, quantitative relation Y A M Y 0 M 0 b, where Y 0 is a standard unit with the same dimension as Y, M 0 is a standard unit of mass, and A is a dimensionless constant. Here we shall take the point of view that death is generally not due to failure of the immune system (see Wiegel & Perelson, 2003 for further discussion of this point), and thus the mammalian immune system should be designed in such a way that it can protect an organism during a lifetime T 0 ~ M 1/4. If the immune system of a larger animal must help keep that animal alive for longer periods than the immune system of a smaller animal, it must be more reliable. A larger animal has more B and T lymphocytes. This implies either more lymphocyte clones, or more cells per clone, or both. This suggests the question: what is the optimal way for the system to balance these two modes of resource allocation - T and B cell diversity versus clone size? Scaling of B and T cell clone size In order to derive the typical size of a lymphocyte clone as a function of M, we follow the model of West et al. (1997), in which the circulatory system is represented by a branching tree. 4
In the West, Brown, and Enquist or WBE model the organism is divided into a certain number of small units, each of which is supplied by a single capillary. These units, called service volumes or service units, are regions that a single capillary can supply with oxygen and remove waste products from. The number of service units scales ~ M 3/4 (West et al.1997; Brown & West, 2000), which implies that the volume of a service unit scales as ~ M 1/4. If we assume that a service unit is spherical; its radius (R) will scale ~ M 1/12. Now to connect the WBE model with immunology, we assume that the service volumes for the blood circulation are also the service volumes for immune surveillance. That is, the capillaries allow lymphocytes to exit the circulatory system and explore regions of tissue for foreign molecules and cells, collectively called antigens. This implies that if each clone of B cells or T cells contains at least ~ M 3/4 cells it can be represented in each of the ~ M 3/4 service volumes. One of the essential ideas in the WBE model is that the capillary that supplies a service volume is universal in its properties such as its diameter. This implies that the amount of blood delivered to the service unit, per unit of time, is independent of M. From the point of view of immune surveillance, most antigens will enter the service unit in the blood. In other words, the number of antigens that enter into a service unit per unit of time is independent of M. The Time to Find an Antigen in a Service Unit. Consider a single antigen and one specific lymphocyte, of some clone that is specific for the antigen, both located in the same service unit. This lymphocyte will 5
crawl within the service unit in a more-or-less random fashion, searching for antigen. How long does it take until it makes contact with the antigen for the first time? If one describes the random walk of the lymphocyte as spatial diffusion with a diffusion coefficient D then one can show that T, the average time until first contact between the lymphocyte and antigen, is given by (Wiegel & Perelson, 2003) Τ 3 R = 3Dl, where l is the sum of the radii of the lymphocyte and the antigen, and R is the radius of the service unit. If the diffusion coefficient D is independent of M, then as R ~ M 1/12, T ~ M 1/4. Thus, if there were only one lymphocyte per clone present in the service unit, the antigen could go undetected by that lymphocyte for a period of time (~ M 1/4 ) that increases with animal size. Since the search time for each clone should scale in the same manner, this result applies to all clones. In order to keep the time until detection a fixed value (smaller than the time during which the antigen could proliferate significantly) the organism has to put ~ M 1/4 copies of the lymphocyte into this service unit; this would reduce the mean time until first detection by a factor of ~ M 1/4 to a value which is independent of M. We conclude that if D is independent of M, the lymphocyte clone size should scale ~M 3/4 M 1/4 = M, where the first factor comes from the number of service units and the second one from the number of lymphocytes needed per unit. In Wiegel and Perelson (2003) we also examine the case in which lymphocyte movement and antigen growth both depend on the basic metabolic rate of an organism and also conclude that for this case clone size ~ M. 6
