diff --git a/_sources/stat/survival_analysis_part2.ipynb b/_sources/stat/survival_analysis_part2.ipynb
index 207b99e..f4d9683 100644
--- a/_sources/stat/survival_analysis_part2.ipynb
+++ b/_sources/stat/survival_analysis_part2.ipynb
@@ -202,16 +202,16 @@
"id": "9adc7d5c",
"metadata": {},
"source": [
- "# Classical machine learning methods\n",
+ "## Classical machine learning methods\n",
"The main advantage - ability to model the non-linear relationships and work with high dimensional data\n",
- "## Decision tree\n",
+ "### Decision tree\n",
"The basic intuition behind the tree models is to recursively partition the data based on a particular splitting criterion, so that the objects that are similar to each other based on the value of interest will be placed in the same node.\n",
"\n",
"We will start with the simplest case - decision tree for classification:\n",
- "### Classification decision tree\n",
+ "#### Classification decision tree\n",
"```{figure} figs/9.PNG\n",
"```\n",
- "#### Probabilities\n",
+ "##### Probabilities\n",
"\n",
"Before the first split:\n",
"\n",
@@ -228,7 +228,7 @@
"$$P(y=\\text{YELLOW}|X > 12) = \\frac{6}{7} \\approx 0.86$$\n",
"\n",
"\n",
- "#### Entropy:\n",
+ "##### Entropy:\n",
"$$\n",
"H(p) = - \\sum_i^K p_i\\log(p_i)\n",
"$$\n",
@@ -246,7 +246,7 @@
"\n",
"$$H_{\\text{total}} = - \\frac{13}{20} 0.96 - \\frac{7}{20} 0.58 \\approx 0.83$$\n",
"\n",
- "#### Information Gain:\n",
+ "##### Information Gain:\n",
"$$\n",
"IG = H(\\text{parent}) - H(\\text{child})\n",
"$$\n",
@@ -263,7 +263,7 @@
"id": "929d56ed",
"metadata": {},
"source": [
- "### Regression decision tree\n",
+ "#### Regression decision tree\n",
"Is a step-wise constant predictor\n",
"\n",
"Lets look at this example:\n",
@@ -294,7 +294,7 @@
"id": "637ecf43",
"metadata": {},
"source": [
- "## Ensembling\n",
+ "### Ensembling\n",
"Ensembling - combining the predictions of multiple base learners to obtain a powerful overall model. The base learners are often very simple models also referred as weak learners. \n",
"Multiple diverse models are created to predict an outcome, either by using many different modeling algorithms or using different training data sets.\n",
"\n",
@@ -320,14 +320,14 @@
"id": "5ada40bb",
"metadata": {},
"source": [
- "### Random forest\n",
+ "#### Random forest\n",
"Random Forest fits a set of Trees independently and then averages their predictions\n",
"\n",
"The general principles as RF: (a) Trees are grown using bootstrapped data; (b) Random feature selection is used when splitting tree nodes; (c) Trees are generally grown deeply (d) Forest ensemble is calculated by averaging terminal node statistics\n",
"\n",
"Importantly, the high number of base learners do not lead to overfitting. \n",
"\n",
- "### Gradient boosting\n",
+ "#### Gradient boosting\n",
"In contrast to random forest gradient boosted model is constructed sequentially in a greedy stagewise fashion\n",
"\n",
"After training decision tree the errors of prediction are obtained and the next decision tree is trained on this prediction errors\n",
@@ -347,7 +347,7 @@
"\n",
"If we want to include high number of base learners we should use very low learning rate to restrict the influence of individual base learners - similar to regularization\n",
"\n",
- "# Survival machine learning\n",
+ "## Survival machine learning\n",
"\n",
"Survival analysis is a type of regression problem as we want to predict a continuous value, but with a twist. It differs from traditional regression by the fact that parts of the data can only be partially observed – they are censored\n",
"\n",
@@ -357,11 +357,11 @@
"Survival machine learning - machine learning methods adapted to work with survival data and censoring!\n",
"\n",
"\n",
- "## 1. Survival random forest\n",
+ "### Survival random forest\n",
"Survival trees are one form of decision trees which are tailored\n",
"to handle censored data. The goal is to split the tree node into left and right daughters with dissimilar event history (survival) behavior.\n",
"\n",
- "### Splitting criterion\n",
+ "#### Splitting criterion\n",
"The primary difference between a survival tree and the standard decision tree is\n",
"in the choice of splitting criterion - the log-rank test. The log-rank test has traditionally been used for two-sample testing of survival data, but it can be used for survival splitting as a means for maximizing between-node survival differences. \n",
"\n",
@@ -403,7 +403,7 @@
"id": "559e3cad",
"metadata": {},
"source": [
- "### Prediction\n",
+ "#### Prediction\n",
"For prediction, a sample is dropped down each tree in the forest until it reaches a terminal node.\n",
"\n",
"Data in each terminal is used to non-parametrically estimate the cumulative hazard function and survival using the Nelson-Aalen estimator and Kaplan-Meier, respectively. \n",
@@ -431,11 +431,11 @@
"id": "0cd849af",
"metadata": {},
"source": [
- "## 2. Survival Gradient boosting\n",
+ "### Survival Gradient boosting\n",
"\n",
"Gradient Boosting does not refer to one particular model, but a framework to optimize loss functions. \n",
"\n",
- "### Cox’s Partial Likelihood Loss\n",
+ "#### Cox’s Partial Likelihood Loss\n",
"The default loss function is the partial likelihood loss of Cox’s proportional hazards model. \n",
"The objective is to maximize the log partial likelihood function, but replacing the traditional linear model with the additive model \n"
]
@@ -456,7 +456,7 @@
"id": "64c915e1",
"metadata": {},
"source": [
- "# Neural networks - Multi-Layer Perceptron Network \n",
+ "## Neural networks - Multi-Layer Perceptron Network \n",
"\n",
"Here is the model of artificial neuron - base element of artificial neural network. The output is computed using activation function on the summation of inputs multiplied by weights and the bias value\n",
"\n",
@@ -481,7 +481,7 @@
"id": "4a762b0f",
"metadata": {},
"source": [
- "## MLP training\n",
+ "### MLP training\n",
"The most popular method for training MLPs is back-propagation. During backpropagation, the output values are compared with the correct answer to compute the value of some predefined error-function. The error is then fed back through the network. Using this information, the algorithm adjusts the weights of each connection in order to reduce the value of the error function by some small amount. After repeating this process for a sufficiently large number of training cycles, the network will usually converge to some state where the error of the calculations is small. In this case, one would say that the network has learned a certain target function. \n"
]
},
@@ -576,11 +576,11 @@
"id": "740e6a49",
"metadata": {},
"source": [
- "# Survival neural networks\n",
+ "## Survival neural networks\n",
"Neural networks methods adapted to work with survival data and censoring!\n",
"Pycox - python package for survival analysis and time-to-event prediction with PyTorch, built on the torchtuples package for training PyTorch models.\n",
"\n",
- "## DeepSurv (CoxPH NN)\n",
+ "### DeepSurv (CoxPH NN)\n",
"Continious-time model. \n",
"\n",
"Nonlinear Cox proportional hazards network. Deep feed-forward neural network with Cox proportional hazards loss function. Can be considered as nonlinear extension of the Cox proportional hazards: can deal with both linear and nonlinear effects from covariates. \n",
@@ -606,7 +606,7 @@
"id": "9a8b7c3b",
"metadata": {},
"source": [
- "## Nnet-survival (Logistic hazard NN)\n",
+ "### Nnet-survival (Logistic hazard NN)\n",
"Discrete-time model, fully parametric survival model\n",
"\n",
"The Logistic-Hazard method parametrize the discrete hazards and optimize the survival likelihood.\n",
@@ -623,21 +623,21 @@
"id": "86a4434a",
"metadata": {},
"source": [
- "## Performance metrics\n",
+ "### Performance metrics\n",
"Our test data is usually subject to censoring too, therefore common metrics like root mean squared error or correlation are unsuitable. Instead, we use specific metrics for survival analysis\n",
- "### 1. Harrell’s concordance index\n",
+ "#### Harrell’s concordance index\n",
"Predictions are often evaluated by a measure of rank correlation between predicted risk scores and observed time points in the test data. Harrell’s concordance index or c-index computes the ratio of correctly ordered (concordant) pairs to comparable pairs\n",
"\n",
"The higher the C-index is - the better model performance is\n",
"\n",
"\n",
- "### 2. Time-dependent ROC AUC\n",
+ "#### Time-dependent ROC AUC\n",
"Extention of the well known receiver operating characteristic curve (ROC curve) to possibly censored survival times. Given a time point, we can estimate how well a predictive model can distinguishing subjects who will experience an event by time \n",
" (sensitivity) from those who will not (specificity).\n",
" \n",
"The higher the ROC AUC is - the better model performance is\n",
"\n",
- " ### 3. TIme-dependent Brier score\n",
+ " #### TIme-dependent Brier score\n",
" The time-dependent Brier score is an extension of the mean squared error to right censored data.\n",
" \n",
" The lower the Brier score is - the better model performance is"
@@ -648,7 +648,7 @@
"id": "b05e6c79",
"metadata": {},
"source": [
- "## Features selection\n",
+ "### Features selection\n",
"Which variable is most predictive?\n",
"\n",
"Different methodologies exist, however we will only talk about one simple but valuable method - permutation importance"
@@ -659,11 +659,11 @@
"id": "2ff2731e",
"metadata": {},
"source": [
- "### 1. Permutation feature importance \n",
+ "#### Permutation feature importance \n",
"Permutation feature importance is a model inspection technique which can be used for any fitted estimator with tabular data. This is especially useful for non-linear estimators. \n",
"\n",
"The permutation feature importance is a decrease in a model score when a single feature value is randomly shuffled. This procedure breaks the relationship between the feature and the target, thus the drop in the model score is indicative of how much the model depends on the feature. \n",
- "# Credits \n",
+ "## Credits \n",
"This notebook was prepared by Margarita Sidorova"
]
}
diff --git a/dyn/complex_systems.html b/dyn/complex_systems.html
index 08d6e4c..70e0385 100644
--- a/dyn/complex_systems.html
+++ b/dyn/complex_systems.html
@@ -6,7 +6,7 @@
- 12. Complex systems approach — Computational aging book
+ 7. Complex systems approach — Computational aging book
@@ -55,8 +55,8 @@
-
-
+
+
@@ -156,7 +156,7 @@
Nothing in aging biology makes sense except in the light of dynamic equilibrium.
