FF1000Research
Invariant death
Steven A. Frank Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92697–2525 USA,[email protected]
In nematodes, environmental or physiological perturbations alter death’s scaling oftime. In human cancer, genetic perturbations alter death’s curvature of time. Thosechanges in scale and curvature follow the constraining contours of death’s invariantgeometry. I show that the constraints arise from a fundamental extension to the theo-ries of randomness, invariance and scale. A generalized Gompertz law follows. Theconstraints imposed by the invariant Gompertz geometry explain the tendency of per-turbations to stretch or bend death’s scaling of time. Variability in death rate arisesfrom a combination of constraining universal laws and particular biological processes.
Keywords: mortality, nematodes, cancer, Gompertz distri-bution, probability theoryPreprint of published version: Frank, S. A. 2016. Invariantdeath. F1000Research 5:2076,doi:10.12688 / f1000research.9456.1.Published under a Creative Commons CC BY 4.0 license. Page 1 of 10 a r X i v : . [ q - b i o . P E ] A ug Introduction
The coil of a snail’s shell expresses the duality of constraintand process. The logarithmic spiral of growth constrainsoverall form. Particular snails modulate the process of shelldeposition, varying the parameters of the logarithmic spi-ral. To interpret the variety of snail shells, one must recog-nize the interplay between broad geometric constraint andthe special modulating processes of individual types .The pattern of death in populations follows the same du-ality of invariant geometric constraint and modulating pro-cess. The invariant geometry of death’s curve arises fromthe intrinsic order of large samples . A large sampleerases underlying randomness, preserving only invariantaggregate values .I extend the large-sample concept to clarify the invariantgeometry of death. I then illustrate the role of particular bi-ological processes in modulating death’s curve: the stretchof death’s time in nematode response to physiological per-turbation and the curvature of cancer’s time in responseto genetic perturbation . The consequences of particularbiological perturbations can only be understood within thegeometry that constrains change to follow invariant con-tours.To restate the puzzle: How can we relate small-scale molec-ular and physiological process to population consequence?The problem remains unsolved. Finch and Crimmins em-phasized: “A key question is how to connect . . . [ linear ] ag-ing processes to the exponential rates of accelerating mor-tality that set life spans. . . . Although we can readily assessmolecular aging, such biomarkers of aging are rarely robustas predictors of individual morbidity and mortality risk inpopulations.” Randomness and invariance
I begin with the relation between small-scale randomnessand large-scale order. The classical theory derives fromthe principles of statistical mechanics , later developedthrough aspects of entropy and information . Here, Ibriefly summarize my own extension of classical resultsbased on geometric principles of invariant measurementand scale . I then show how the abstract geome-try constrains the relation between biological process andthe pattern of death’s curve.To understand the probability of dying at a particular age,we begin with the geometry of probability patterns . Foran underlying quantity, z , the probability of observing avalue near to z is the rectangular area with height q z , widthd ψ z , and area q d ψ . A probability pattern is a curve withcoordinates (cid:0) ψ z , q z (cid:1) defined parametrically with respect to z . For the curve of death, the input, z , may be age or time.Two invariances constrain the geometry of probabilitycurves. First, total probability is invariantly one. Invariant total probability implies that the height of the probabilitycurve has a natural exponential expression q z = k a e − λ ( a + T z ) = ke − λ T z , (1)in which k = k a e λ a remains constant for any a to satisfy therequirement that total probability is invariant. The expo-nential form for the height of the probability curve, q z , im-plies that the probability curve remains invariant to a shiftof the fundamental metric, T z (see Frank ).In general, we seek metrics for which it does not matterwhere we set our zero reference point. In geometry, a circleshifted in space retains its invariant form. Similarly, propergeometric scaling for probability patterns is shift invariant.In terms of death, any transformation of time, z , into afundamental time metric for probability pattern, T z , mustmeasure time such that a shift a + T z does not alter death’scurve. That shift-invariant requirement leads to the expo-nential expression in eqn 1.The second key invariance is that a uniform stretching orshrinking of the fundamental metric does not alter proba-bility pattern q z = ke − λ b bT z = ke − λ T z , (2)in which λ = λ b b remains constant for any b , causing theprobability pattern to be invariant to stretch of T z . Stretchinvariance is equivalent to invariance of λ 〈 T 〉 ψ , the valueof λ multiplied by the average value of T when probability, q d ψ , is measured on the scale, ψ (see Frank ).To summarize, probability curves remain invariant to shiftand stretch of the fundamental metric, T z , such that T (cid:55)→ a + bT ∼ T , (3)in which ‘ ∼ ’ means invariant with respect to shift andstretch. In geometry, invariance with respect to shift andstretch is affine invariance.Affine invariance leads to probability pattern described bya sequence of rectangular areas q d ψ = ke − λ T d ψ ,in which k and λ are constants that adjust to satisfy, re-spectively, invariant total probability and invariant averagevalue, λ 〈 T 〉 ψ . Many different approaches and interpreta-tions all arrive at this same basic form. Consequences of affine invariance