Scaling of the lymphocyte repertoire Next, we ask how many different clones of lymphocytes should be present in an animal of mass M. The number of different clones is called the size of the immune repertoire and determines how many different antigens the immune system can recognize. We assume that antigens mainly enter the body through our intake of food, liquids and air. The rate at which a mammal consumes food and air is governed by its metabolic rate. One can show that the lifetime total metabolic activity of a mammal scales as ~ M (Wiegel & Perelson, 2003), suggesting that a mammal needs to deal with cm antigens during its lifetime, where c is some constant. In order to assess the probability that an immune system with a repertoire of size N can recognize an antigen, Perelson and Oster (1979) introduced the idea of shapespace. In this theory it is assumed each lymphocyte has a receptor that can recognize antigens in a volume v 0 of shape space, which has a total volume V. If we let ε be the probability of the immune system, i.e., all N different clones, failing to recognize an antigen then N 0 v0 v ε = 1 exp N V V The probability of a successful immune response to an infection is then 1 ε, and the probability, P s, that the organism will successfully repel all cm infections during its lifespan is given by P s = ( 1 ε) cm exp ( εcm ). This probability should be very near to unity, so we require ε cm << 1 or 7
v0 1 ε = exp N << V cm. This in turn implies N >> V v 0 ln ( cm ), which shows that N, the repertoire size, should scale ~ ln (cm). Thus, this theory suggests that there should only be a weak dependence of repertoire size on a mammal s size. Scaling and the anatomical features of the immune system The lymphocytes of the mammalian immune system not only circulate throughout the body but also accumulate in the spleen and lymph nodes. These tissues act as filters, with the spleen trapping antigens from the blood and lymph nodes trapping antigens that enter the tissues. Because antigens are there, lymphocytes search for and interact with antigens in the secondary lymphoid tissues. Interesting questions then are how should the size or mass of the spleen and lymph nodes scale with body size? Each lymph node drains a certain volume of tissue. Thus, as animals get larger do lymph nodes simply get larger or are they more numerous? An appropriate scaling theory of the immune system should be able to answer these questions. Here, as a first step, we look to see what data are available and if there is any indication of scaling that is more complex than that of scaling simply by mass, i.e. ~ M. Stahl (1965) found that spleen mass scaled with body mass as ~ M 1.02 across diverse mammal species, or as ~ M 0.85 when limited to primates only. More recently, Nunn (2002a) found that spleen mass scaled with body mass as ~ M 1.17 across primates, 8
based on an analysis that included some of the data used by Stahl (1965), but which used mean mass values for species as data points, rather than values for individual animals. Estimates for the number of lymph nodes in dogs, humans, horses and cows are available (Altman & Dittmer, 1974), and in Fig. 1 are plotted against adult body masses typical for these species. While there is some suggestion of an increase in the number of lymph nodes with mass, it is difficult to evaluate the slope reliably with data for so few species, and where body masses and lymph node counts were not available for the same individuals. Concentrations of lymphocytes in the blood have been compared among species of primates, using data compiled by the International Species Information System (Nunn, Gittleman, & Antonovics, 2000; Nunn, 2002b). Using a similar database (International Species Information System, Minnesota Zoological Garden, Apple Valley, MN: Reference ranges for physiological values in captive wildlife, 2002 Edition), we find that among diverse mammal species, adult blood lymphocyte concentrations scale with adult body mass as ~ M -0.07 (n = 138, F = 18.53, P < 0.001, r 2 = 0.12), and as ~ M -.10 (n = 45, F = 18.78, P < 0.001, r 2 = 0.30) when limited to primates. This implies that if blood volume scales with mass as ~ M 1.02 (Stahl, 1967), total number of lymphocytes will scale as ~ M 1.0 M -0.1 ~ M 0.9. Data are also available for lymphocyte output from the thoracic duct (Altman & Dittmer, 1971), which is the major lymphatic vessel through which lymph enters the circulatory system (Goldsby, Kindt, & Osbourne, 2000). Among nine species of mammals, concentration of lymphocytes in the lymph decreases with body mass as 9