Aging biology today is not engineering but science at its early step. This expresses that we frequently cannot observe direct causal and numerically tractable relationships (as in physics) but we are doomed to see mostly associations and patterns. However, it may also be a good starting point for dynamic modeling. Large volumes of data and the absence of an organizing paradigm are major challenges of modern aging biology. Complex systems approaches provide some computational instruments and necessary vocabulary for simplification, understanding, and identification of the essential features of data. Throughout this chapter, you will come across many new words: system, complexity, networks, emergence, resilience, and critical transitions. They form an important minimal vocabulary necessary for developing complex systems intuition which (we believe) can be greatly useful for any researcher. We organize them into subsections and uncover them one by one. This chapter partly relies on the recent work of [Cohen et al., 2022] where a corresponding philosophical framework, as well as important computational examples, were proposed.
Before the consideration of complex systems, we should understand what we had before. Typical two common approaches in sciences are Top-Down and Bottom-Up. If the object of study is a human organism, then the Top-Down approach asks the question of which large-scale observations are associated with small scale. For example, a patient’s state change is associated with changing in a level of some protein. And vice versa, the Bottom-Up approach asks whether a large-scale patient’s state changes after perturbation of some protein level. Moving Bottom-Up generates something we call metabolic pathways, gene ontologies, etc. - i.e. combining small-scale entities into new higher-scale entities. In turn, moving Top-Down we can differentiate new entities such as organs, tissues, cells, organelles, and nuclei at a smaller scale (Fig. 12.1). You might have noticed that science greatly advanced by applying only these two approaches. However, an encounter with living systems forces us to develop a new methodology. The problem with that two approaches is that they do not take into account interactions between feedback/feedforward loops at too different scales trying to provide a mechanistic explanation of processes in a level-by-level manner. They provide theories like A causes B and B causes C whilst in real complex systems we observe behaviors like A causes B, B causes A, B causes C and C causes A and B. Do you agree that this no more presumes a one-dimensional schematic explanation of the mechanism of action? Such an abundance of causal relationships forces us to use the concept of networks for building a more correct body of knowledge.
Before the consideration of complex systems, we should understand what we had before. Typical two common approaches in sciences are Top-Down and Bottom-Up. If the object of study is a human organism, then the Top-Down approach asks the question of which large-scale observations are associated with small scale. For example, a patient’s state change is associated with changing in a level of some protein. And vice versa, the Bottom-Up approach asks whether a large-scale patient’s state changes after perturbation of some protein level. Moving Bottom-Up generates something we call metabolic pathways, gene ontologies, etc. - i.e. combining small-scale entities into new higher-scale entities. In turn, moving Top-Down we can differentiate new entities such as organs, tissues, cells, organelles, and nuclei at a smaller scale (Fig. 7.1). You might have noticed that science greatly advanced by applying only these two approaches. However, an encounter with living systems forces us to develop a new methodology. The problem with that two approaches is that they do not take into account interactions between feedback/feedforward loops at too different scales trying to provide a mechanistic explanation of processes in a level-by-level manner. They provide theories like A causes B and B causes C whilst in real complex systems we observe behaviors like A causes B, B causes A, B causes C and C causes A and B. Do you agree that this no more presumes a one-dimensional schematic explanation of the mechanism of action? Such an abundance of causal relationships forces us to use the concept of networks for building a more correct body of knowledge.
Note
For example, depending on their cellular context, calpains may either activate caspase 3, leading to apoptosis, or degrade caspase 3, preventing apoptosis [Łopatniuk and Witkowski, 2011].
Fig. 12.1 Example of Top-Down and Bottom-Up organization of knowledge.#
+
Fig. 7.1 Example of Top-Down and Bottom-Up organization of knowledge.#
-
Cohen and colleagues [Cohen et al., 2022] provide a great example of such a switch to the complexity that happened in ecological sciences recently (Fig. 12.2). You can see that a previously popular food chains theory was changed by a system approach where interactions between species form a complex network where it is not obvious how the system will respond to removing a particular species. Currently, we observe a similar transformation in the aging domain. Two great examples (and already classical) of such a transformation are Hallmarks of aging [López-Otín et al., 2013] and Pillars of aging [Kennedy et al., 2014] papers trying to highlight a set of key entities related to aging and connecting them into a network of interacting elements. Thinking in a complex systems paradigm challenges a traditional perspective that single-molecule interventions will be found that substantially slow aging (indeed many researchers are pursuing such approaches). Instead, we should focus on a combinatorial approach requiring, however, a robust model of the system under treatment.
+
Cohen and colleagues [Cohen et al., 2022] provide a great example of such a switch to the complexity that happened in ecological sciences recently (Fig. 7.2). You can see that a previously popular food chains theory was changed by a system approach where interactions between species form a complex network where it is not obvious how the system will respond to removing a particular species. Currently, we observe a similar transformation in the aging domain. Two great examples (and already classical) of such a transformation are Hallmarks of aging [López-Otín et al., 2013] and Pillars of aging [Kennedy et al., 2014] papers trying to highlight a set of key entities related to aging and connecting them into a network of interacting elements. Thinking in a complex systems paradigm challenges a traditional perspective that single-molecule interventions will be found that substantially slow aging (indeed many researchers are pursuing such approaches). Instead, we should focus on a combinatorial approach requiring, however, a robust model of the system under treatment.
-
Fig. 12.2 Shift to complex systems approaches in ecology and aging biology.#
+
Fig. 7.2 Shift to complex systems approaches in ecology and aging biology.#
What can be one of the most common characteristics of aging? Can we express the plethora of our observations of an aging organism by one cumulative term? It turns out, we can consider the topological characteristics of complex living systems comparing them at different time moments (at young and old ages). The complexity was proposed as such characteristic in [Lipsitz and Goldberger, 1992] where authors suggested that aging is associated with “loss of complexity”. It is tough to define “complexity” explicitly and the authors do not give such a definition. Instead, they measure complexity with some mathematical instruments relying on an intuitive understanding of this concept. One example is depicted at Fig. 12.3, where you can see a comparison between young (left) and old (right) neurons. If we evaluate fractal dimensions of left and right structures we will spot a decrease in the measure. Intuitively it’s correct: the left structure looks more complex than the right counterpart, at least, because the left arbor has more branches.
What can be one of the most common characteristics of aging? Can we express the plethora of our observations of an aging organism by one cumulative term? It turns out, we can consider the topological characteristics of complex living systems comparing them at different time moments (at young and old ages). The complexity was proposed as such characteristic in [Lipsitz and Goldberger, 1992] where authors suggested that aging is associated with “loss of complexity”. It is tough to define “complexity” explicitly and the authors do not give such a definition. Instead, they measure complexity with some mathematical instruments relying on an intuitive understanding of this concept. One example is depicted at Fig. 7.3, where you can see a comparison between young (left) and old (right) neurons. If we evaluate fractal dimensions of left and right structures we will spot a decrease in the measure. Intuitively it’s correct: the left structure looks more complex than the right counterpart, at least, because the left arbor has more branches.
-
Fig. 12.3 Age-related loss of fractal structure in the dendritic arbor of the giant pyramidal Betz cell of the motor cortex.#
+
Fig. 7.3 Age-related loss of fractal structure in the dendritic arbor of the giant pyramidal Betz cell of the motor cortex.#
-
But maybe another example from the same paper will be more intelligible. Look at the Fig. 12.4, where you can see the heart rate dynamics of young (top) and old (bottom) individuals. We see that the top and bottom signals are statistically different. From the spectral theory point of view, we could say that the top signal has more high frequencies in its spectrum. This means that the heart rate signal tends to lose high-frequency components as we age. Another way to express this behavior of signal is to measure its Shannon’s entropy [Shannon, 1948]. Since entropy by definition is a measure of the amount of information needed to predict the future state of the system, the higher the entropy signal has, the higher its complexity is. The fact is that the top signal has higher entropy than the lower one, so, within the aforementioned terms, we again observe a loss of complexity with aging. The observation of complexity decreasing can inspire us to make the next step in studying the phenomenon. For example, in [Lipsitz and Goldberger, 1992] mentioned that the age-related decline in heart rate variability is likely due to dropout of sinus node cells, altered \(\beta\)-adrenoceptor responsiveness, and reduction in parasympathetic tone.
+
But maybe another example from the same paper will be more intelligible. Look at the Fig. 7.4, where you can see the heart rate dynamics of young (top) and old (bottom) individuals. We see that the top and bottom signals are statistically different. From the spectral theory point of view, we could say that the top signal has more high frequencies in its spectrum. This means that the heart rate signal tends to lose high-frequency components as we age. Another way to express this behavior of signal is to measure its Shannon’s entropy [Shannon, 1948]. Since entropy by definition is a measure of the amount of information needed to predict the future state of the system, the higher the entropy signal has, the higher its complexity is. The fact is that the top signal has higher entropy than the lower one, so, within the aforementioned terms, we again observe a loss of complexity with aging. The observation of complexity decreasing can inspire us to make the next step in studying the phenomenon. For example, in [Lipsitz and Goldberger, 1992] mentioned that the age-related decline in heart rate variability is likely due to dropout of sinus node cells, altered \(\beta\)-adrenoceptor responsiveness, and reduction in parasympathetic tone.