Here, by emphasizing the fundamental invariances, we cantake the next key step in understanding the geometry ofprobability patterns and the curves of death. In partic-ular, each successive application of the affine transforma-tion (eqn 3) to T leaves the probability pattern unchanged, Page 2 of 10 defining an invariant group of metrics T = β (cid:128) e β w − (cid:138) ∼ e β w , (4)with β → T → w . Here, w ( z ) is a scale for theunderlying values, z , such that a shift in that scale, w (cid:55)→ α + w , only changes T by a constant multiple, and thereforedoes not change the probability pattern.To find the proper metric, T , for a particular probabilitypattern, we only need to find the proper base scale w forwhich the probability pattern is shift invariant. If, for ex-ample, z is time or age, then we only need to discover thescaling, w ( z ) , for which q d ψ = ke − λ e β w d ψ (5)is invariant to a shift in w , when allowing adjustment of λ .When β →
0, then q → e − λ w .Eqn 5 expresses the abstract form of common probabilitypatterns . The abstraction does not specify the two keyscaling relations ψ ( z ) and w ( z ) that define the coordinatesof the parametric probability curve (cid:0) ψ , q (cid:1) with respect to z . However, the invariances that define the geometry doimpose strong constraints, leading to a limited set of formsfor almost all of the commonly observed probability pat-terns .We have two scaling relations ψ and w , but only a singleparametric probability curve (cid:0) ψ , q (cid:1) with associated proba-bility q d ψ in each increment. Thus, many different scalescan express the same probability pattern. For each applica-tion, there is often a natural scale that has a simple, under-standable form for its scaling relations. The invariant ticking of death’s clock
A natural scale corresponds to an additional invariancewith a simple interpretation. That additional invariancesets the underlying metric for the pair of scaling relations.For death, we can set the probability of dying to be invari-ant in each increment of the scale, d ψ , so that ψ representsthe uniform metric of mortality—the invariant ticking ofdeath’s clock. This uniform metric extends the theory of ex-treme values and time to failure to a more abstractand general understanding of the invariances that shape allof the common probability patterns .Invariant probability in each increment can be written as q d ψ = − d q and thus ψ = − log q , in which d q is a con-stant incremental fraction of the total probability. I use aminus sign as a convention to express the total probabilityas declining with an increase in ψ .With regard to dying, we may think of the total probabilityof being alive as declining by a constant increment of death, − d q , in each increment d ψ . In classic epidemiology, this definition of q would be expressed as q ( z ) ≡ S ( z ) , in which S ( z ) is the probability of survival to time z . However, itis important to consider the classic definition as a specialcase of the deeper abstract geometry, which leads to a moregeneral understanding of the constraints that shape death’scurve. Universal Gompertz geometry
Given the exponential form for q z in eqn 2, a constant prob-ability q d ψ in each increment requires d ψ = d T . Using thegeneral form of T in eqn 4, we have d T = e β w d w = T (cid:48) d w ,in which T (cid:48) > T with respect to w .With ˆ q d w = q d T for the constant probability in each incre-ment, we have ˆ q d w = ke log T (cid:48) − λ T d w = ke β w − λ e β w d w . (6)This probability pattern is expressed on the scale w , inwhich w defines the natural shift-invariant metric. In otherwords, for some underlying observable value z , such as timeor age, w ( z ) transforms z to a scale, w , that expresses aninvariant total probability ˆ q d w in each increment, and forwhich shifts in the scale w (cid:55)→ α + w , do not change theprobability pattern.The probability pattern in eqn 6 has the familiar Gompertzform. I derived that form solely from a few simple geomet-ric invariances. The simple invariances elevate the gener-alized Gompertzian form to a universal geometric principlefor probability patterns . By contrast, the Gompertzianpattern is usually derived from descriptive statistics or fromparticular assumptions about processes of failure or growth. Pattern on the observed scale
We may express the probability pattern on the scale of theunderlying observable value, z . For that scale, d w = w (cid:48) d z ,in which w (cid:48) > w with respect z . Theabstract Gompertzian geometry in w becomes the explicitform with respect to the directly measured value z as˜ q d z = ke log w (cid:48) T (cid:48) − λ T d z = kw (cid:48) e β w − λ e β w d z . (7) The hazard of death