~ M -0.16 (though this relationship is not significant at the P = 0.05 level; n = 9, F = 3.49, P = 0.10, r 2 = 0.33), while flow from the thoracic duct increases with body mass as ~ M 0.89 (n = 9, F = 85.10, P < 0.001, r 2 = 0.92). It then follows that total lymphocyte output per unit time scales with body mass as ~ M 0.73 (n = 9, F = 44.40, P < 0.001, r 2 = 0.86; Fig. 2), which is intriguingly similar to the M 3/4 scaling laws found in the WBE theory. CONCLUSIONS The use of scaling laws in immunology is a field in its infancy. Here we have tried to show that scaling laws can provide insights into the properties of immune systems in animals of different sizes. Although we focused our attention on the number and size of lymphocyte clones needed to provide protection to animals of different masses, there is a great need to also understand the anatomical features of immune systems in different mammals. The lymphatic system, with its chains of lymph nodes, is organized to some extent as an inverse branching network, with collecting lymphatics joining together to form larger vessels. Along the way are lymph nodes that filter the fluid as it returns to the circulation. The scaling relations that govern the operation of the circulatory system have been well-studied (cf. West et al., 1997). An analogous theory for the lymphatic system still needs to be developed. Hopefully such a theory will address questions such as how does the size and number of lymph nodes scale with animal size? Do big animals have more lymph nodes or bigger lymph nodes than smaller mammals? Also, if scaling 10
relations can be developed they may open a window into better understanding of the relationship between the human immune system and that of other species, which are commonly used as model systems for studying the effects of drugs and immune system modifiers. A more informed approach to such studies can be nothing but helpful. Acknowledgements We thank Geoffrey West for valuable conversations about this work. Portions of this work were done under the auspices of the U.S. Department of Energy and supported under contract W-7405-ENG-36. This work was also facilitated by interactions at the Santa Fe Institute and supported at the Santa Fe Institute by a grant from the Joseph P. and Jeanne M. Sullivan Foundation. 11
Figure 1. Number of lymph nodes per individual as a function of body mass for dog, human, horse and cow (in order of mass). Figure 2. Total lymphocyte output from the thoracic duct plotted as a function of body mass for the hamster, rat, guinea pig, cat, rabbit, monkey, dog, goat and human (in order of mass). Equation for the fitted line is y = 211930x 0.73. 12
REFERENCES Altman, P. L. & Dittmer, D. S. (1971). Biological handbooks: Respiration and circulation. Bethesda, MD: Federation of American Societies for Experimental Biology. Altman, P. L. & Dittmer, D. S. (1974). Biology data book, 2nd edn, vol. 3. Bethesda, MD: Federation of American Societies for Experimental Biology. Brown, J. H., & West, G. B. (2000). Scaling in Biology. New York: Oxford University Press. Goldsby, R. A., Kindt, T. J., & Osbourne, B. A. (2000). Kuby Immunology, 4th edn. New York: W. H. Freeman. Langman, R. E., & Cohn M. (1987). The elephant-tadpole paradox necessitates the concept of a unit of B cell function: the protecton. Molecular Immunol. 24, 675-697. Nunn, C. L. (2002a). Spleen size, disease risk and sexual selection: a comparative study in primates. Evol. Ecol. Res. 4, 91-107. 13
Nunn, C. L. (2002b). A comparative study of leukocyte counts and disease risk in primates. Evolution 56, 177-190. Nunn, C. L., Gittleman, J. L., & Antonovics, J. (2000). Promiscuity and the primate immune system. Science 290, 1168-1170. Perelson, A. S., & Oster, G. (1979). Theoretical studies of clonal selection. J. Theoret. Biol. 81, 645-670. Stahl, W. R. (1965). Organ weights in primates and other mammals. Science 150, 1039-1042. Stahl, W. R. (1967). Scaling of respiratory variables in mammals. J. Appl. Physiol. 22, 453-460. West, G. B., Brown, J. H., & Enquist, B. J. (1997). A general model for the origin of allometric scaling laws in biology. Science 276, 122-126. Wiegel, F. W., & Perelson, A. S. (2003). Some scaling principles for the immune system. (submitted). 14
10 5 Lymph nodes (number) 10 4 10 3 10 2 10 1 10 4 10 5 10 6 Body mass (g) [log 10 scale] Fig. 1 15
10 9 Lymphocyte output (cells hr -1 ) [log 10 scale] 10 8 10 7 10 6 10 1 10 2 10 3 10 4 10 5 Body mass (g) [log 10 scale] Fig. 2 16