-
Fig. 12.4 Heart rate time series for a 22-year-old female subject (top panel) and a 73-year-old male subject (bottom panel).#
+
Fig. 7.4 Heart rate time series for a 22-year-old female subject (top panel) and a 73-year-old male subject (bottom panel).#
Biological systems can be viewed as complex networks of interacting components. This simple idea may seem quite naive until we start to consider the structural properties of these networks. Bow-tie or hourglass structure is a common structural feature found in many biological systems. A bow-tie means that a large number of inputs are converted to a small number of intermediates, which then fan out to generate a large number of outputs (Fig. 12.5) [Cohen et al., 2022] (looks like autoencoder isn’t it?). The important principle lying behind bow-tie formation is called degeneracy[Edelman and Gally, 2001]. In a biological sense, this means the ability of structurally different biological elements (e.g. proteins or metabolites) to perform the same function. Two examples of bow-tie include metabolic networks, where the large range of nutrients consumed by the organism is decomposed into 12 universal precursors (including pyruvate, G6P, AKG, ACCOA) from which the organism builds all of its biomass including carbohydrates, nucleic acids and proteins; the human visual system having \(\sim 10^8\) input photoreceptors in the retina which fans into only about \(\sim 10^6\) ganglion cells whose axons form the optic nerve going to visual cortex that detects a pattern, color, depth and movement [Friedlander et al., 2015]. In the context of aging biology bow-tie structure expresses the ability of the same pathways to be simultaneously implicated in inflammation, regulating oxidative stress, cancer, reproduction and metabolism (Fig. 12.5). This is a natural consequence of the network’s degeneracy by integrating information from multiple inputs and inflating it to multiple outputs. There is a lot of evidence that degeneracy is an important property of evolving systems and may be even a prerequisite for evolution [Edelman and Gally, 2001].
Biological systems can be viewed as complex networks of interacting components. This simple idea may seem quite naive until we start to consider the structural properties of these networks. Bow-tie or hourglass structure is a common structural feature found in many biological systems. A bow-tie means that a large number of inputs are converted to a small number of intermediates, which then fan out to generate a large number of outputs (Fig. 7.5) [Cohen et al., 2022] (looks like autoencoder isn’t it?). The important principle lying behind bow-tie formation is called degeneracy[Edelman and Gally, 2001]. In a biological sense, this means the ability of structurally different biological elements (e.g. proteins or metabolites) to perform the same function. Two examples of bow-tie include metabolic networks, where the large range of nutrients consumed by the organism is decomposed into 12 universal precursors (including pyruvate, G6P, AKG, ACCOA) from which the organism builds all of its biomass including carbohydrates, nucleic acids and proteins; the human visual system having \(\sim 10^8\) input photoreceptors in the retina which fans into only about \(\sim 10^6\) ganglion cells whose axons form the optic nerve going to visual cortex that detects a pattern, color, depth and movement [Friedlander et al., 2015]. In the context of aging biology bow-tie structure expresses the ability of the same pathways to be simultaneously implicated in inflammation, regulating oxidative stress, cancer, reproduction and metabolism (Fig. 7.5). This is a natural consequence of the network’s degeneracy by integrating information from multiple inputs and inflating it to multiple outputs. There is a lot of evidence that degeneracy is an important property of evolving systems and may be even a prerequisite for evolution [Edelman and Gally, 2001].
Note
Bow-tie - network structure assuming that a large number of inputs are converted to a small number of intermediates, which then fan out to generate a large number of outputs.
@@ -615,14 +615,14 @@
What technical outcomes we can obtain from bow-tie networks? So, the presence of a low-dimensional intermediate layer hints us that application of dimensionality reduction techniques to biological data is a good decision. It is the fact that many biological entries (e.g. expression level of different genes) are highly correlated with each other [Podolskiy et al., 2015] and a huge dimensionality of inputs can be satisfactorily described by a small number of latent parameters. The latent parameter is a mathematical term describing a low-dimensional linear (or nonlinear) combination of inputs. This is not a big deal to compute a latent parameter according to a model. The more important thing is to give a correct biological interpretation for this parameter. Sometimes latent parameters do not have a clear interpretation but in other cases, a particular latent parameter may catch a clear biological process such as the expression of one of the metabolic precursors mentioned above.
The natural property of human consciousness (and, need to say, its bias) is a tendency to simplify things by extracting their common pattern. We call a property of a system emergent if we do not see this property of any system component. By discovering emergent properties of the system we cope with its complexity. Classical examples of emergent properties in physical systems are temperature of a gas and fluidity of a fluid - in both cases, we do not see such properties in system components (molecules). Revealed emergent properties are especially useful when we can measure them. Biological systems are abundant by emergences. Moreover, the concept of Life is a typical example of emergence. One common way that emergence arises in biological systems is through canalization, the tendency of the system to converge toward one of a limited number of discrete states [Cohen et al., 2022]. Once again, if the system can take distinguishable states and these states seem not to be emanating from properties of its components then these states are emergent. An immediate example is acquiring somatic cell state (or cell fate) through differentiation (Fig. 12.6) from pluripotent one [Takahashi and Yamanaka, 2016]. It is quite remarkable, despite the huge complexity of a human cell as a system it has a rather small number of discrete (distinguishable) somatic states that were shown experimentally [Huang et al., 2005].
The natural property of human consciousness (and, need to say, its bias) is a tendency to simplify things by extracting their common pattern. We call a property of a system emergent if we do not see this property of any system component. By discovering emergent properties of the system we cope with its complexity. Classical examples of emergent properties in physical systems are temperature of a gas and fluidity of a fluid - in both cases, we do not see such properties in system components (molecules). Revealed emergent properties are especially useful when we can measure them. Biological systems are abundant by emergences. Moreover, the concept of Life is a typical example of emergence. One common way that emergence arises in biological systems is through canalization, the tendency of the system to converge toward one of a limited number of discrete states [Cohen et al., 2022]. Once again, if the system can take distinguishable states and these states seem not to be emanating from properties of its components then these states are emergent. An immediate example is acquiring somatic cell state (or cell fate) through differentiation (Fig. 7.6) from pluripotent one [Takahashi and Yamanaka, 2016]. It is quite remarkable, despite the huge complexity of a human cell as a system it has a rather small number of discrete (distinguishable) somatic states that were shown experimentally [Huang et al., 2005].
Note
Emergence - a high-level property of a system, which is not a property of any component of that system. This intractability can be variously interpreted. In weak emergence interpretations, the inexplicability is merely a practical one due to the sheer complexity of the computation required to reach an explanation. In strong emergence interpretations, the emergent property possesses causal autonomy independently of its constituents [O’Byrne and Jerbi, 2022].
@@ -631,26 +631,26 @@
Fig. 12.6 Waddington landscape - a metaphor of cell fate transition during differentiantion as a ball rolling down from the top of Waddington’s “mountain” (pluripotency) to the bottom of a “valley” (somatic state, e.g. skin, muscle). The opposite process is called reprogramming.#
+
Fig. 7.6 Waddington landscape - a metaphor of cell fate transition during differentiantion as a ball rolling down from the top of Waddington’s “mountain” (pluripotency) to the bottom of a “valley” (somatic state, e.g. skin, muscle). The opposite process is called reprogramming.#
Canalization and emergence often arise because biological networks can achieve robustness through degeneracy, and so become stable against perturbations - they can maintain dynamic functional stability even when environmental or internal conditions change. This degeneracy, as we said above, means that the same functional result can be achieved by multiple alternative pathways, strategies or system states. For example, during skeletal muscle aging, impaired chaperone-mediated autophagy appears to be compensated by upregulation of macro-autophagy [Cohen et al., 2022]. Other aging-related examples of canalization include cellular senescence which is a relatively discrete state, an alternative to being a healthy cell or to apoptosis. Senescence is thus an attractor state, with many intermediate states within its “attractor basin”. On the level of the organism, we can discover other emergences like frailty - a clinically recognizable state of increased vulnerability resulting from the aging-associated decline in reserve and function across multiple physiologic systems such that the ability to cope with everyday or acute stressors is comprised [Xue, 2011]. Finally, aging is another example of emergent organism property and as it was mentioned above, we highly favor attempts to measure this property.
It is time to discuss the dynamic equilibrium mentioned in the epigraph of this chapter. Dynamic equilibrium has two synonyms: resilience and stability - the first is more biological and the latter is more physical. But in the majority of contexts, they mean the ability of a system to return to the equilibrium state preceding a perturbation. The common metaphor for understanding resilience is a ball in the hole model (Fig. 12.7a,d). You can see that a small perturbation barely can knock out the ball from a basin with sharp and high walls (so-called potential barrier) which is associated with the high resilience of the system. On the other hand, low walls do not save the ball from even a small perturbation. The event when a ball leaves its current local minimum state and cross the boundary of its basin is called critical transition which we discuss below. Resilience can be measured and different measures of a resilience (including mathematically tractable) were proposed [Varadhan et al., 2008], [Gavrilov and Gavrilova, 2005], [Gijzel et al., 2017], [Pyrkov et al., 2021], [Schosserer et al., 2019]. It is especially important to measure resilience in older adults to aware of the risks which they bear. Moreover, it was proposed that aging itself is a progressing loss of resilience [Promislow et al., 2022].
-
The resilience concept is tightly bounded with critical transition concept which has synonyms of (more physical) phase transition and (more mathematical) bifurcation. It is usually defined as a qualitative change in macroscopic properties of the system [O’Byrne and Jerbi, 2022]. One important class of critical transitions is a fold catastrophic transition (which is also tempted to associate with the death of the organism) [Scheffer et al., 2009]. This transition is characterized by the “sharp” falling of the system parameter into a new state (we will cover the mathematical details of this bifurcation in the next chapter). We are especially interested in this type of bifurcation because of the presence of early warning signals preceding the system’s critical transition which we can directly observe in the data. Suppose that our aging-related dataset contains a system variable characterizing its macroscopic state (examples include mentioned heart rate (Fig. 12.4) or self-rated health evaluations [Gijzel et al., 2017]) and this variable is presented in a form of one-dimensional time series for patients of different ages. Then, we can calculate variances and autocorrelations for all participants in the dataset. The bifurcation theory predicts that approaching a critical transition boundary (aka tipping point) is accompanied by the system critical slowing down and an increase in variance and temporal autocorrelations of the system state variable (Fig. 12.7b,c,e,f) [Scheffer et al., 2009] and this is something that we expect to observe in real aging data.