Only living individuals can die. Thus, the hazard of deathis the probability of dying in an incremental metric of timedivided by the probability of being alive. The incrementalmetric scale, ψ z , transforms the observed value, z , whichmay be time or age, into the abstract incremental scale,d ψ . The abstract expression for the hazard of dying in anincrement d ψ is h ( ψ ) d ψ = q d ψ − (cid:82) q d ψ . (8) Page 3 of 10
In each increment, the probability of dying is q d ψ . Theintegral in the bottom is the sum of the probabilities of dy-ing over the period from a starting point until the currentperiod, in which the time metric is described by ψ ( z ) .Three different metrics transform the observable time in-put, or other measurable input, z , into the scale of analysis: T , w , and z itself. Those three metrics yield three equiv-alent forms for the hazard, each emphasizing different as-pects of the underlying geometric invariances h ( T ) d T = λ d T ∝ d T (9) ˆ h ( w ) d w = λ T (cid:48) d w ∝ e β w d w (10)˜ h ( z ) d z = λ w (cid:48) T (cid:48) d z ∝ w (cid:48) e β w d z (11)in which ‘ ∝ ’ denotes proportionality. The top form ex-presses the most general and abstract invariance of death.By transforming time into a general metric, z (cid:55)→ T , the haz-ard is invariantly λ in each increment of the metric, d T .The metric T defines the scale on which the probability pat-tern is invariant to the affine transformation T (cid:55)→ a + bT .We know the scale of death’s curve when we can transformour underlying observation, z , such as age, to the affine-invariant scale, T . Often, adding a constant to age or mul-tiplying age by a constant, z (cid:55)→ a + bz , changes the patternof death’s curve, so using age itself as the metric is usuallynot sufficient. We must find some transformation of age.The middle expression in eqn 10 describes the generalizedGompertzian geometry in the most direct way. In this case,when we transform z (cid:55)→ w , changing an observation suchas age, z , to the metric, w ( z ) , we only require that death’scurve be invariant to a shift, w (cid:55)→ a + w . The force of death and the curvature oftime
We are partitioning the scaling of death’s curve into twosteps, z (cid:55)→ w (cid:55)→ T . Once we have the shift-invariant scalingof time, w , then T = e β w changes w into the ultimate affine-invariant scaling, T . To make that last change, we need toknow β , which is β = T (cid:48)(cid:48) T (cid:48) = d log ˆ h d w , (12)in which each prime denotes the derivative with respect to w . This value of β defines the curvature for the geometry ofdeath and time. Once we have the shift-invariant scaling fortime, z (cid:55)→ w , we can consider death’s invariant curvature inthe transformation w (cid:55)→ T .The expression T (cid:48)(cid:48) is the acceleration, or absolute curva-ture, of T . The expression T (cid:48) is the rate or velocity at which T is changing. Thus, T (cid:48)(cid:48) / T (cid:48) can be thought of as the accel-eration relative to the velocity. Acceleration, curvature and force are ultimately equivalent.In terms of death, for a given velocity or rate at a particu-lar age, T (cid:48) , the value of β is the relative force that bendsdeath’s curve. The bending of death’s curve may also bedescribed as (cid:65) = d log ˆ h d w = ˆ h (cid:48) ˆ h , (13)which is the change in the hazard of death relative to thecurrent hazard. The hazard, ˆ h , is the rate, or velocity, ofdeath on the scale w . Thus, the change in relative velocity, (cid:65) , describes the acceleration of mortality in terms of therelative bending of death’s rate.The invariant geometry of death’s curve in eqn 12 may beexpressed as a balance, β − (cid:65) =
0, between force and ac-celeration. That balance is roughly analogous to Newton’ssecond law of motion, F = m (cid:65) , relating force to accelera-tion. Inference
The invariant geometry does not tell us the form for theshift-invariant scaling of death’s time, w , or the value ofthe invariant force, β , that bends death’s curve. However,the invariances strongly constrain the likely form of death’scurve and the meaningful metrics of death’s time. Impor-tantly, these expressions allow us to transform data aboutrates or motions into expressions that emphasize force andcausal interpretations . In biology, we rarely can pre-dict trajectories. Instead, we focus on interpreting thechanges in observed trajectories with respect to hypothe-sized forces .The abstract geometry is correct unto itself. In application,the geometry provides a tool that we may use for particularproblems. A tool is neither right nor wrong. Instead, a toolis helpful or not according to its aid in providing insight.Below I discuss some examples. A few comments preparefor the discussion.If we knew the correct scaling for age, w ( z ) , then withinthat frame of reference, the force, β , and acceleration, (cid:65) ,of mortality would be constant with respect to w . Thus,the frame of reference, w , provides valuable insight. How-ever, w may turn out to be a weirdly nonlinear scaling ofmeasured time, z , in which the form of w is difficult to de-termine directly. In practice, we can derive w from eqn 11by relating the hazard, ˜ h ( z ) , to w by β w = log (cid:90) ˜ h d z , (14)or T z ∼ (cid:82) ˜ h d z , the affine similarity of T z to the accumulatedhazard on the z scale.I now discuss the time scaling of mortality in nematodesand cancer. I consider these applications only to illus-trate general aspects of mortality’s temporal geometry. See Page 4 of 10