It is time to discuss the dynamic equilibrium mentioned in the epigraph of this chapter. Dynamic equilibrium has two synonyms: resilience and stability - the first is more biological and the latter is more physical. But in the majority of contexts, they mean the ability of a system to return to the equilibrium state preceding a perturbation. The common metaphor for understanding resilience is a ball in the hole model (Fig. 7.7a,d). You can see that a small perturbation barely can knock out the ball from a basin with sharp and high walls (so-called potential barrier) which is associated with the high resilience of the system. On the other hand, low walls do not save the ball from even a small perturbation. The event when a ball leaves its current local minimum state and cross the boundary of its basin is called critical transition which we discuss below. Resilience can be measured and different measures of a resilience (including mathematically tractable) were proposed [Varadhan et al., 2008], [Gavrilov and Gavrilova, 2005], [Gijzel et al., 2017], [Pyrkov et al., 2021], [Schosserer et al., 2019]. It is especially important to measure resilience in older adults to aware of the risks which they bear. Moreover, it was proposed that aging itself is a progressing loss of resilience [Promislow et al., 2022].
+
The resilience concept is tightly bounded with critical transition concept which has synonyms of (more physical) phase transition and (more mathematical) bifurcation. It is usually defined as a qualitative change in macroscopic properties of the system [O’Byrne and Jerbi, 2022]. One important class of critical transitions is a fold catastrophic transition (which is also tempted to associate with the death of the organism) [Scheffer et al., 2009]. This transition is characterized by the “sharp” falling of the system parameter into a new state (we will cover the mathematical details of this bifurcation in the next chapter). We are especially interested in this type of bifurcation because of the presence of early warning signals preceding the system’s critical transition which we can directly observe in the data. Suppose that our aging-related dataset contains a system variable characterizing its macroscopic state (examples include mentioned heart rate (Fig. 7.4) or self-rated health evaluations [Gijzel et al., 2017]) and this variable is presented in a form of one-dimensional time series for patients of different ages. Then, we can calculate variances and autocorrelations for all participants in the dataset. The bifurcation theory predicts that approaching a critical transition boundary (aka tipping point) is accompanied by the system critical slowing down and an increase in variance and temporal autocorrelations of the system state variable (Fig. 7.7b,c,e,f) [Scheffer et al., 2009] and this is something that we expect to observe in real aging data.
-
Fig. 12.7 a-c, Far from the bifurcation point (a), resilience is large in two respects: the basin of attraction is large and the rate of recovery from perturbations is relatively high. If such a system is stochastically forced, the resulting dynamics are characterized by low correlation between the states at subsequent time intervals (b, c). d–f, When the system is closer to the transition point (d), resilience decreases in two senses: the basin of attraction shrinks and the rate of recovery from small perturbations is lower. As a consequence of this slowing down, the system has a longer memory for perturbations, and its dynamics in a stochastic environment are characterized by a larger variance and a stronger correlation between subsequent states (e, f).#
+
Fig. 7.7 a-c, Far from the bifurcation point (a), resilience is large in two respects: the basin of attraction is large and the rate of recovery from perturbations is relatively high. If such a system is stochastically forced, the resulting dynamics are characterized by low correlation between the states at subsequent time intervals (b, c). d–f, When the system is closer to the transition point (d), resilience decreases in two senses: the basin of attraction shrinks and the rate of recovery from small perturbations is lower. As a consequence of this slowing down, the system has a longer memory for perturbations, and its dynamics in a stochastic environment are characterized by a larger variance and a stronger correlation between subsequent states (e, f).#
-
One interesting example of the critical slowing down can be found in frail patients (Fig. 12.8) [Kalyani et al., 2012]. The figure represents the resulting dynamics of glucose in oral glucose tolerance test (OGTT) demonstrating slower recovery in glucose level after perturbation in frail patients by comparing with non-frail and pre-frail. Indeed, the slowdown in metabolism, proliferation and information processing is a major feature of aging [Ukraintseva et al., 2021]. Consequently, frail and non-frail individuals may differ in how they respond dynamically to stressors or drug perturbations.
+
One interesting example of the critical slowing down can be found in frail patients (Fig. 7.8) [Kalyani et al., 2012]. The figure represents the resulting dynamics of glucose in oral glucose tolerance test (OGTT) demonstrating slower recovery in glucose level after perturbation in frail patients by comparing with non-frail and pre-frail. Indeed, the slowdown in metabolism, proliferation and information processing is a major feature of aging [Ukraintseva et al., 2021]. Consequently, frail and non-frail individuals may differ in how they respond dynamically to stressors or drug perturbations.
-
Fig. 12.8 Glucose dynamics during oral glucose tolerance test by frailty status. Mean ± SE (error bars) for glucose values respectively after a 75 g glucose load.#
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Fig. 7.8 Glucose dynamics during oral glucose tolerance test by frailty status. Mean ± SE (error bars) for glucose values respectively after a 75 g glucose load.#
Is the living organism the complex dynamic network of interacting elements acquiring discrete emergent states through the canalization, forming modules experiencing loss of resilience through the loss of complexity during aging and approaching its critical transition point ultimately falling into the death state? - the question is open to You!
Aging biology is, in particular, so challenging because of the absence of longitudinal highly dense data: chemical signals are not so easy for measurements like electrical signals in neurons, moreover, data quality is rather poor frequently.
Complex systems framework do not answer directly what molecule or intervention will repair the system by getting it back to the equilibrium state, but rather provide a useful methodology to evaluate this state and more precisely measure, control and study effects of any new molecules used for treatment. It also may point to what essential components of the system undergoes changes that manifest themselves as aging. Might it be an extracellular matrix degradation, genomic instability or mitochondrial dysfunction? What system state parameter may unequivocally indicate the loss of resilience and explain the reason for the loss? How can we define those elements of the system which contribute more to aging? - these are open questions. What we can do now is to make your dynamic systems intuition deeper in the next section.
The differential equation is a very useful instrument for understanding and modeling real biological processes. Once you develop a basic intuition of the differential equation you will see the world as a set of dynamic systems. What is a dynamic system? Well, everything! Everything that we may observe evolving in time could be (and possibly must be) described with this instrument.
where f(x, t) - given function of time \(t\) and state variable \(x\) itself. Note, that \(x=x(t)\) is also a function of time, hence, \(\dot{x} = dx(t)/dt = dx/dt\) - we omit \((t)\) for simplicity. A huge body of knowledge of differential equations theory has been elaborated for ages. In this tutorial, we only touch on the most important topics needed for understanding complex systems resilience introduced in the previous chapter.
where constant \(C\) is defined from initial conditions. This is the simplest case of a separable equation (s.t. you can separate variable \(x\) from \(t\) on both sides) that describes linear growth of \(x\). For example, the amount of water accumulated in a water tank fed by a pipe. The next case:
This equation describes the dynamics of so-called exponential growth/death depending on the sign of the coefficient \(a\). One common example of a system described by this equation is the number of bacteria in a petri dish in the case when resources for growth are unlimited. Indeed, you know that number of bacteria the next day depends on the number of bacteria on the previous day. Thus, the differential equation has the form \(\dot{x} = ax\) and has a solution in a form of exponential growth.
Two important observations: (1) the solution of equation (13.2) always diverges, i.e there is no non-zero constant \(Const\) such that \(x(\infty) \to Const\); (2) the solution of equation (13.3) has three regimes: one at positive constant \(a\) - divergence (exponential growth), one at negative \(a\) - convergence to zero (exponential decay), and one at \(a=0\) - trivial case with no dynamics. Adjusting constant \(a\) continuously leads to switching between regimes which is something that we call bifurcation and will study further. In other words. they say a system undergoes a bifurcation as \(a\) approaches zero (from the left or right side). Of course, in this case, the system changes its behavior qualitatively.
+
Two important observations: (1) the solution of equation (8.2) always diverges, i.e there is no non-zero constant \(Const\) such that \(x(\infty) \to Const\); (2) the solution of equation (8.3) has three regimes: one at positive constant \(a\) - divergence (exponential growth), one at negative \(a\) - convergence to zero (exponential decay), and one at \(a=0\) - trivial case with no dynamics. Adjusting constant \(a\) continuously leads to switching between regimes which is something that we call bifurcation and will study further. In other words. they say a system undergoes a bifurcation as \(a\) approaches zero (from the left or right side). Of course, in this case, the system changes its behavior qualitatively.
Linear differential equations and systems of them provide a lot of possibilities for modeling technical and living systems. However, already in the case of bacteria growth modeling with limited resources, we force the need to use quadratic models. This is where the logistic model appears:
What regimes are possible for this solution? Let’s suppose that carrying capacity \(K=1\), initial population \(x_0=0.5\) and explore graphically the dynamics at different \(r\):
The class of integrable differential equations is very small, even so, they are extremely useful for describing complex systems dynamics. In some cases, however, you may develop a more sophisticated model of a process under study and the corresponding differential equation does not have a closed-form solution. What can we do in this case? Consider the following complication of the logistic model:
now our logistic model contains two additional assumptions: (i) the growth rate changes as a sinusoidal function of time \(r = r(t) = r_0 sin^2(t)\) - it models a photo-sensitivity of bacteria allowing them to reproduce effectively in the day time with maximum growth rate or \(r_0\); (ii) the carrying capacity slowly increases with time by square root law starting from 1, namely \(K = K(t) = 1 + \sqrt{t}\). You can check that this new differential equation is non-integrable in elementary functions, so we need to obtain the solution numerically.