Stroustrup et al. for details about nematodes and Frank for details about cancer. Nematode mortality and the stretch oftime
Stroustrup et al. conclude from their study of nematodemortality: [W]e observe that interventions as diverse aschanges in diet, temperature, exposure to ox-idative stress, and disruption of [various] genes . . . all alter lifespan distributions by an apparentstretching or shrinking of time. To produce suchtemporal scaling, each intervention must alter tothe same extent throughout adult life all physi-ological determinants of the risk of death. I begin with the apparent stretching or shrinking of time . Iwill arrive at the same description of the nematode mor-tality pattern as given by Stroustrup et al. , but framedwithin my more general understanding of mortality’s in-variant geometry. From that broader perspective, the ob-served stretching or shrinking of time in the nematodestudy can be seen as a special case of the various temporaldeformations that arise with respect to mortality’s invariantscale.The perspective of my general framing calls into questionthe second conclusion that each intervention must alter tothe same extent throughout adult life all physiological deter-minants of the risk of death . I present a simple counterex-ample consistent with the observed patterns. My counterex-ample may not be the correct description of process in ne-matode mortality. The counterexample does, however, em-phasize key aspects of the logic by which we must evaluatethe relations between pattern and process in mortality.My framework analyzes the sequence of transformations z (cid:55)→ w (cid:55)→ T . The initial input, z , typically represents whatwe measure, such as a standard description of time or age.We then seek a transformation, w ( z ) , such that the para-metric curve, (cid:0) w , ˆ q (cid:1) , for observed or assumed probabilitypattern is shift-invariant with respect to w (eqn 6). In otherwords, the shift w (cid:55)→ α + w does not alter the probabil-ity curve. When we find the shift-invariant scale for w , wehave an expression for the probability pattern in terms ofthe Gompertzian geometry of eqn 6.A probability pattern that remains the same except fora constant stretching or shrinking of time corresponds to w ( z ) = log z , because a constant stretch or shrink of timeby a = e α yields w ( az ) = α + w ( z ) . If we express the asso-ciated parametric probability curve as the relation betweentime and probability, (cid:0) z , ˜ q (cid:1) , as in eqn 7 with w = log z , weobtain a curve that is invariant to a constant stretching or shrinking of the temporal scale, z , as˜ q ∝ z β − e − λ z β ,in which the parameter λ and the constant of proportion-ality both adjust to cancel any stretch or shrink of time by a > ). This curve is the Fréchet probabil-ity distribution, corresponding to the power law hazard ineqn 11 as ˜ h ∝ z β − .Stroustrup et al. concluded that the Fréchet distributionis the best overall match to their nematode studies. How-ever, they invoked the Gompertz-Fréchet family of distribu-tions by appeal to traditional epidemiology and by appealto the general form of extreme value distributions for fail-ure times. By contrast, I derived those distributions simplyas the inevitable consequence of basic assumptions aboutthe invariant geometry of meaningful scales . The deformation of death’s time
Stroustrup et al. discussed the stretching or shrinking ofdeath’s time by a single constant value. My framework gen-eralizes the deformation of time in relation to death. Webegin with T , the universal frame of reference for the scal-ing of death’s time. On the temporal scale, T , the hazard ofdeath, h ( T ) , remains constant at all times (eqn 9). Thus, T represents the invariant ticking of mortality’s clock.Given that universal frame of reference for time, we maythen consider other temporal scales in terms of the way inwhich they deform the invariant frame of reference. In thiscase, we work inversely, by starting with T in eqn 4, andthen inferring the deformations with respect to the under-lying scale of description, z . We can then think of the shapeof the curve (cid:0) T z , z (cid:1) as describing how measured time, z ,is deformed in relation to the universal invariant scale ofmortality’s time, T z .Ideally, we first infer the shift-invariant scale, w ( z ) , andthen use w in eqn 4 to determine the relation between T and z . In the nematode case, w ( z ) = log z achieved shiftinvariance. Thus T z ∼ e β w = z β . The power law curve (cid:128) z β , z (cid:138) , with curvature determined by β , describes the de-formation of time. The different experimental treatmentsdid not significantly alter the curvature associated with β .We can relate increments of the measured input, d z , to in-crements of mortality’s universal measure, d T , by startingwith eqn 11 as ˜ h ∝ d T / d z , and then writingd z ∝ d T / ˜ h . (15)For a case such as the nematodes in which T z ∼ z β , themeasured temporal increments, d z , scale in relation to theuniversal temporal frame as d z ∝ d T / z β − . For β >