-
We introduce the Forward-Euler scheme of integration. without going into the details of the proof, forward Euler prescribes just to iteratively compute each next step as a sum of results from the previous step and the computed value of the function on the previous step. Let’s rewrite our new logistic equation (13.5) in the general form:
+
We introduce the Forward-Euler scheme of integration. without going into the details of the proof, forward Euler prescribes just to iteratively compute each next step as a sum of results from the previous step and the computed value of the function on the previous step. Let’s rewrite our new logistic equation (8.5) in the general form:
Now we start to develop the core dynamical system property - its stability. It is a case when the mathematical framework has a strict everyday physical analogy. Let’s discuss the picture below:
We see a ball in three positions. You can easily say which position is stable, unstable, or metastable. Indeed, the first position denoted \(x^*_1\) - is a stable position. Mathematically speaking, the position of \(x\) is called stable if after any arbitrarily small perturbation (\(y\)) the system returns the initial state \(x^*_1\). The opposite case for \(x^*_2\) where any small perturbation forces the system to leave the previous state. The third case \(x^*_3\) is metastable (or marginally stable) that is this position is a midpoint between stable and unstable regimes of the system (remember equation (13.3) with \(a=0\)). This third case is also called critical and it is very important for our discussion of system resilience. One minor technical thing remained. We need to understand how to obtain this so-called potential landscape depicted at Fig. 13.1, i.e. some function whose maxima, minima and saddles (or flat valleys) describe the full set of fixed (or critical) points \(\{x^* | \frac{dx}{dt}|_{x=x^*}=0\}\), and, what is more important, characterizes stability properties of these points.
+
We see a ball in three positions. You can easily say which position is stable, unstable, or metastable. Indeed, the first position denoted \(x^*_1\) - is a stable position. Mathematically speaking, the position of \(x\) is called stable if after any arbitrarily small perturbation (\(y\)) the system returns the initial state \(x^*_1\). The opposite case for \(x^*_2\) where any small perturbation forces the system to leave the previous state. The third case \(x^*_3\) is metastable (or marginally stable) that is this position is a midpoint between stable and unstable regimes of the system (remember equation (8.3) with \(a=0\)). This third case is also called critical and it is very important for our discussion of system resilience. One minor technical thing remained. We need to understand how to obtain this so-called potential landscape depicted at Fig. 8.1, i.e. some function whose maxima, minima and saddles (or flat valleys) describe the full set of fixed (or critical) points \(\{x^* | \frac{dx}{dt}|_{x=x^*}=0\}\), and, what is more important, characterizes stability properties of these points.
Let’s use an already familiar logistic model for all derivations.
-(13.6)#\[
+(8.6)#\[
\frac{dy}{dt} \approx \lambda_\theta y
\]
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We will refer to the obtained as a perturbation equation. A simple analysis is now applied to determine whether the perturbation grows or decays as time evolves (remember equation (13.3) with \(\lambda_\theta\) instead of \(a\)). If \(\lambda_\theta > 0\) the initially arbitrarily small perturbation grows in time and dynamics become unstable. On the other hand, if \(\lambda_\theta < 0\) the perturbation decays in time and stable dynamics is observed. In the intermediate case, \(\lambda_\theta = 0\) system is in a critical state and nothing can be said about system stability until the value of \(\lambda_\theta\) changes.
+
We will refer to the obtained as a perturbation equation. A simple analysis is now applied to determine whether the perturbation grows or decays as time evolves (remember equation (8.3) with \(\lambda_\theta\) instead of \(a\)). If \(\lambda_\theta > 0\) the initially arbitrarily small perturbation grows in time and dynamics become unstable. On the other hand, if \(\lambda_\theta < 0\) the perturbation decays in time and stable dynamics is observed. In the intermediate case, \(\lambda_\theta = 0\) system is in a critical state and nothing can be said about system stability until the value of \(\lambda_\theta\) changes.
Let’s look how \(\lambda_\theta\) depends on its parameters \(\theta\):
Once we studied a system’s behavior around fixed points, we are ready to formulate a potential function definition. It is rather straightforward but let us step back and discuss what drives any dynamics system. It is a funny coincidence (or not?) that symbol of function \(f\) is denoted with the letter “f” just like a physical force \(F\). Indeed, any mechanistic dynamical system drives with some force \(F\):
In this paragraph, we develop our first dynamical model of organismal aging. This will be, certainly, a toy model but it helps us to formulate the aging problem in the learned dynamical system framework, study its potential functions and get acquainted with a bifurcation concept. We develop our model from scratch and step by step, based on the techniques we learned above.
First, we need to model the growth of an organism. We already know how to do that because the organism is a bag of cells with limited bag sizes. The important difference between a colony of unicellular organisms and a multicellular is now intercellular communication is of great importance. Following the spirit of previous chapters of this book, we will model biological age with an evolving system variable \(x\). The growth term can be defined as before in the logistic model \((r-x)x\), where \(r\) is a growth rate. Note, that the first term model does not contain the carrying capacity term explicitly, but you may easily guess that it is equal to \(1\), check it out! The second term is something new for us. We want to model costs for intercellular communication which: (i) grow with the number of cells as a square of cell number (due to pairwise interactions), (ii) becomes larger with age (change in \(\delta\) parameter) due to the increase in the number of breakdowns, protein misfolding and epigenetic alterations within cells, an extracellular matrix degradation, other deregulations which are consequences of the previous two. We propose to model all this stuff with a term \(\delta x^2\), where \(\delta\) reflects costs related to an organism’s maintenance. Writing them together we have:
You may notice some ambiguity in this model. On the one hand, we model biological age \(x\), but terms model the number of cells and costs on their pairwise communications. The reason is that biological age is a *latent variable describing organism’s state. This variable may be constructed, in principle, as a combination of other explicit variables that has direct physical meaning. Therefore, a latent variable may be correlated with an actual number of cells at a one-time interval and with costs for an organism interval at another time interval. Unfortunately, it is very difficult to formulate aging as a unimodal explicit physical process (otherwise it would have already been done), so it is convenient to use latent representation.
Exercise 3
-
Compute critical points, potential function and analytical solution for the equation (13.7).
+
Compute critical points, potential function and analytical solution for the equation (8.7).
Fig. 13.2 Bifurcation diagram for the toy aging model.#
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Fig. 8.2 Bifurcation diagram for the toy aging model.#
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The solid blue line on the diagram is a set of stable solutions for the system (13.7). The dashed blue line at point \(x=0\), otherwise, a set of unstable solutions. The set of initial conditions taken above the stable line results in convergence to the stable solution. The set of initial conditions taken between the stable and the unstable lines results in convergence to the stable solution. Black arrows point in the direction of system evolution. When we increase \(\delta\) starting from \(0\) the stable solution increases in magnitude until we achieve point \(\delta_c\) which is a critical threshold after which a bifurcation occurs. After crossing the critical threshold no physically stable solutions remain (here we assume that biological age cannot accept negative values, so the red line cannot be a solution) and the system is doomed to diverge.
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The solid blue line on the diagram is a set of stable solutions for the system (8.7). The dashed blue line at point \(x=0\), otherwise, a set of unstable solutions. The set of initial conditions taken above the stable line results in convergence to the stable solution. The set of initial conditions taken between the stable and the unstable lines results in convergence to the stable solution. Black arrows point in the direction of system evolution. When we increase \(\delta\) starting from \(0\) the stable solution increases in magnitude until we achieve point \(\delta_c\) which is a critical threshold after which a bifurcation occurs. After crossing the critical threshold no physically stable solutions remain (here we assume that biological age cannot accept negative values, so the red line cannot be a solution) and the system is doomed to diverge.
Okay, let’s gather two insights from the bifurcation diagram about our model:
It is not necessary to cross the critical threshold to obtain an arbitrarily large biological age. It is enough that \(\delta\) would be close to \(1\).
Solve the equation (13.8) analytically and draw a bifurcation diagram (on a sheet of paper). What difference with the initial model?
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Solve the equation (8.8) analytically and draw a bifurcation diagram (on a sheet of paper). What difference with the initial model?
This a good first step to the world of really complex aging models. But we already can say much even with these one-dimensional models. More insights are waiting for us in the multidimensional world which we will discuss below.
It’s time to expand our understanding of dynamical systems to a multidimensional case. We first introduce a new concept of state vector\(x\) which is a replacement for state variable \(x\) used above for unidimensional dynamic systems. State vector is composed from \(n\) vector components\(\{x_i\}_n\) which represents different dynamical variables of a system (organism). State vector, in principle, can be composed from any physiological state variables (blood pressure, body temperature, heart rate, etc.), but it is important to choose such of them which describes the system most “completely”. For example, when you try to build a dynamical model of a running human, it is desirable to include heart rate as variable into a state vector. On the other hand, such variable as hair length is apparently redundant to include. The same logic works for a model of aging. When we try to model it on a physiological level, tissue level or molecular level, different variables should be used. However, some models can comprise state variables from different levels of system organization. In this case, we need to bind them properly with functional relations.
In this section we will consider molecular model of human aging building it based on the concept of gene regulatory network, which we discuss below.
Linear systems of differential equations allows to model a countless number of complex phenomena. We start with a simple model of gene regulation, which includes feedback control loop (Fig. 13.3). This is a scalable model proposed in [] for modeling a number of genes. We restrict ourselves to a consideration only a one gene-rna-protein cascade where the protein acts as a regulator of the gene transcription.
Linear systems of differential equations allows to model a countless number of complex phenomena. We start with a simple model of gene regulation, which includes feedback control loop (Fig. 8.3). This is a scalable model proposed in [] for modeling a number of genes. We restrict ourselves to a consideration only a one gene-rna-protein cascade where the protein acts as a regulator of the gene transcription.
Let’s combine both state variables into a state vector \(x = [r, p]^T\) (we will use column notation for vectors). Then the right part of the equation (8.9) can be written as a matrix \(M\) of coefficients multiplied by the state vector \(x\):
We start with rather simple but still tractable Michaelis-Menten gene regulatory model []. We first formulate this model for \(n\) genes where each gene \(x_i\) is regulated by others including itself:
Here, \(B\) - degradation rate in a linear term we have seen earlier, we assume it is the same for all genes; \(A_{ij}\) the regulatory constant which can take values \(0\) or \(1\) indicating the presence or absence of an activatory regulation. An element of matrix \(A_{ij}\) is read as \(A_{i\leftarrow j}\) meaning that \(j\)-th gene (column index) acts to \(i\)-th gene (row index). But wait, why the second non-linear term has so strange functional form? It is a saturating function converging to \(1\) as \(x_j\) increases. It turns out that it reflects quite simple idea of promoter regulation, namely, if some product of gene \(j\) binds to a promoter of gene \(i\) then this promoter is no more free for binding. Thus, the activatory effect is restricted with a number of different promoter binding sites for a particular gene what is reflected in the saturating functional form of the term.
In this section, we will demonstrate exactly how complex systems can be reduced to simple universal laws. We will see that the system that we considered above obeys a universal law that allows us to investigate its stability and resilience (yes, in this section we will consider the difference between these two concepts). We will follow a framework developed in the paper [] which in short can be formulated as following. Consider a system consisting of \(n\) components (nodes of a network) \(x = (x_1, ..., x_n)^T\). The components of systems evolves according to the following general system of non-linear (in general) differential equations:
The first term on the right-hand side of the equation describes the self-dynamics of each component, while the second term describes the interactions between component \(i\) and its interacting partners. The nonlinear functions \(F(x_i)\) and \(G(x_i, x_j)\) represent the dynamical laws that govern the system’s components, while the weighted connectivity matrix \(A_{ij} > 0\) captures the positive interactions between the nodes. This is quite general form of non-linear equation and a lot of multidimensional dynamical systems can be described with it.
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As we learned earlier, the stability of a system can be compromised if many system parameters are changed. This distinguishes multidimensional systems (networks) from one-dimensional ones they have a definite topology of interaction network. The brilliant discovery of the paper under consideration is that we can compute universal number which characterize a total network topology of equation (13.12). The main result of the paper is that we can rewrite (13.12) in one-dimensional form as follows:
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As we learned earlier, the stability of a system can be compromised if many system parameters are changed. This distinguishes multidimensional systems (networks) from one-dimensional ones they have a definite topology of interaction network. The brilliant discovery of the paper under consideration is that we can compute universal number which characterize a total network topology of equation (8.12). The main result of the paper is that we can rewrite (8.12) in one-dimensional form as follows:
where \(x_{eff}\) - is an effective dynamic component representing “average” dynamics of the whole system, and \(\beta_{eff}\) - is a effective topological characteristic of the network which is actually a macroscopic description of \(n^2\) microscopic parameters of matrix \(A\). They can be computed as follows:
-(13.15)#\[
+(8.15)#\[
\beta_{eff} = \langle s\rangle + SH
\]
We already know \(\langle s \rangle\) is a average degree of elements, but it has another name - density of a network; \(S\) represents the symmetry of network \(A\), and \(H\) represents heterogeneity. They can be computed as follows:
Fig. 13.4 The topological characteristics of a network.#
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Fig. 8.4 The topological characteristics of a network.#
The density property is quite self-explainable - it is just an average number of in/out degrees in a network. The symmetry means that most of nodes have bi-directed edges to each other. An asymmetric network is one where nodes with a large in-degree tend to have a small out-degree. The large heterogeneity means that some nodes may have large in/out degrees and others - small. In the network with small heterogeneity, all nodes have almost equal degrees. Let’s compute them and check that hey are actually sum into \(\beta_{eff}\).
These nice network characteristics confer us an instrument for improvement \(\beta_{eff}\) value. You remember expression (13.14)? It is time to plot bifurcation diagram!
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These nice network characteristics confer us an instrument for improvement \(\beta_{eff}\) value. You remember expression (8.14)? It is time to plot bifurcation diagram!
We have stable non-zero solution while \(\beta_{eff} > \beta_c\). The critical value of \(\beta_c=1\) is a root of the second expression in (13.14). Of course, crossing the \(\beta_c\) from right to left leads to a bifurcation and a cell dies. What else interesting about value of \(\beta_{eff}\) is that it represents the reserve of resilience in the system. If some topological perturbation occurs (e.g. some edges are removed) then it is better to have larger value \(\beta_{eff}\), otherwise the system is under risk of a killing bifurcation.
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How can we improve the resilience? From the equation (13.15) we see several options:
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We have stable non-zero solution while \(\beta_{eff} > \beta_c\). The critical value of \(\beta_c=1\) is a root of the second expression in (8.14). Of course, crossing the \(\beta_c\) from right to left leads to a bifurcation and a cell dies. What else interesting about value of \(\beta_{eff}\) is that it represents the reserve of resilience in the system. If some topological perturbation occurs (e.g. some edges are removed) then it is better to have larger value \(\beta_{eff}\), otherwise the system is under risk of a killing bifurcation.
+
How can we improve the resilience? From the equation (8.15) we see several options:
Make the network more dense;
If \(S\) is negative - decrease its absolute value;
From the previous model, we saw that there are many ways to reduce the resilience of the system to such an extent that the system (organism) can no longer cope with the next perturbation and a catastrophic bifurcation occurs. The hypothesis of aging as a complex system in which resilience decreases is not very optimistic, in the sense that it immediately follows that aging is mostly stochastic. However, the optimistic side of this hypothesis is that if we can find a systemic effect on the body that improves its resilience, then we may well be able to avoid aging. Thus, we could improve the dynamic model by supplementing it with the mechanisms of action of various metabolites on the gene regulatory network and get a playground for testing drugs that have a systemic effect on a body resilience. In the end, it’s possible that the complex system model is simply wrong and we need to look for something better.
“The bifurcation theory predicts that approaching to a critical transition boundary (aka tipping point) is accompained with the system critical slowing down and an increase in variance and temporal autocorrelations of system state variable” - do your remember this citation from the previous chapter? In this section we discuss the explicit mathematical background of the phenomenon. We also introduce one additional instrument which is widely used in dynamical modelling, namely, stochasticity. In this section we will mostly rely on the theoretical results form the paper [Scheffer et al., 2009].
In contrast to the previous section, we return to the one dimensional models, in addition, we understood that some of multidimensional models can be reduced to one-dimensional allowing to analyze system resilience in more compact form. We start with a new model which is a combination of what we learned above:
We have seen all the parts of this equation. The first quadratic term describes the growth of something with a carrying capacity \(K\). The second term is similar to what we have seen in the previous section (especially if you did the exercise), it describes some saturating removal of something from the system with rate \(c\). For example, this model can describe a number of senescent cells in the organism which has a limited capacity for the growth and they are removed by the immune system. The last term represents the noise in the growth condition by a plephora of random factors which we do not know explicitly and just model them by random gaussian noise \(\epsilon \sim \mathbf{N}(0, 1)\) with average \(0\) and standard deviation of \(1\). After multiplication with \(\sigma\) the standard deviation is scaled. The presented stochastic model is simple and do not require a lot of specific knowledge for modeling as we will see it below.
Exercise
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Compute the potential function for the deterministic part of (13.16).
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Compute the potential function for the deterministic part of (8.16).
Critical slowing down arises in dynamical systems which undergoes changing in their parameters. If some parameter changes in such way that it moves the system to the bifurcation point (critical point), several interesting effects in actual dynamics can be observed. There are three such effects:
If the system is driven by stochastic force, the critical slowing down manifests itself in:
We discuss the first and the second hallmarks of critical slowing down that can be easily observed in one-dimensional system. We will change the parameter \(c\) from the system above and move, therefore, system to the bifurcation. Below we write some code for simulating the equation (13.16) and the potential function for the deterministic part of the equation.
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We discuss the first and the second hallmarks of critical slowing down that can be easily observed in one-dimensional system. We will change the parameter \(c\) from the system above and move, therefore, system to the bifurcation. Below we write some code for simulating the equation (8.16) and the potential function for the deterministic part of the equation.
We have studied quite useful tools for data analysis. Can we use them to create an organism resilience indicator? The good resilience indicator should satisfy not only the all three properties mentioned above, but how one-dimensional variable can satisfy the third one? There is one way to do that by constructing a new latent variable from initial organism state variables. Moreover, the good resilience indicator is expected to be negatively correlated with age. How to deal with all of that? Let’s do it step by step.
First let’s get the real data of mouse complete blood count (CBC) analysis. We will use one prepared dataset from the []. As authors mentioned, the samples were obtained from NIH Swiss male and female mice from Charles River Laboratories (Wilmington, MA) - please refer to the original paper for the details. We will use only one cohort of male mice to avoid batch effect.
This chapter has been entirely devoted to developing an intuition about how aging can be perceived through the lens of the criticality and resilience of systems.
We have studied the very foundations of the theory of dynamical systems, got acquainted with their one-dimensional and multidimensional incarnations. We saw how multidimensional ones can be reduced to one-dimensional ones, the stability of which is much easier to study.
In addition, we have learned to better understand the language of complex systems by linking its most important concepts (criticality, bifurcation, phase transition, network, etc.) with mathematical counterparts. It is good to understand this new language, but it is important to use it. If possible, try to model biological processes explicitly, instead of just saying “associated”, “correlates”, “induces”.
Be quantitative, numerical relationships between variables deepen your understanding of the process. However, sometimes numbers can be confusing - don’t be a slave to numbers, think twice!
Imagine that a genie gives you a magic device that shows people’s health as a number from zero to one hundred. For healthy young people, the device shows 100. Over time, the number decreases and when it reaches zero, the person dies. How could such a device be used?
First, it could speed up and cheapen clinical trials. If you are looking for a geroprotector, the primary endpoint of the trial is lifespan. People live on average 72 years, so it takes a lot of money and time to do the research, so you have all the chances to die yourself before you get the results. For example, now funds are being raised for the TAME [Barzilai et al., 2016] clinical trial of metformin for aging. In the last seven years, TAME – Targeting Aging with Metformin – has raised only 11 million dollars out of the necessary 40, and patient observation will last 6 years. If we had such a magic device, we wouldn’t necessarily have to wait so long. Instead of the primary endpoint (prolonging lifespan), we could use a surrogate endpoint. If the device shows an improvement in health, this would be a good sign that the geroprotector is working and we could stop the trial earlier. This would save money and time.