1, mea-sured temporal increments, d z , shrink as time passes rela-tive to the constant ticking of mortality’s clock at d T . When Page 5 of 10 we think of d T as mortality’s constant temporal frame ofreference, then the deformation of measured time isd z ∝ z β − .The shrinking of measured time corresponds to the increasein the rate of measured mortality, in other words, the sameamount of mortality, d T , is squeezed into smaller tempo-ral increments, d z , increasing the density of mortality permeasured unit.In other cases, the relation of measured inputs, z , to mortal-ity’s universal scale, T z , will have different functional forms.Those different functional forms may correspond to non-uniform stretching and shrinking of the observed temporalscale at different magnitudes of z relative to the universalframe of reference for mortality on the scale T z . If possible,we first infer the shift-invariant scale, w ( z ) , for exampleby eqn 14, and then use w to determine the relation be-tween T and z , as in the nematode example. However, inpractice, it may be easier to go directly from the invariantclock, T , to the deformed time scale, z , by using the rela-tion d z ∝ d T / ˜ h . The following critique of the conclusionsby Stroustrup et al. about nematode mortality provides anexample. Invariant pattern and underlying process
Stroustrup et al. claimed that all physiological determi-nants of the risk of death change in the same way with eachexperimental intervention. I present simple counterexam-ples. Although my counterexamples may not describe thetrue underlying process, they do highlight two importantpoints. First, commonly observed patterns often express in-variances that are consistent with many alternative under-lying processes . Second, consideration of the alterna-tive processes with the same observable invariances leads totestable predictions about the underlying causal processes.In these examples, suppose that death follows a multistageprocess, as is often discussed in cancer progression . Fol-lowing Frank , p. 98, we may write the dynamics of pro-gression toward mortality as a sequence of transitions˙ x ( z ) = − u x ( z ) ˙ x i ( z ) = u i − x i − ( z ) − u i x i ( z ) i =
2, . . . , n − x n ( z ) = u n − x n − ( z ) ,where x i ( z ) is the fraction of the initial population born attime z = i at measured time, z . Assumethat when the cohort is born, at z =
0, all individuals are instage 0, that is, x ( ) =
1, and the fraction of individuals inother stages is zero.As time passes, some individuals move into later stages ofprogression toward death. The rate of transition from stage i to stage i + u i . The ˙ x ’s are the derivatives of x with respect to z . Death occurs when individuals transition intostage n . A fraction x n ( z ) of individuals has died at time z ,and the rate of death at time z is ˙ x n ( z ) ≡ ˜ q , in which ˜ q hasthe probability interpretation of eqn 7.If the transition rates are constant and equal, u i = u forall i , then we can obtain an explicit solution for the mul-tistage model . This solution provides a special case thathelps to interpret more complex assumptions. The solutionis x i ( z ) = e − uz ( uz ) i / i ! for i =
0, . . . , n −
1, with the initialcondition that x ( ) = x i ( ) = i >
0. Note thatthe x i ( z ) follow the Poisson distribution for the probabilityof observing i events when the expected number of eventsis uz .In the multistage model above, the derivative of x n ( z ) isgiven by ˙ x n ( z ) = ux n − ( z ) . From the solution for x n − ( z ) ,we have ˜ q = ˙ x n ( z ) = ue − uz ( uz ) n − / n − n thevent, in which each event occurs at constant rate, u . How-ever, many other processes lead to the same gamma distri-bution.Age-specific incidence is the hazard ˜ h ( z ) = ˙ x n ( z ) − x n ( z ) = u ( uz ) n − / n − (cid:80) n − i = ( uz ) i / i ! . (16)We can express the scaling of measured time, d z , relativeto the constant ticking of mortality’s time, d T , from eqn 15,by taking d T as constant and thusd z ∝ ( uz ) n − (cid:130) + uz + ( uz ) + . . . + ( uz ) n − n − (cid:140) . (17) Simultaneity and temporal deformation
When measured time, z , is small, during the initial periodof the process, the deformation of time is approximatelythe same as the Fréchet pattern, d z ∝ / z n − . This defor-mation in the gamma process describes the force of simul-taneity. Early in the process, all components that protectagainst mortality remain in the initial working state. Thus,mortality requires the nearly simultaneous failure of n in-dependent events, which creates a force that deforms theconstant ticking of mortality’s clock by rescaling the mea-sured increments, d z .As measured time increases, the increments d z shrink, com-pressing the same amount of mortality, d T , into smallermeasured temporal increments. As z becomes larger, theincrements d z in eqn 17 shrink less, because of the re-duced force of simultaneity that deforms mortality’s con-stant clock. With larger z , the higher-power terms of the Page 6 of 10 sum increasingly dominate, until the largest power termdominates and d z then ticks at a constant rate, with d z ∝ d T .The changing deformation of d z and the associated force ofmortality can be thought of roughly as follows . Early inthe gamma process, mortality requires the nearly simulta-neous