Secondly, the magic device would help us search for new interventions against aging. For example, we could collect a large dataset of device readings for different people and try to find associations between lifestyle and better health. Or if our magic device is a computer program that predicts a person’s health based on their indicators, we could run an optimization process: we would select such indicators to maximize health, and then find a way to similarly change human indicators. We will see how this idea helped find new geroprotectors that extended the life of worms [Janssens et al., 2019].
The most common causes of death in developed countries are cancer, cardiovascular disease, type 2 diabetes, Alzheimer’s disease, and Parkinson’s disease. These diseases have different mechanisms, but the strongest risk factor for all of them is age, so researchers of aging believe that aging is not just an association but a cause of many diseases, so slowing aging would reduce the likelihood of illness. This is called the geroscience hypothesis.
During transcription, RNA is produced based on the DNA sequence, which usually leads to protein synthesis. Different proteins are required under different conditions, so there are different ways to control transcription. One of these methods is DNA methylation, in which a methyl group is attached to a nucleotide, which typically leads to suppression of transcription.
In mammals, methylation usually occurs in special DNA regions consisting of “CG” nucleotides, called CpG sites. The average level of cytosine methylation in these regions can be experimentally measured as a number between 0 and 1, where 1 means that all cytosines in the site are methylated on both alleles and 0 means that all cytosines are unmethylated.
Ordinary biological clocks are trained to predict chronological age, expecting the model error to explain the mortality difference. But where does this assumption come from? If we could train a perfect oracle model that always correctly predicted chronological age, such a model would be useless. That is why we could predict how many years a person has left to live instead of chronological age.
In [Levine et al., 2018] the authors first trained a Cox proportional regression model to predict mortality using clinical biomarkers and age from the NHANES database. With it, they made a prediction for a dataset with blood methylation, and then trained the model to directly predict mortality from methylation data.
Acceleration of such a phenotypic epigenetic age is positively associated with mortality – an important property that was not present in previous clocks, so such clocks are called second generation clocks.
The GrimAge[Lu et al., 2019] clock was constructed in a similar manner. As proxies for predicting mortality, the authors took the concentration of blood proteins and the number of smoked cigarette packs, which were predicted from methylation. As a result, a model was obtained that better predicted the date of death and the onset of age-related diseases than previous models.
DNA transcription is the process by which mRNA molecules, which are used to create proteins, are synthesized. By analyzing the full transcriptome, or all of the RNA molecules, of a cell, we can sequence and compare the transcriptomes of healthy individuals with those of sick individuals. This helps to uncover the mechanisms behind diseases and potentially discover new treatments[Yang et al., 2020].
We could learn more about aging by comparing the transcriptomes of old people with those of young people. One way to do this is to build aging clocks on the transcriptome and try to understand why the model turned out to be that way. This is what was done in [Mamoshina et al., 2018], where the authors trained clocks on the transcriptomes of human muscles. The clocks were less accurate (MAE 6.19 years) than methylome-based clocks, but feature importance techniques proposed new targets for potential geroprotectors.
In addition to interpretability, transcriptome-based clocks have another advantage. C. elegans roundworms do not have 5mC nucleotides, so epigenetic clocks cannot be built for them in the traditional sense. Therefore, transcriptome-based clocks were built for c. elegans [Meyer and Schumacher, 2021, Tarkhov et al., 2019] that work for worms of different strains living in different conditions.
Transcriptome and methylome are obtained through expensive sequencing. It would be more convenient for clinical practice to build an aging clock on cheap complete blood count. This was done in [Putin et al., 2016], and the feature importance technique showed that the most important predictors of age are albumin, glucose, alkaline phosphatase, urea, and erythrocytes. The model can be conveniently used through the website http://aging.ai.
In the following article [Mamoshina et al., 2018], the authors improved the accuracy of the model and reduced the number of necessary measurements. The authors found that the quality of the model varies between patients from different countries: Canada, Eastern Europe, and South Korea.
Acceleration of biological age meets the criteria for a marker of aging. For example, it predicts mortality from various causes[Marioni et al., 2015]. An increase in epigenetic clock of the first generation is not associated with a deterioration of physical and cognitive functions[Marioni et al., 2015], but is associated for clocks of the second generation[Maddock et al., 2020].
In 2006, the CMAP dataset was published [Lamb et al., 2006], in which the authors measured how the transcriptome of cells changes after exposure to 1300 different drugs. Using this dataset and biological clocks, it is possible to find new drugs. For this, in [Janssens et al., 2019] the authors trained transcriptomic clocks on 51 human tissue and then searched in CMAP for drugs that, according to the model, reduce biological age. As a result, known geroprotectors and several new molecules were found, which were tested on worms c. elegans. The experiment confirmed that inhibitors of the protein Hsp90 extend life.
During the search for drugs that lower the predicted model age, we find geroprotectors that extend life. Does this mean a causal relationship? Can we call a molecule that reduces biological age a geroprotector, but for which a costly, long-term clinical trial has not yet been conducted that proves a reduction in mortality? There are several problems [Bell et al., 2019, Schork et al., 2022].
Predictions of different clocks are weakly correlated with each other. Epigenetic clocks are trained on different CpG sites.
Does this mean that clocks measure different aspects of aging? Or that clock predictions are very noisy? The authors of [Liu et al., 2020] show that clocks capture different aspects of aging. If this is the case, it would be important to find these separate aspects and make separate clocks for them.
As we can see, developing biological clocks is an important task in aging biology. Most biological clock models predict either chronological age or risk of death based on a person’s parameters. But how does this technically happen? How can we make our own biological clocks for our own data?
Almost all biological clocks are linear regression. Linear regression is ubiquitous, so we assume that the reader is already familiar with it, so we will only remind you of the main ideas. For a first introduction to linear regression, we recommend practical tutorials[1], [2], [3].
In the linear regression model, we assume that the predicted value is linearly dependent on the input data. Let \(y\) be the chronological age that we are trying to predict. \(x_{1}, x_{2}, \dots x_{n}\) are known input parameters, such as blood test results. Then in the linear regression model, we approximate \(y\) as a linear function:
The more parameters a model has, and the less data in the training dataset, the easier it is for the model to simply memorize the entire training dataset. In this case, the linear function approximating chronological age will poorly describe the real world, and we will see a large error on the test dataset. This is called overfitting and can easily occur if you have a small dataset and a large number of parameters.
To deal with overfitting, regularization can be applied – that is, imposing some constraints to make the model simpler. For example, if you overfit a linear regression model, you will notice that its coefficients become very large:
A model can be regularized in different ways, but the two most common methods are L1 and L2 regularization. L2 regularization differs in that the error is not just the absolute value of the coefficients, but also their squares added to the error:”
Fig. 6.3 Effect of L1-regularization on the coefficients. The more penalty coefficient (alpha in picture), the less the coefficients. From sklearn tutorial.#
Aging clocks are an important area of research that has the potential to greatly improve our understanding of the aging process and lead to the development of interventions to promote healthy aging. These clocks are biological markers that can be used to measure and predict the biological age of an individual. They can be based on a variety of factors, such as epigenetic modifications, blood tests, and gene expression patterns. While there has been significant progress in understanding and utilizing aging clocks, there are still many open questions and areas for further research.
One important aspect of aging clock research is the use of statistical analysis to make these clocks. Linear regression is a commonly used statistical method in this context, as it allows researchers to identify correlations and trends in the data. However, it is important to recognize that basic linear regression has limitations and additional regularizations may be necessary in certain circumstances.
Overall, the study of aging clocks has the potential to provide insights into the underlying mechanisms of aging and may lead to the development of interventions that can promote healthy aging and extend the human lifespan. While there is still much to learn about aging clocks and the aging process, the potential benefits of this research make it an important area of study for the scientific community.
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Biology of Aging book","1. Introduction to Aging Biology","12. Complex systems approach","13. System resilience","Welcome to the Computational Biology of Aging coursebook","3. Omics Data Analysis in Aging: Methylomics","Weighted gene correlation network analysis (WGCNA)","11. Aging Clocks","Dynamic Organism State Indicator","Meta-analysis of aging transcriptomes","Computational Model of Parkinson\u2019s Disease","Unsupervised aging","Project report template","2. Differential expression analysis","4. Survival analysis","5. Survival analysis. 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Biology of Aging book","1. Introduction to Aging Biology","7. Complex systems approach","8. System resilience","Welcome to the Computational Biology of Aging coursebook","3. Omics Data Analysis in Aging: Methylomics","Weighted gene correlation network analysis (WGCNA)","6. Aging Clocks","Dynamic Organism State Indicator","Meta-analysis of aging transcriptomes","Computational Model of Parkinson\u2019s Disease","Unsupervised aging","Project report template","2. Differential expression analysis","4. Survival analysis","5. Survival analysis. Advanced methods."],titleterms:{"1":[2,3,6,13,14,15],"2":[3,6,13,14],"2011":7,"2013":7,"2018":7,"2019":7,"3":[3,6,13,14],"4":[6,14],"5":[3,6,14],"6":14,"7":14,"8":14,"abstract":[0,4],"case":3,"do":11,"function":[3,6,13],"import":[6,15],A:7,In:10,The:[3,6,9,11],accumul:1,acknowledg:[4,14],across:9,add:0,advanc:15,ag:[0,1,2,3,4,5,7,8,9,10,11],aim:10,alter:1,amplitud:10,an:7,analysi:[5,6,9,13,14,15],analyz:5,antagonist:1,applic:7,approach:[2,9,11],area:10,assembl:5,assig:6,auc:15,author:11,autophagi:1,band:10,basal:10,beta:10,between:9,bgtc:10,bifurc:3,biolog:[2,7],biologi:[0,1,4],biomark:7,bisulfit:5,blood:7,book:0,boost:15,brain:9,brier:15,build:6,call:11,canal:2,catastroph:3,cdf:14,cell:[1,7],cellular:1,challeng:14,check:10,chronic:1,cite:[0,4],classic:15,classif:15,clock:7,cluster:13,cmi:8,colab:6,color:6,combin:[2,6],common:9,comparison:[9,14],complex:[2,3],compon:13,comput:[0,4,10],concept:[2,14],conclus:[1,3,5,7,14],concord:15,condit:8,constant:14,construct:[3,6],content:[1,2,3,4,5,7],contribut:[0,4],control:[5,9,13],correl:6,cours:[4,10],coursebook:4,cox:[8,15],coxph:15,credit:[1,2,3,5,7,8,9,10,11,12,13,14,15],criterion:15,critic:[2,3],curv:14,data:[3,5,9,13,14],dataset:9,decis:15,deepsurv:15,degeneraci:2,densiti:10,depend:[0,15],differ:10,differenti:[3,9,13],disabl:1,discuss:[8,9,10,11,12],diseas:10,disrupt:1,dna:[1,5,7],dnam:5,dosi:8,down:3,downregul:9,dowstream:6,dynam:[4,8],dysbiosi:1,dysfunct:1,each:6,elasticnet:7,emerg:2,enrich:[6,9],ensembl:15,entropi:15,environ:6,epigenet:[1,7],equat:3,estim:14,evolutionari:1,exercis:[2,3,6,13,14],exhaust:1,exponenti:14,express:[9,13],extern:6,f:7,featur:15,find:6,fitt:14,forest:15,format:10,from:[3,5],gain:15,ganglia:10,genag:9,gene:[6,9,13],genom:[1,7],go:[6,9,13],gompertz:14,googl:6,gradient:15,grimag:7,hallmark:1,harrel:15,hazard:[8,15],health:8,healthspan:7,heart:9,heterochromatin:1,home:6,how:[0,1,5],hub:6,human:7,hypothes:10,i:6,identifi:6,ii:6,iii:6,improv:[9,11],index:15,indic:[3,8],inflamm:1,inform:[6,15],initi:6,instabl:1,integr:[1,3],intercellular:1,interpret:5,intersect:9,intersectio:9,introduc:[3,10],introduct:[1,8,9,10,11,12],intuit:3,iv:6,kaplan:14,kind:[],l1:7,l2:7,law:[],layer:15,learn:[2,3,4,13,14,15],lifespan:[7,8],likelihood:15,linear:[3,7,15],list:9,load:6,logist:15,logrank:14,look:5,loss:[1,2,15],machin:[4,15],main:14,maximum:8,measur:5,meier:14,mention:5,meta:9,method:[5,14,15],methyl:[1,5,6,7],methylom:5,metric:15,microarrai:5,mitochondri:1,mlp:15,model:[3,7,10,14,15],modul:6,more:[3,13,14],mortal:14,multi:15,multidimension:3,muscl:9,mutat:1,network:[2,6,15],neural:15,nn:15,nnet:15,non:3,nonparametr:14,normal:13,note:3,numer:3,nutrient:1,object:6,omic:5,ontolog:6,open:7,oracl:7,ordinari:3,organ:8,other:7,output:[8,15],overfit:7,overview:5,oxid:1,paper:[5,9],paradigm:11,parametr:[14,15],parkinson:10,part:6,partial:15,pca:[8,9],pd:10,pdf:14,peak:10,perceptron:15,perform:15,permut:15,perspect:2,ph:[8,15],phenoag:7,point:[9,11],potenti:3,power:10,practic:5,predict:[7,8,15],predictor:7,prepar:13,preprocess:9,prerequisit:4,previous:5,primari:1,princip:13,probabl:15,process:5,profil:[5,7],project:[4,12],proport:[8,15],proteostasi:1,psd:10,pywgcna:6,q:7,qualiti:[5,9,13],quantit:7,question:[6,7],random:15,rate:7,reactiv:1,real:[6,14],recap:3,refer:[1,2,5,9,10,12,13],regress:[7,15],regular:7,relat:6,remark:3,report:12,resili:[2,3],result:[8,9,10,11,12],reveal:7,risk:[8,14],roc:15,role:1,rpy2:13,s:[10,15],sampl:6,save:6,score:15,section:5,select:15,semi:15,senesc:1,sequenc:5,set:[6,13],shorten:1,signal:1,site:6,slow:3,some:15,spectral:10,split:15,stabil:3,state:8,statist:[4,5,9,13],stem:1,stochast:3,stress:1,strongli:7,student:4,suggest:9,summari:2,surviv:[14,15],system:[2,3,4,10],task:[],telomer:1,templat:12,term:10,test:[7,9,13,14],them:6,theori:[3,8,15],thi:[4,5,10],three:10,time:15,tissu:7,train:15,transcriptom:[7,9],transit:2,transposon:1,tree:15,type:7,univers:3,unsupervis:11,up:[6,13],updat:6,upregul:9,us:[0,4,6],valid:5,view:[4,7],visual:6,visualis:5,vs:15,we:10,weak:[9,11],weibul:14,weight:6,welcom:4,wgcna:6,what:[5,7],who:4,why:[5,11],wide:7,work:[6,10],your:6}})
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coefficients in a form of loghazard ratios or hazard ratios. The loghazard ratio represents the increase in the expected log of the relative hazard for each one unit increase in the predictor covariate, holding other covariates constant. The hazard ratios are computed by exponentiating loghazard ratios for more interpretability and can show the increase in the expected hazard in percents.
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The basic intuition behind the tree models is to recursively partition the data based on a particular splitting criterion, so that the objects that are similar to each other based on the value of interest will be placed in the same node.
We will start with the simplest case - decision tree for classification:
Ensembling - combining the predictions of multiple base learners to obtain a powerful overall model. The base learners are often very simple models also referred as weak learners.
Multiple diverse models are created to predict an outcome, either by using many different modeling algorithms or using different training data sets.
Random Forest fits a set of Trees independently and then averages their predictions
The general principles as RF: (a) Trees are grown using bootstrapped data; (b) Random feature selection is used when splitting tree nodes; (c) Trees are generally grown deeply (d) Forest ensemble is calculated by averaging terminal node statistics
Importantly, the high number of base learners do not lead to overfitting.
Survival analysis is a type of regression problem as we want to predict a continuous value, but with a twist. It differs from traditional regression by the fact that parts of the data can only be partially observed – they are censored
Rather than focusing on predicting a single point in time of an event, the prediction step in survival analysis often focuses on predicting a function: the survival or hazard function.
However for the survival models that do not rely on the proportional hazards assumption, it is impossible to estimate survival or cumulative hazard function. Their predictions are risk scores. If samples are ordered according to their predicted risk score (in ascending order), one obtains the sequence of events, as predicted by the model.
Survival machine learning - machine learning methods adapted to work with survival data and censoring!
Survival trees are one form of decision trees which are tailored
to handle censored data. The goal is to split the tree node into left and right daughters with dissimilar event history (survival) behavior.
The primary difference between a survival tree and the standard decision tree is
in the choice of splitting criterion - the log-rank test. The log-rank test has traditionally been used for two-sample testing of survival data, but it can be used for survival splitting as a means for maximizing between-node survival differences.
Let \(X\) denote a specific variable. A proposed split using \(X\) is \(X≤a\)
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For prediction, a sample is dropped down each tree in the forest until it reaches a terminal node.
Data in each terminal is used to non-parametrically estimate the cumulative hazard function and survival using the Nelson-Aalen estimator and Kaplan-Meier, respectively.
The default loss function is the partial likelihood loss of Cox’s proportional hazards model.
The objective is to maximize the log partial likelihood function, but replacing the traditional linear model with the additive model
Here is the model of artificial neuron - base element of artificial neural network. The output is computed using activation function on the summation of inputs multiplied by weights and the bias value
The most popular method for training MLPs is back-propagation. During backpropagation, the output values are compared with the correct answer to compute the value of some predefined error-function. The error is then fed back through the network. Using this information, the algorithm adjusts the weights of each connection in order to reduce the value of the error function by some small amount. After repeating this process for a sufficiently large number of training cycles, the network will usually converge to some state where the error of the calculations is small. In this case, one would say that the network has learned a certain target function.
Neural networks methods adapted to work with survival data and censoring!
Pycox - python package for survival analysis and time-to-event prediction with PyTorch, built on the torchtuples package for training PyTorch models.
Nonlinear Cox proportional hazards network. Deep feed-forward neural network with Cox proportional hazards loss function. Can be considered as nonlinear extension of the Cox proportional hazards: can deal with both linear and nonlinear effects from covariates.
The network propagates the inputs through a number of hidden layers with weights θ. The hidden layers of the network consist of a fully-connected layer of nodes, followed by a dropout layer. The output of the network is a single node with a linear activation which estimates the log-risk function \(h(X)\) in the Cox model
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Discrete-time model, fully parametric survival model
The Logistic-Hazard method parametrize the discrete hazards and optimize the survival likelihood.
The model is trained with the maximum likelihood method using mini-batch stochastic gradient descent. The model incorporates time-varying baseline hazard rate and non-proportional hazards
Our test data is usually subject to censoring too, therefore common metrics like root mean squared error or correlation are unsuitable. Instead, we use specific metrics for survival analysis
Predictions are often evaluated by a measure of rank correlation between predicted risk scores and observed time points in the test data. Harrell’s concordance index or c-index computes the ratio of correctly ordered (concordant) pairs to comparable pairs
The higher the C-index is - the better model performance is
Extention of the well known receiver operating characteristic curve (ROC curve) to possibly censored survival times. Given a time point, we can estimate how well a predictive model can distinguishing subjects who will experience an event by time
(sensitivity) from those who will not (specificity).
The higher the ROC AUC is - the better model performance is
Permutation feature importance is a model inspection technique which can be used for any fitted estimator with tabular data. This is especially useful for non-linear estimators.
The permutation feature importance is a decrease in a model score when a single feature value is randomly shuffled. This procedure breaks the relationship between the feature and the target, thus the drop in the model score is indicative of how much the model depends on the feature.
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