failure of n independent events, creating a force ofsimultaneity such that d z ∝ / z n − . As time passes, manyindividuals suffer failure of some of the n processes, leav-ing in aggregate the equivalent of n − z ∝ / z n − . As more time passes, additionalcomponents fail, and the remaining force of simultaneitydiminishes, until eventually only one protective componentremains for those still alive, at which point d z then ticks ata constant rate, so that d z ∝ d T .We may also express the scaling of time on the shift-invariant Gompertzian scale, w , in which β is a relativemeasure of the acceleration of mortality (eqn 12), by usingthe general expression in eqn 14 and the specific form of ˜ h in eqn 16 to yield β w = log (cid:90) ˜ h d z = log log Γ( n , uz ) − ,in which Γ is the incomplete gamma function. Alternative models of nematode mortal-ity
With this understanding of the gamma process, we can con-sider alternative interpretations of the nematode data . Ipresent these alternatives to illustrate the logic of mortal-ity’s temporal scaling and the potential relation to under-lying process. The data do not provide information aboutwhether or not these alternative interpretations are the cor-rect description of nematode mortality. The point here isthat these alternatives, or some other structurally similaralternative, might be correct, and therefore the strong con-clusions of the original article may be false.To repeat the key conclusion from Stroustrup et al. , eachintervention must alter to the same extent throughout adultlife all physiological determinants of the risk of death .That conclusion is true for the simple gamma process, assummarized by eqn 17. In that equation, the value of u represents the rate at which each of the n processes failsand contributes to overall mortality. If we substitute uz (cid:55)→ ξ , then the scaling of measured time, expressed as d z ∝ d ξ ,changes only by a constant of proportionality as the rate, u ,changes.I now consider two variations on the underlying gammaprocess for mortality. Each of these variations leads toa constant rescaling of time, d z . However, that con- stant rescaling arises from underlying processes of mortal-ity that change in different ways in response to perturba-tions. These examples show that the constant rescaling oftime does not imply that an intervention alters to the sameextent throughout adult life all physiological determinantsof the risk of death.The first example considers two distinct sets of underlyingprocesses that influence mortality, each set composed of n processes. Mortality occurs only after the failure of all 2 n processes. Before experimental perturbation, one set of n processes has a relatively slow failure rate per process of u .The other set of n processes has a relatively fast failure rateof u (cid:48) (cid:29) u .In this case, the fast processes will all tend to fail earlyin life, almost always before all of the slow processes fail.Thus, the fast processes have little influence on mortality.The mortality rate will closely follow the gamma processwith n steps, each step at rate u , as analyzed above .Now suppose that an intervention influences all of the slowsteps but none of the fast steps. The intervention changesthe previously slow rate processes into fast processes, u (cid:55)→ u (cid:48)(cid:48) (cid:29) u (cid:48) . After intervention, the mortality rate will closelyfollow the gamma process with n steps, with each stepat rate u (cid:48) . The mortality pattern remains unchanged, ex-cept for a constant rescaling of time. However, the un-derlying physiological processes that determine mortalityhave changed completely. Previously unimportant rate pro-cesses with respect to mortality now completely dominate,and previously important rate processes no longer influencemortality.The second example considers a set of n underlying pro-cesses that influence mortality. Each process has a differentfailure rate of u i for i =
1, . . . , n , with u i < u i + . As before,mortality occurs only after the failure of all n processes.Frank presented numerical studies for this heterogeneousrate process model. Typically, the faster rate processes failearly in life and have relatively little influence. The slowerprocesses dominate the overall temporal pattern.With n equal rate processes, the curvature declines withtime, as in eqn 17. With a heterogeneous set of rate pro-cesses, the curvature tends to decline more quickly as thefastest processes fail earlier, typically leaving a progres-sively smaller set of remaining protective mechanisms astime passes, reducing the force of simultaneity.Now suppose that the heterogeneous set of n processes hasa simple hierarchy of rates, such that u i + = γ u i , in which γ > u , thenthe hierarchy is effectively altered only by multiplying eachrate in the set by γ , because the fastest processes typicallyhave almost no influence on pattern.Once again, the overall mortality pattern will change only Page 7 of 10 by a constant rescaling of time, even though the underlyingphysiological processes have changed significantly with re-spect to their influence on mortality. In this case, the mostimportant process that limited mortality before interven-tion was in effect knocked out after intervention, whereasall other processes did not change.
Different processes lead to same invari-ances
The actual biology of nematodes will, of course, not followexactly either of these two example cases. The examples doshow, however, that a constant rescaling of measured timefor mortality can arise by heterogeneous changes in the un-derlying physiological determinants of the risk of death.How should we interpret the match between variants of themultistage gamma process model and the observed scalingof nematode mortality? The correct view is that the invari-ances expressed by the gamma model are approximatelythe same as the invariances that arise by the true physio-logical processes. Those invariances dominate the shape ofthe observed patterns. The examples of the gamma mod-els are helpful, because they show the sorts of underlyingprocesses that generate the required invariances.Ultimately, the theoretical challenge is to understand thefull set of underlying processes that lead to the same in-variances and thus the same observed pattern . The em-pirical challenge is, of course, to figure out which particu-lar processes occur in each particular case. Success in theempirical challenge will likely depend on further progresson the theoretical challenge, because the theoretical framestrongly influences how one goes about solving the empiri-cal problem. Cancer incidence and the curvature oftime
I now turn to genetic knockouts in cancer that changethe curvature of time. Cancer incidence often follows apattern roughly consistent with a multistage gamma pro-cess . Again, that match does not mean that the underly-ing physiological processes truly follow the assumptions ofthe gamma model. Instead, the correct view is that theinvariances expressed by the gamma model are approxi-mately the same as the invariances that arise by the truephysiological processes.Consider the simplest gamma process, in which cancerarises only after n protective mechanisms fail. Each mech-anism fails at the same rate, u . I gave the explicit solutionfor that process earlier. In interpreting that solution for can-cer, it is important to note an essential distinction betweenmortality and cancer incidence. Everyone dies but only a small fraction of individuals de-velop a particular form of cancer. Thus, we must analyzemortality by running the measured time, or age, z , out toa large enough value so that the cumulative probability ofdying approaches one. In the gamma models above, thatmeans letting uz increase significantly above one. By con-trast, if only a small fraction of the population develops can-cer before dying of other causes, then we must run z onlyup to a time at which the cumulative probability of cancerremains small. That limit on total incidence typically meanscapping uz below one.With a small maximum value of uz , the age-specific haz-ard simplifies approximately to ˜ h ∝ z n − , and the scalingof measured time simplifies to d z ∝ / z n − . Those scalingsmatch the Fréchet model when we equate the curvature oftime, β , with the number of steps, n , and we interpret forceand curvature with respect to the shift-invariant scaling oftime, w ( z ) = log z .We can think of n = β as the force imposed on the loga-rithmic time scale, log z , caused by the requirement for thenearly simultaneous failure of n protective processes. Thegreater n , the greater the protective force, and the greaterthe bending of observed time relative to cancer’s invariantclock, ticking in increments of d T .All of that may seem to be a very abstract theory in relationto the actual physiological processes of cancer. However,certain empirical studies suggest that the simple geometrictheory of cancer’s time does in fact capture key aspects ofcancer’s real physiology and genetics. In particular, certaininherited genetic mutations correspond almost exactly tothe predicted theoretical change in the force of simultaneityand the temporal curvature of incidence.If a mutational knockout reduces the number of protectivemechanisms by one, such that n (cid:55)→ n −
1, then the ap-proximate pattern of incidence changes from ˜ h ∝ z n − to˜ h ∝ z n − . In other words, the force and associated curva-ture, β , is reduced by one.Two classic studies of cancer incidence made exactly thatcomparison. Ashley compared colorectal cancer inci-dence between groups with and without an inherited muta-tion that predisposes to the disease. Similarly, Knudson compared retinoblastoma incidence between groups withand without a predisposing inherited mutation.I analyzed those same cancers with additional data thatbecame available after the original studies . My analysisshowed that, in each case, the groups carrying the inher-ited predisposing mutation had a pattern of incidence thatchanged relative to the control groups by reducing the esti-mated value of β by approximately one. Thus, the geneticknockouts reduced time’s curvature by almost exactly theamount predicted by the reduced force of diminished si-multaneity in the protective mechanisms. Page 8 of 10
Conclusion
A few simple invariances shape the patterns of death. Thatgeometry does not tell us exactly how biological mecha-nisms influence mortality. But the geometry does set theconstraints within which we must analyze the relation be-tween pattern and process.I started with the temporal frame of reference, d T , onwhich mortality has a constant rate, or velocity. That tem-poral frame, with unchanging rate, expresses the ticking ofmortality’s clock in the absence of any apparent force thatwould change velocity.Given that frame without apparent force, we can then eval-uate other temporal scales in terms of the forces that mustbe applied to change mortality’s rate relative to the force-free scale. That approach focuses attention on the forces ofmortality, rather than the incidence or “motion” alone, be-cause the pattern of motion is inherently confounded withthe particular temporal frame of reference .Mortality’s temporal frame leads to a natural expression ofinvariant death with respect to a universal Gompertzian ge-ometry. That geometric expression separates the uniformapplication of force from the additional distortions of timewith respect to observed pattern.The examples of nematodes and cancer illustrated how toparse observable deformations of mortality’s clock with re-spect to invariant aspects of pattern and potential underly-ing explanations about process.Until biologists can see the constraints of Gompertzian ge-ometry on the curves of death as clearly as they can see theconstraints of the logarithmic spiral on the growth curvesof snail shells and goats’ horns, we will not be able to readproperly the relations between the molecular causes of fail-ure and the observable patterns of death.Put another way, geometry does not tell one how to build abridge. But one would not want to build a bridge withoutunderstanding the constraints of geometry. Properly inter-preting the duality of constraint and process with respectto pattern is among the most difficult and most importantaspects of science. Author contributions
SAF did all the research and wrote the article.
Competing interests
No competing interests were disclosed.
Grant information
National Science Foundation grant DEB–1251035 supportsmy research.
References [ ] D. W. Thompson.
On Growth and Form, 2nd ed (Cantoreprint of 1961 revised edition) . Cambridge University Press,Cambridge, 1992. [ ] R. P. Feynman.
Statistical Mechanics: A Set Of Lectures . West-view Press, New York, 2nd edition, 1998. [ ] B. V. Gnedenko and A. N. Kolmogorov.
Limit Distributionsfor Sums of Independent Random Variables . Addison-Wesley,Reading, MA, 1968. [ ] E. T. Jaynes.
Probability Theory: The Logic of Science . Cam-bridge University Press, New York, 2003. [ ] N. Stroustrup et al. The temporal scaling of
Caenorhabditiselegans ageing.
Nature , 530:103–107, 2016. [ ] S. A. Frank. Age-specific incidence of inherited versus spo-radic cancers: A test of the multistage theory of carcinogen-esis.
Proceedings of the National Academy of Sciences USA ,102:1071–1075, 2005. [ ] S. A. Frank.
Dynamics of Cancer: Incidence, Inheritance, andEvolution . Princeton University Press, Princeton, NJ, 2007. [ ] C. E. Finch and E. M. Crimmins. Constant molecular agingrates vs. the exponential acceleration of mortality.
Proceed-ings of the National Academy of Sciences , 113:1121–1123,2016. [ ] S. Pressé, K. Ghosh, J. Lee, and K. A. Dill. Principles of max-imum entropy and maximum caliber in statistical physics.
Reviews of Modern Physics , 85:1115–1141, 2013. [ ] S. A. Frank and E. Smith. Measurement invariance, entropy,and probability.
Entropy , 12:289–303, 2010. [ ] S. A. Frank and E. Smith. A simple derivation and classifi-cation of common probability distributions based on infor-mation symmetry and measurement scale.
Journal of Evolu-tionary Biology , 24:469–484, 2011. [ ] S. A. Frank. How to read probability distributions as state-ments about process.
Entropy , 16:6059–6098, 2014. [ ] S. A. Frank. Common probability patterns arise from simpleinvariances.
Entropy , 18:192, 2016. [ ] P. Embrechts, C. Kluppelberg, and T. Mikosch.
Modeling Ex-tremal Events: For Insurance and Finance . Springer Verlag,Heidelberg, 1997. [ ] S. Kotz and S. Nadarajah.
Extreme Value Distributions: The-ory and Applications . World Scientific, Singapore, 2000. [ ] S. Coles.
An Introduction to Statistical Modeling of ExtremeValues . Springer, New York, 2001. [ ] E. J. Gumbel.
Statistics of Extremes . Dover Publications, NewYork, 2004. [ ] C. Lanczos.
The Variational Principles of Mechanics . DoverPublications, New York, 4th edition, 1986.
Page 9 of 10 [ ] S. A. Frank. d’Alembert’s direct and inertial forces actingon populations: The Price equation and the fundamentaltheorem of natural selection.
Entropy , 17:7087–7100, 2015. [ ] S. A. Frank. The inductive theory of natural selection. arXiv:1412.1285 , 2014. [ ] S. A. Frank. The common patterns of nature.
Journal ofEvolutionary Biology , 22:1563–1585, 2009. [ ] S. A. Frank. Generative models versus underlying symme-tries to explain biological pattern.
Journal of EvolutionaryBiology , 27:1172–1178, 2014. [ ] P. Armitage and R. Doll. The age distribution of cancer anda multi-stage theory of carcinogenesis.
British Journal ofCancer , 8:1–12, 1954. [ ] S. A. Frank. A multistage theory of age-specific accelerationin human mortality.
BMC Biology , 2:16, 2004. [ ] D. J. Ashley. Colonic cancer arising in polyposis coli.
Journalof Medical Genetics , 6:376–378, 1969. [ ] A. G. Knudson. Mutation and cancer: statistical study ofretinoblastoma.
Proceedings of the National Academy of Sci-ences of the United States of America , 68:820–823, 1971., 68:820–823, 1971.